Quantum Mechanics

Quantum Mechanics An Introductory Framework

PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Fri, 07 Dec 2012 21:34:09 UTC

Contents Articles 1. Introductory Principles

1

History of Quantum Mechanics

1

Basic Concepts of Quantum Mechanics

7

Introduction to Quantum Mechanics

2. The Quantum Theories

26 45

Old Quantum Theory

45

Quantum Mechanics after 1925

52

3. The Interpretation of Quantum Mechanics

72

Interpretations of Quantum Mechanics

72

The Copenhagen Interpretation

85

4. Einstein's Objections

93

Principle of Locality

93

EPR Paradox

96

Bell's Theorem

5. Schrödinger's Objections Schrödinger's Cat

6. Measurement Problems

106 121 121 127

The Measurement Problem

127

Measurement in Quantum Mechanics

129

7. Advanced Concepts

137

Quantum Number

137

Quantum Information

142

Quantum Statistical Mechanics

145

8. Advanced Topics

148

Quantum Field Theory

148

String Theory

159

Quantum Gravity

178

Appendix

188

Quantum

188

Quantum state

190

References Article Sources and Contributors

196

Image Sources, Licenses and Contributors

201

Article Licenses License

202

1

1. Introductory Principles History of Quantum Mechanics The history of quantum mechanics is a fundamental part of the history of modern physics. Quantum mechanics' history, as it interlaces with the history of quantum chemistry, began essentially with a number of different scientific discoveries: the 1838 discovery of cathode rays by Michael Faraday; the 1859-1860 winter statement of the black body radiation problem by Gustav Kirchhoff; the 1877 suggestion by Ludwig Boltzmann that the energy states of a physical system could be discrete; the discovery of the photoelectric effect by Heinrich Hertz in 1887; and the 1900 quantum hypothesis by Max Planck that any energy-radiating atomic system can theoretically be divided into a number of discrete "energy elements" ε (epsilon) such that each of these energy elements is proportional to the frequency ν with which each of them individually radiate energy, as defined by the following formula:

where h is a numerical value called Planck's constant. Then, Albert Einstein in 1905, in order to explain the photoelectric effect previously reported by Heinrich Hertz in 1887, postulated consistently with Max Planck's quantum hypothesis that light itself is made of individual quantum particles, which in 1926 came to be called photons by Gilbert N. Lewis. The photoelectric effect was observed upon shining light of particular wavelengths on certain materials, such as metals, which caused electrons to be ejected from those materials only if the light quantum energy was greater than the Fermi level (work function) in the metal. The phrase "quantum mechanics" was first used in Max Born's 1924 paper "Zur Quantenmechanik". In the years to follow, this theoretical basis slowly began to be applied to chemical structure, reactivity, and bonding.

10 influential figures in the history of quantum mechanics. Left to right:Max Planck, Albert Einstein,Niels Bohr, Louis de Broglie,Max Born, Paul Dirac,Werner Heisenberg, Wolfgang Pauli,Erwin Schrödinger, Richard Feynman.

History of Quantum Mechanics

2

Overview Ludwig Eduard Boltzmann suggested in 1877 that the energy levels of a physical system, such as a molecule, could be discrete. He was a founder of the Austrian Mathematical Society, together with the mathematicians Gustav von Escherich and Emil Müller. Boltzmann's rationale for the presence of discrete energy levels in molecules such as those of iodine gas had its origins in his statistical thermodynamics and statistical mechanics theories and was backed up by mathematical arguments, as it will also be the case twenty years later with the first quantum theory put forward by Max Planck.

Ludwig Boltzmann’s diagram of the I2 molecule proposed in 1898 showing the atomic "sensitive region" (α, β) of overlap.

In 1900, the German physicist Max Planck reluctantly introduced the idea that energy is quantized in order to derive a formula for the observed frequency dependence of the energy emitted by a black body, called Planck's Law, that included a Boltzmann distribution (applicable in the classical limit). Planck's law[1] can be stated as follows:

where:

I(ν,T) is the energy per unit time (or the power) radiated per unit area of emitting surface in the normal direction per unit solid angle per unit frequency by a black body at temperature T; h is the Planck constant; c is the speed of light in a vacuum; k is the Boltzmann constant; ν is the frequency of the electromagnetic radiation; and T is the temperature of the body in degrees Kelvin. The earlier Wien approximation may be derived from Planck's law by assuming

.

Moreover, the application of Planck's quantum theory to the electron allowed Ștefan Procopiu in 1911—1913, and subsequently Niels Bohr in 1913, to calculate the magnetic moment of the electron, which was later called the "magneton"; similar quantum computations, but with numerically quite different values, were subsequently made possible for both the magnetic moments of the proton and the neutron that are three orders of magnitude smaller than that of the electron.

Photoelectric effect

The emission of electrons from a metal plate caused by light quanta (photons) with energy greater than the Fermi level of the metal.

History of Quantum Mechanics

3 The photoelectric effect reported by Heinrich Hertz in 1887, and explained by Albert Einstein in 1905. Low-energy phenomena: Photoelectric effect Mid-energy phenomena: Compton scattering High-energy phenomena: Pair production

In 1905, Einstein explained the photoelectric effect by postulating that light, or more generally all electromagnetic radiation, can be divided into a finite number of "energy quanta" that are localized points in space. From the introduction section of his March 1905 quantum paper, "On a heuristic viewpoint concerning the emission and transformation of light", Einstein states: "According to the assumption to be contemplated here, when a light ray is spreading from a point, the energy is not distributed continuously over ever-increasing spaces, but consists of a finite number of 'energy quanta' that are localized in points in space, move without dividing, and can be absorbed or generated only as a whole." This statement has been called the most revolutionary sentence written by a physicist of the twentieth century.[2] These energy quanta later came to be called "photons", a term introduced by Gilbert N. Lewis in 1926. The idea that each photon had to consist of energy in terms of quanta was a remarkable achievement; it effectively solved the problem of black body radiation attaining infinite energy, which occurred in theory if light were to be explained only in terms of waves. In 1913, Bohr explained the spectral lines of the hydrogen atom, again by using quantization, in his paper of July 1913 On the Constitution of Atoms and Molecules. These theories, though successful, were strictly phenomenological: during this time, there was no rigorous justification for quantization, aside, perhaps, from Henri Poincaré's discussion of Planck's theory in his 1912 paper Sur la théorie des quanta.[3][4] They are collectively known as the old quantum theory. The phrase "quantum physics" was first used in Johnston's Planck's Universe in Light of Modern Physics (1931). In 1924, the French physicist Louis de Broglie put forward his theory of matter waves by stating that particles can exhibit wave characteristics and vice versa. This theory was for a single particle and derived from special relativity theory. Building on de Broglie's approach, modern quantum mechanics was born in 1925, when the German physicists Werner Heisenberg and Adam Jonathon Davis[5][6] developed matrix mechanics and the Austrian physicist Erwin Schrödinger invented wave mechanics and the non-relativistic Schrödinger equation as an approximation to the generalised case of de Broglie's theory.[7] Schrödinger subsequently showed that the two approaches were equivalent.

With decreasing temperature, the peak of the blackbody radiation curve shifts to longer wavelengths and also has lower intensities. The blackbody radiation curves (1862) at left are also compared with the early, classical limit model of Rayleigh and Jeans (1900) shown at right. The short wavelength side of the curves was already approximated in 1896 by the Wien distribution law.

Heisenberg formulated his uncertainty principle in 1927, and the Copenhagen interpretation started to take shape at about the same time. Starting around 1927, Paul Dirac began the process of unifying quantum mechanics with special relativity by proposing the Dirac equation for the electron. The Dirac equation achieves the relativistic description of the wavefunction of an electron that Schrödinger failed to obtain. It predicts electron spin and led Dirac to predict the existence of the positron. He also pioneered the use of operator theory,

History of Quantum Mechanics

4

including the influential bra-ket notation, as described in his famous 1930 textbook. During the same period, Hungarian polymath John von Neumann formulated the rigorous mathematical basis for quantum mechanics as the theory of linear operators on Hilbert spaces, as described in his likewise famous 1932 textbook. These, like many other works from the founding period, still stand, and remain widely used. The field of quantum chemistry was pioneered by physicists Walter Heitler and Fritz London, who published a study of the covalent bond of the hydrogen molecule in 1927. Quantum chemistry was subsequently developed by a large number of workers, including the American theoretical chemist Linus Pauling at Caltech, and John C. Slater into various theories such as Molecular Orbital Theory or Valence Theory. Beginning in 1927, researchers made attempts at applying quantum mechanics to fields instead of single particles, resulting in quantum field theories. Early workers in this area include P.A.M. Dirac, W. Pauli, V. Weisskopf, and P. Jordan. This area of research culminated in the formulation of quantum electrodynamics by R.P. Feynman, F. Dyson, J. Schwinger, and S.I. Tomonaga during the 1940s. Quantum electrodynamics describes a quantum theory of electrons, positrons, and the electromagnetic field, and served as a model for subsequent Quantum Field theories.[8][5][6]

Niels Bohr's 1913 quantum model of the atom, which incorporated an explanation of Johannes Rydberg's 1888 formula, Max Planck's 1900 quantum hypothesis, i.e. that atomic energy radiators have discrete energy values (ε = hν), J. J. Thomson's 1904 plum pudding model, Albert Einstein's 1905 light quanta postulate, and Ernest Rutherford's 1907 discovery of the atomic nucleus. Note that the electron does not travel along the black line when emitting a photon. It jumps, disappearing from the outer orbit and appearing in the inner one and cannot exist in the space between orbits 2 and 3.

The theory of Quantum Chromodynamics was formulated beginning in the early 1960s. The theory as we know it today was formulated by Politzer, Gross and Wilczek in 1975. Building on pioneering work by Schwinger, Higgs and Goldstone, the physicists Glashow, Weinberg and Salam independently showed how the weak nuclear force and quantum electrodynamics could be merged into a single electroweak force, for which they received the 1979 Nobel Prize in Physics. Feynman diagram of gluon radiation in Quantum Chromodynamics

Founding experiments

History of Quantum Mechanics • • • • • • • • • • •

Thomas Young's double-slit experiment demonstrating the wave nature of light. (c1805) Henri Becquerel discovers radioactivity. (1896) J. J. Thomson's cathode ray tube experiments (discovers the electron and its negative charge). (1897) The study of black body radiation between 1850 and 1900, which could not be explained without quantum concepts. The photoelectric effect: Einstein explained this in 1905 (and later received a Nobel prize for it) using the concept of photons, particles of light with quantized energy. Robert Millikan's oil-drop experiment, which showed that electric charge occurs as quanta (whole units). (1909) Ernest Rutherford's gold foil experiment disproved the plum pudding model of the atom which suggested that the mass and positive charge of the atom are almost uniformly distributed. (1911) Otto Stern and Walther Gerlach conduct the Stern-Gerlach experiment, which demonstrates the quantized nature of particle spin. (1920) Clinton Davisson and Lester Germer demonstrate the wave nature of the electron[9] in the Electron diffraction experiment. (1927) Clyde L. Cowan and Frederick Reines confirm the existence of the neutrino in the neutrino experiment. (1955) Clauss Jönsson`s double-slit experiment with electrons. (1961)

• The Quantum Hall effect, discovered in 1980 by Klaus von Klitzing. The quantized version of the Hall effect has allowed for the definition of a new practical standard for electrical resistance and for an extremely precise independent determination of the fine structure constant. • The experimental verification of quantum entanglement by Alain Aspect. (1982) • The Mach-Zehnder Interferometer experiment conducted by Paul Kwiat, Harold Wienfurter, Thomas Herzog, Anton Zeilinger, and Mark Kasevich, providing experimental verification of the Elitzur-Vadiman bomb tester, proving Interaction-free measurement is possible. (1994)

References [1] M. Planck (1914). The theory of heat radiation, second edition, translated by M. Masius, Blakiston's Son & Co, Philadelphia, pages 22, 26, 42, 43. [2] Folsing, Albrecht (1997), Albert Einstein: A Biography, trans. Ewald Osers, Viking [3] McCormmach, Russell (Spring, 1967), "Henri Poincaré and the Quantum Theory", Isis 58 (1): 37–55, doi:10.1086/350182 [4] Irons, F. E. (August, 2001), "Poincaré's 1911–12 proof of quantum discontinuity interpreted as applying to atoms", American Journal of Physics 69 (8): 879–884, Bibcode 2001AmJPh..69..879I, doi:10.1119/1.1356056 [5] David Edwards,The Mathematical Foundations of Quantum Mechanics, Synthese, Volume 42, Number 1/September, 1979, pp. 1–70. [6] D. Edwards, The Mathematical Foundations of Quantum Field Theory: Fermions, Gauge Fields, and Super-symmetry, Part I: Lattice Field Theories, International J. of Theor. Phys., Vol. 20, No. 7 (1981). [7] Hanle, P.A. (December 1977), "Erwin Schrodinger's Reaction to Louis de Broglie's Thesis on the Quantum Theory.", Isis 68 (4): 606–609, doi:10.1086/351880 [8] S. Auyang, How is Quantum Field Theory Possible?, Oxford University Press, 1995. [9] The Davisson-Germer experiment, which demonstrates the wave nature of the electron (http:/ / hyperphysics. phy-astr. gsu. edu/ hbase/ quantum/ davger2. html)

Further reading • Bacciagaluppi, Guido; Valentini; Valentini, Antony (2009), Quantum theory at the crossroads: reconsidering the 1927 Solvay conference, Cambridge, UK: Cambridge University Press, pp. 9184, arXiv:quant-ph/0609184, Bibcode 2006quant.ph..9184B, ISBN 978-0-521-81421-8, OCLC 227191829 • Bernstein, Jeremy (2009), Quantum Leaps (http://books.google.com/?id=j0Me3brYOL0C& printsec=frontcover), Cambridge, Massachusetts: Belknap Press of Harvard University Press, ISBN 978-0-674-03541-6 • Jammer, Max (1966), The conceptual development of quantum mechanics, New York: McGraw-Hill, OCLC 534562

5

History of Quantum Mechanics • Jammer, Max (1974), The philosophy of quantum mechanics: The interpretations of quantum mechanics in historical perspective, New York: Wiley, ISBN 0-471-43958-4, OCLC 969760 • F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Deformation theory and quantization I,and II, Ann. Phys. (N.Y.), 111 (1978) pp. 61–110, 111-151. • D. Cohen, An Introduction to Hilbert Space and Quantum Logic, Springer-Verlag, 1989. This is a thorough and well-illustrated introduction. • Finkelstein, D., "Matter, Space and Logic", Boston Studies in the Philosophy of Science V: 1969. • A. Gleason. Measures on the Closed Subspaces of a Hilbert Space, Journal of Mathematics and Mechanics, 1957. • R. Kadison. Isometries of Operator Algebras, Annals of Mathematics, Vol. 54, pp. 325–338, 1951 • G. Ludwig. Foundations of Quantum Mechanics, Springer-Verlag, 1983. • G. Mackey. Mathematical Foundations of Quantum Mechanics, W. A. Benjamin, 1963 (paperback reprint by Dover 2004). • R. Omnès. Understanding Quantum Mechanics, Princeton University Press, 1999. (Discusses logical and philosophical issues of quantum mechanics, with careful attention to the history of the subject). • N. Papanikolaou. Reasoning Formally About Quantum Systems: An Overview, ACM SIGACT News, 36(3), pp. 51–66, 2005. • C. Piron. Foundations of Quantum Physics, W. A. Benjamin, 1976. • Hermann Weyl. The Theory of Groups and Quantum Mechanics, Dover Publications, 1950. • A. Whitaker. The New Quantum Age: From Bell's Theorem to Quantum Computation and Teleportation, Oxford University Press, 2011, ISBN 978-0-19-958913-5

External links • A History of Quantum Mechanics (http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/ The_Quantum_age_begins.html) • A Brief History of Quantum Mechanics (http://www.oberlin.edu/physics/dstyer/StrangeQM/history.html) • Homepage of the Quantum History Project (http://quantum-history.mpiwg-berlin.mpg.de/)

6

Basic Concepts of Quantum Mechanics

7

Basic Concepts of Quantum Mechanics Quantum mechanics is the body of scientific principles that explains the behavior of matter and its interactions with energy on the scale of atoms and subatomic particles and how these phenomena could be related to everyday life (see: Schrodinger's cat). Classical physics explains matter and energy at the macroscopic level of the scale familiar to human experience, including the behavior of astronomical bodies. It remains the key to measurement for much of modern science and technology; but at the end of the 19th Century observers discovered phenomena in both the large (macro) and the small (micro) worlds that classical physics could not explain.[1] Coming to terms with these limitations led to the development of quantum mechanics, a major revolution in physics. This article describes how physicists discovered the limitations of classical physics and developed the main concepts of the quantum theory that replaced them in the early decades of the 20th century.[2] These concepts are described in roughly the order they were first discovered; for a more complete history of the subject, see History of quantum mechanics. Some aspects of quantum mechanics can seem counter-intuitive or even paradoxical, because they describe behavior quite different than that seen at larger length scales, where classical physics is an excellent approximation. In the words of Richard Feynman, quantum mechanics deals with "nature as She is — absurd."[3] Many types of energy, such as photons (discrete units of light), behave in some respects like particles and in other respects like waves. Radiators of photons (such as neon lights) have emission spectra that are discontinuous, in that only certain frequencies of light are present. Quantum mechanics predicts the energies, the colours, and the spectral intensities of all forms of electromagnetic radiation. Quantum mechanics ordains that the more closely one pins down one measure (such as the position of a particle), the less precise another measurement pertaining to the same particle (such as its momentum) must become. Put another way, measuring position first and then measuring momentum does not have the same outcome as measuring momentum first and then measuring position; the act of measuring the first property necessarily introduces additional energy into the micro-system being studied, thereby perturbing that system.

From above and from left to right:Max Planck, Albert Einstein,Niels Bohr, Louis de Broglie,Max Born, Paul Dirac,Werner Heisenberg, Wolfgang Pauli,Erwin Schrödinger, Richard Feynman.

Even more disconcerting, pairs of particles can be created as "entangled twins." As is described in more detail in the article on Quantum entanglement, entangled particles seem to exhibit what Einstein called "spooky action at a distance," matches between states that classical physics would insist must be random even when distance and the speed of light ensure that no physical causation could account for these correlations.[4]

Basic Concepts of Quantum Mechanics

8

The first quantum theory: Max Planck and black body radiation

Hot metalwork from a blacksmith. The yellow-orange glow is the visible part of the thermal radiation emitted due to the high temperature. Everything else in the picture is glowing with thermal radiation as well, but less brightly and at longer wavelengths than the human eye can detect. A far-infrared camera can observe this radiation.

Thermal radiation is electromagnetic radiation emitted from the surface of an object due to the object's temperature. If an object is heated sufficiently, it starts to emit light at the red end of the spectrum — it is red hot. Heating it further causes the colour to change from red to yellow to blue to white, as light at shorter wavelengths (higher frequencies) begins to be emitted. It turns out that a perfect emitter is also a perfect absorber. When it is cold, such an object looks perfectly black, because it absorbs all the light that falls on it and emits none. Consequently, an ideal thermal emitter is known as a black body, and the radiation it emits is called black body radiation.

In the late 19th century, thermal radiation had been fairly well-characterized experimentally. How the wavelength at which the radiation is strongest changes with temperature is given by Wien's displacement law, and the overall power emitted per unit area is given by the Stefan–Boltzmann law. However, classical physics was unable to explain the relationship between temperatures and predominant frequencies of radiation. In fact, at short wavelengths, classical physics predicted that energy will be emitted by a hot body at an infinite rate. This result, which is clearly wrong, is known as the ultraviolet catastrophe. Physicists were searching for a single theory that explained why they got the experimental results that they did. The first model that was able to explain the full spectrum of thermal radiation was put forward by Max Planck in 1900.[5] He modeled the thermal radiation as being in equilibrium, using a set of harmonic oscillators. To reproduce the experimental results he had to assume that each oscillator produced an integer number of units of energy at its single characteristic frequency, rather than being able to emit any arbitrary amount of energy. In other words, the energy of each oscillator was "quantized."[6] The quantum of energy for each oscillator, according to Planck, was proportional to the frequency of the oscillator; the constant of proportionality is now known as the Planck constant. The Planck constant, usually written as h, has the value 663 × 10−34 J s, and so the energy E of an oscillator of frequency f is given by

Correct values (green) contrasted against the classical values (Rayleigh-Jeans law, red and Wien approximation, blue).

[7]

Planck's law was the first quantum theory in physics, and Planck won the Nobel Prize in 1918 "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta."[8] At the time, however, Planck's view was that quantization was purely a mathematical trick, rather than (as we now know) a fundamental change in our understanding of the world.[9]

Basic Concepts of Quantum Mechanics

9

Photons: the quantisation of light In 1905, Albert Einstein took an extra step. He suggested that quantisation was not just a mathematical trick: the energy in a beam of light occurs in individual packets, which are now called photons.[10] The energy of a single photon is given by its frequency multiplied by Planck's constant:

For centuries, scientists had debated between two possible theories of light: was it a wave or did it instead comprise a stream of tiny particles? By the 19th century, the debate was generally considered to have been settled in favour of the wave theory, as it was able to explain observed effects such as refraction, diffraction and polarization. James Clerk Maxwell had shown that electricity, magnetism and light are all manifestations of the same phenomenon: the Einstein's portrait by Harm electromagnetic field. Maxwell's equations, which are the complete set of laws of Kamerlingh Onnes at the University of Leiden in 1920 classical electromagnetism, describe light as waves: a combination of oscillating electric and magnetic fields. Because of the preponderance of evidence in favour of the wave theory, Einstein's ideas were met initially with great scepticism. Eventually, however, the photon model became favoured; one of the most significant pieces of evidence in its favour was its ability to explain several puzzling properties of the photoelectric effect, described in the following section. Nonetheless, the wave analogy remained indispensable for helping to understand other characteristics of light, such as diffraction.

The photoelectric effect In 1887 Heinrich Hertz observed that light can eject electrons from metal.[11] In 1902 Philipp Lenard discovered that the maximum possible energy of an ejected electron is related to the frequency of the light, not to its intensity; if the frequency is too low, no electrons are ejected regardless of the intensity. The lowest frequency of light that causes electrons to be emitted, called the threshold frequency, is different for every metal. This observation is at odds with classical electromagnetism, which predicts that the electron's energy should be proportional to the intensity of the radiation.[12]:24

Light (red arrows, left) is shone upon a metal. If the light is of sufficient frequency (i.e. sufficient energy), electrons are ejected (blue arrows, right).

Einstein explained the effect by postulating that a beam of light is a stream of particles (photons), and that if the beam is of frequency f then each photon has an energy equal to hf.[11] An electron is likely to be struck only by a single photon, which imparts at most an energy hf to the electron.[11] Therefore, the intensity of the beam has no effect;[13] only its frequency determines the maximum energy that can be imparted to the electron.[11] To explain the threshold effect, Einstein argued that it takes a certain amount of energy, called the work function, denoted by φ, to remove an electron from the metal.[11] This amount of energy is different for each metal. If the energy of the photon is less than the work function then it does not carry sufficient energy to remove the electron from the metal. The threshold frequency, f0, is the frequency of a photon whose energy is equal to the work function: If f is greater than f0, the energy hf is enough to remove an electron. The ejected electron has a kinetic energy EK which is, at most, equal to the photon's energy minus the energy needed to dislodge the electron from the metal:

Basic Concepts of Quantum Mechanics Einstein's description of light as being composed of particles extended Planck's notion of quantised energy: a single photon of a given frequency f delivers an invariant amount of energy hf. In other words, individual photons can deliver more or less energy, but only depending on their frequencies. However, although the photon is a particle it was still being described as having the wave-like property of frequency. Once again, the particle account of light was being "compromised".[14][15] The relationship between the frequency of electromagnetic radiation and the energy of each individual photon is why ultraviolet light can cause sunburn, but visible or infrared light cannot. A photon of ultraviolet light will deliver a high amount of energy—enough to contribute to cellular damage such as occurs in a sunburn. A photon of infrared light will deliver a lower amount of energy—only enough to warm one's skin. So an infrared lamp can warm a large surface, perhaps large enough to keep people comfortable in a cold room, but it cannot give anyone a sunburn. If each individual photon had identical energy, it would not be correct to talk of a "high energy" photon. Light of high frequency could carry more energy only because of flooding a surface with more photons arriving per second. Light of low frequency could carry more energy only for the same reason. If it were true that all photons carry the same energy, then if you doubled the rate of photon delivery, you would double the number of energy units arriving each second. Einstein rejected that wave-dependent classical approach in favour of a particle-based analysis where the energy of the particle must be absolute and varies with frequency in discrete steps (i.e. is quantised). All photons of the same frequency have identical energy, and all photons of different frequencies have proportionally different energies. In nature, single photons are rarely encountered. The sun emits photons continuously at all electromagnetic frequencies, so they appear to propagate as a continuous wave, not as discrete units. The emission sources available to Hertz and Lennard in the 19th century shared that characteristic. A sun that radiates red light, or a piece of iron in a forge that glows red, may both be said to contain a great deal of energy. It might be surmised that adding continuously to the total energy of some radiating body would make it radiate red light, orange light, yellow light, green light, blue light, violet light, and so on in that order. But that is not so for otherwise larger suns and larger pieces of iron in a forge would glow with colours more toward the violet end of the spectrum. To change the color of such a radiating body it is necessary to change its temperature, and increasing its temperature changes the quanta of energy that are available to excite individual atoms to higher levels and permit them to emit photons of higher frequencies. The total energy emitted per unit of time by a sun or by a piece of iron in a forge depends on both the number of photons emitted per unit of time and also on the amount of energy carried by each of the photons involved. In other words, the characteristic frequency of a radiating body is dependent on its temperature. When physicists were looking only at beams of light containing huge numbers of individual and virtually indistinguishable photons it was difficult to understand the importance of the energy levels of individual photons. So when physicists first discovered devices exhibiting the photoelectric effect, the effect that makes the light meters of modern cameras work, they initially expected that a higher intensity of light would produce a higher voltage from the photoelectric device. They discovered that strong beams of light toward the red end of the spectrum might produce no electrical potential at all, and that weak beams of light toward the violet end of the spectrum would produce higher and higher voltages. Einstein's idea that individual units of light may contain different amounts of energy depending on their frequency made it possible to explain the experimental results that hitherto had seemed quite counter-intuitive. Although the energy imparted by photons is invariant at any given frequency, the initial energy-state of the electrons in a photoelectric device prior to absorption of light is not necessarily uniform. Therefore anomalous results may occur in the case of individual electrons. An electron that was already excited above the equilibrium level of the photoelectric device might be ejected when it absorbed uncharacteristically low frequency illumination. Statistically, however, the characteristic behavior of a photoelectric device will reflect the behavior of the vast majority of its electrons, which will be at their equilibrium level. This point is helpful in comprehending the distinction between the study of individual particles in quantum dynamics and the study of massed particles in classical physics.

10

Basic Concepts of Quantum Mechanics

11

The quantisation of matter: the Bohr model of the atom By the dawn of the 20th century, it was known that atoms comprise a diffuse cloud of negatively-charged electrons surrounding a small, dense, positively-charged nucleus. This understanding suggested a model in which the electrons circle around the nucleus like planets orbiting a sun.[16] However, it was also known that the atom in this model would be unstable: according to classical theory orbiting electrons are undergoing centripetal acceleration, and should therefore give off electromagnetic radiation, the loss of energy also causing them to spiral toward the nucleus, colliding with it in a fraction of a second. A second, related, puzzle was the emission spectrum of atoms. When a gas is heated, it gives off light only at discrete frequencies. For example, the visible light given off by hydrogen consists of four different colours, as shown in the picture below. By contrast, white light consists of a continuous emission across the whole range of visible frequencies.

Emission spectrum of hydrogen. When excited, hydrogen gas gives off light in four distinct colours (spectral lines) in the visible spectrum, as well as a number of lines in the infra-red and ultra-violet.

In 1885 the Swiss mathematician Johann Balmer discovered that each wavelength λ (lambda) in the visible spectrum of hydrogen is related to some integer n by the equation

where B is a constant which Balmer determined to be equal to 364.56 nm. Thus Balmer's constant was the basis of a system of discrete, i.e. quantised, integers. In 1888 Johannes Rydberg generalized and greatly increased the explanatory utility of Balmer's formula. He predicted that λ is related to two integers n and m according to what is now known as the Rydberg formula:[17]

where R is the Rydberg constant, equal to 0.0110 nm−1, and n must be greater than m. Rydberg's formula accounts for the four visible wavelengths of hydrogen by setting m = 2 and n = 3, 4, 5, 6. It also predicts additional wavelengths in the emission spectrum: for m = 1 and for n > 1, the emission spectrum should contain certain ultraviolet wavelengths, and for m = 3 and n > 3, it should also contain certain infrared wavelengths. Experimental observation of these wavelengths came two decades later: in 1908 Louis Paschen found some of the predicted infrared wavelengths, and in 1914 Theodore Lyman found some of the predicted ultraviolet wavelengths.[17]

Basic Concepts of Quantum Mechanics

12

Bohr's model In 1913 Niels Bohr proposed a new model of the atom that included quantized electron orbits.[18] In Bohr's model, electrons could inhabit only certain orbits around the atomic nucleus. When an atom emitted (or absorbed) energy, the electron did not move in a continuous trajectory from one orbit around the nucleus to another, as might be expected classically. Instead, the electron would jump instantaneously from one orbit to another, giving off the emitted light in the form of a photon.[19] The possible energies of photons given off by each element were determined by the differences in energy between the orbits, and so the emission spectrum for each element would contain a number of lines.[20] Bohr theorised that the angular momentum, L, of an electron is quantised:

The Bohr model of the atom, showing an electron quantum jumping to ground state n = 1.

where n is an integer and h is the Planck constant. Starting from this assumption, Coulomb's law and the equations of circular motion show that an electron with n units of angular momentum will orbit a proton at a distance r given by , where ke is the Coulomb constant, m is the mass of an electron, and e is the charge on an electron. For simplicity this is written as

where a0, called the Bohr radius, is equal to 0.0529 nm. The Bohr radius is the radius of the smallest allowed orbit. The energy of the electron[21] can also be calculated, and is given by . Thus Bohr's assumption that angular momentum is quantised means that an electron can only inhabit certain orbits around the nucleus, and that it can have only certain energies. A consequence of these constraints is that the electron will not crash into the nucleus: it cannot continuously emit energy, and it cannot come closer to the nucleus than a0 (the Bohr radius). An electron loses energy by jumping instantaneously from its original orbit to a lower orbit; the extra energy is emitted in the form of a photon. Conversely, an electron that absorbs a photon gains energy, hence it jumps to an orbit that is farther from the nucleus. Each photon from glowing atomic hydrogen is due to an electron moving from a higher orbit, with radius rn, to a lower orbit, rm. The energy Eγ of this photon is the difference in the energies En and Em of the electron:

Since Planck's equation shows that the photon's energy is related to its wavelength by Eγ = hc/λ, the wavelengths of light that can be emitted are given by

This equation has the same form as the Rydberg formula, and predicts that the constant R should be given by

Basic Concepts of Quantum Mechanics

13

Therefore the Bohr model of the atom can predict the emission spectrum of hydrogen in terms of fundamental constants.[22] However, it was not able to make accurate predictions for multi-electron atoms, or to explain why some spectral lines are brighter than others.

Wave-particle duality In 1924, Louis de Broglie proposed the idea that just as light has both wave-like and particle-like properties, matter also has wave-like properties.[23] The wavelength, λ , associated with a particle is related to its momentum, p through the Planck constant h :[24][25]

The relationship, called the de Broglie hypothesis, holds for all types of matter. Thus all matter exhibits properties of both particles and waves. Three years later, the wave-like nature of electrons was demonstrated by showing that a beam of electrons could exhibit diffraction, just like a beam of light. At the University of Aberdeen, George Thomson passed a beam of electrons through a thin metal film and observed the predicted diffraction patterns. At Bell Labs, Davisson and Germer guided their beam through a crystalline grid. Similar wave-like phenomena were later shown for atoms and even small molecules. De Broglie was awarded the Nobel Prize for Physics in 1929 for his hypothesis; Thomson and Davisson shared the Nobel Prize for Physics in 1937 for their experimental work. The concept of wave-particle duality says that neither the classical concept of "particle" nor of "wave" can fully describe the behavior of quantum-scale objects, either photons or matter. Indeed, astrophysicist A.S. Eddington proposed in 1927 that "We can scarcely describe such an entity as a wave or as a particle; perhaps as a compromise we had better call it a 'wavicle' ".[26] (This term was later popularised by mathematician Banesh Hoffmann.)[27]:172 Wave-particle duality is an example of the principle of complementarity in quantum physics. An elegant example of wave-particle duality, the double slit experiment, is discussed in the section below. De Broglie's treatment of quantum events served as a jumping off point for Schrödinger when he set about to construct a wave equation to describe quantum theoretical events.

The double-slit experiment

Light from one slit interferes with light from the other, producing an interference pattern (the 3 fringes shown at the right).

In the double-slit experiment as originally performed by Thomas Young and Augustin Fresnel in 1827, a beam of light is directed through two narrow, closely spaced slits, producing an interference pattern of light and dark bands on a screen. If one of the slits is covered up, one might naively expect that the intensity of the fringes due to interference would be halved everywhere. In fact, a much simpler pattern is seen, a simple diffraction pattern. Closing one slit results in a much simpler pattern diametrically opposite the open slit. Exactly the same behaviour can be demonstrated in water waves, and so the double-slit experiment was seen as a demonstration of the wave nature of light.

Basic Concepts of Quantum Mechanics

14

The double-slit experiment has also been performed using electrons, atoms, and even molecules, and the same type of interference pattern is seen. Thus it has been demonstrated that all matter possesses both particle and wave characteristics. Even if the source intensity is turned down so that only one particle (e.g. photon or electron) is passing through the apparatus at a time, the same interference pattern develops over time. The quantum particle acts as a wave when passing through the double slits, but as a particle when it is detected. This is a typical feature of quantum complementarity: a quantum particle will act as a wave when we do an experiment to measure its wave-like properties, and like a particle when we do an experiment to measure its particle-like properties. Where on the detector screen any individual particle shows up will be the result of an entirely random process.

The diffraction pattern produced when light is shone through one slit (top) and the interference pattern produced by two slits (bottom). The interference pattern from two slits is much more complex, demonstrating the wave-like propagation of light.

Application to the Bohr model De Broglie expanded the Bohr model of the atom by showing that an electron in orbit around a nucleus could be thought of as having wave-like properties. In particular, an electron will be observed only in situations that permit a standing wave around a nucleus. An example of a standing wave is a violin string, which is fixed at both ends and can be made to vibrate. The waves created by a stringed instrument appear to oscillate in place, moving from crest to trough in an up-and-down motion. The wavelength of a standing wave is related to the length of the vibrating object and the boundary conditions. For example, because the violin string is fixed at both ends, it can carry standing waves of wavelengths 2l/n, where l is the length and n is a positive integer. De Broglie suggested that the allowed electron orbits were those for which the circumference of the orbit would be an integer number of wavelengths.

Development of modern quantum mechanics In 1925, building on de Broglie's hypothesis, Erwin Schrödinger developed the equation that describes the behaviour of a quantum mechanical wave. The equation, called the Schrödinger equation after its creator, is central to quantum mechanics, defines the permitted stationary states of a quantum system, and describes how the quantum state of a physical system changes in time.[28] In the paper that introduced Schrödinger's cat, he says that the psi-function featured in his equation provides the "means for predicting probability of measurement results," and that it therefore provides "future expectation[s] , somewhat as laid down in a catalog."[29] Schrödinger was able to calculate the energy levels of hydrogen by treating a hydrogen atom's electron as a classical wave, moving in a well of electrical potential created by the proton. This calculation accurately reproduced the energy levels of the Bohr model.

Erwin Schrödinger, about 1933, age 46

At a somewhat earlier time, Werner Heisenberg was trying to find an explanation for the intensities of the different lines in the hydrogen emission spectrum. By means of a series of mathematical analogies, Heisenberg wrote out the quantum mechanical analogue for the classical computation of intensities. Shortly afterwards, Heisenberg's colleague Max Born realised that Heisenberg's method of calculating the

Basic Concepts of Quantum Mechanics

15

probabilities for transitions between the different energy levels could best be expressed by using the mathematical concept of matrices.[30] In May 1926, Schrödinger proved that Heisenberg's matrix mechanics and his own wave mechanics made the same predictions about the properties and behaviour of the electron; mathematically, the two theories were identical. Yet the two men disagreed on the interpretation of their mutual theory. For instance, Heisenberg saw no problem in the theoretical prediction of instantaneous transitions of electrons between orbits in an atom, but Schrödinger hoped that a theory based on continuous wave-like properties could avoid what he called (in the words of Wilhelm Wien[31]) "this nonsense about quantum jumps."

Copenhagen interpretation Bohr, Heisenberg and others tried to explain what these experimental results and mathematical models really mean. Their description, known as the Copenhagen interpretation of quantum mechanics, aimed to describe the nature of reality that was being probed by the measurements and described by the mathematical formulations of quantum mechanics. The main principles of the Copenhagen interpretation are: 1. A system is completely described by a wave function,

. (Heisenberg)

2. How changes over time is given by the Schrödinger equation. 3. The description of nature is essentially probabilistic. The probability of an event — for example, where on the screen a particle will show up in the two slit experiment — is related to the square of the amplitude of its wave function. (Born rule, due to Max Born, which gives a physical meaning to the wavefunction in the Copenhagen interpretation: the probability amplitude) 4. It is not possible to know the values of all of the properties of the system at the same time; those properties that are not known with precision must be described by probabilities. (Heisenberg's uncertainty principle) 5. Matter, like energy, exhibits a wave-particle duality. An experiment can demonstrate the particle-like properties of matter, or its wave-like properties; but not both at the same time. (Complementarity principle due to Bohr) 6. Measuring devices are essentially classical devices, and measure classical properties such as position and momentum. 7. The quantum mechanical description of large systems should closely approximate the classical description. (Correspondence principle of Bohr and Heisenberg) Various consequences of these principles are discussed in more detail in the following subsections.

Basic Concepts of Quantum Mechanics

16

Uncertainty principle Suppose that we want to measure the position and speed of an object — for example a car going through a radar speed trap. Naively, we assume that the car has a definite position and speed at a particular moment in time, and how accurately we can measure these values depends on the quality of our measuring equipment — if we improve the precision of our measuring equipment, we will get a result that is closer to the true value. In particular, we would assume that how precisely we measure the speed of the car does not affect the measurement of its position, and vice versa. In 1927, Heisenberg proved that these assumptions are not correct.[33] Quantum mechanics shows that certain pairs of physical properties, like position and speed, cannot both be known to arbitrary precision: the more precisely one property is known, the less precisely the other can be known. This statement is known as the uncertainty principle. The uncertainty principle isn't a statement about the accuracy of our measuring equipment, but about the nature of the system itself — our naive assumption that the car had a definite position and speed was incorrect. On a scale of cars and people, these uncertainties are too small to notice, but when dealing with atoms and electrons they become critical.[34]

Werner Heisenberg at the age of 26. Heisenberg won the Nobel Prize in Physics in 1932 for the work that he [32] did at around this time.

Heisenberg gave, as an illustration, the measurement of the position and momentum of an electron using a photon of light. In measuring the electron's position, the higher the frequency of the photon the more accurate is the measurement of the position of the impact, but the greater is the disturbance of the electron, which absorbs a random amount of energy, rendering the measurement obtained of its momentum increasingly uncertain (momentum is velocity multiplied by mass), for one is necessarily measuring its post-impact disturbed momentum, from the collision products, not its original momentum. With a photon of lower frequency the disturbance - hence uncertainty - in the momentum is less, but so is the accuracy of the measurement of the position of the impact.[35] The uncertainty principle shows mathematically that the product of the uncertainty in the position and momentum of a particle (momentum is velocity multiplied by mass) could never be less than a certain value, and that this value is related to Planck's constant.

Wave function collapse Wave function collapse is a forced term for whatever happened when it becomes appropriate to replace the description of an uncertain state of a system by a description of the system in a definite state. Explanations for the nature of the process of becoming certain are controversial. At any time before a photon "shows up" on a detection screen it can only be described by a set of probabilities for where it might show up. When it does show up, for instance in the CCD of an electronic camera, the time and the space where it interacted with the device are known within very tight limits. However, the photon has disappeared, and the wave function has disappeared with it. In its place some physical change in the detection screen has appeared, e.g., an exposed spot in a sheet of photographic film, or a change in electric potential in some cell of a CCD.

Basic Concepts of Quantum Mechanics

Eigenstates and eigenvalues For a more detailed introduction to this subject, see: Introduction to eigenstates Because of the uncertainty principle, statements about both the position and momentum of particles can only assign a probability that the position or momentum will have some numerical value. Therefore it is necessary to formulate clearly the difference between the state of something that is indeterminate, such as an electron in a probability cloud, and the state of something having a definite value. When an object can definitely be "pinned-down" in some respect, it is said to possess an eigenstate.

The Pauli exclusion principle In 1924, Wolfgang Pauli proposed a new quantum degree of freedom (or quantum number), with two possible values, to resolve inconsistencies between observed molecular spectra and the predictions of quantum mechanics. In particular, the spectrum of atomic hydrogen had a doublet, or pair of lines differing by a small amount, where only one line was expected. Pauli formulated his exclusion principle, stating that "There cannot exist an atom in such a quantum state that two electrons within [it] have the same set of quantum numbers."[36] A year later, Uhlenbeck and Goudsmit identified Pauli's new degree of freedom with a property called spin. The idea, originating with Ralph Kronig, was that electrons behave as if they rotate, or "spin", about an axis. Spin would account for the missing magnetic moment, and allow two electrons in the same orbital to occupy distinct quantum states if they "spun" in opposite directions, thus satisfying the exclusion principle. The quantum number represented the sense (positive or negative) of spin.

Application to the hydrogen atom Bohr's model of the atom was essentially two-dimensional — an electron orbiting in a plane around its nuclear "sun." However, the uncertainty principle states that an electron cannot be viewed as having an exact location at any given time. In the modern theory the orbit has been replaced by an atomic orbital, a "cloud" of possible locations. It is often depicted as a three-dimensional region within which there is a 95 percent probability of finding the electron.[37] Schrödinger was able to calculate the energy levels of hydrogen by treating a hydrogen atom's electron as a wave, represented by the "wave function" Ψ, in a electric potential well, V, created by the proton. The solutions to Schrödinger's equation are distributions of probabilities for electron positions and locations. Orbitals have a range of different shapes in three dimensions. The energies of the different orbitals can be calculated, and they accurately reproduce the energy levels of the Bohr model. Within Schrödinger's picture, each electron has four properties: 1. An "orbital" designation, indicating whether the particle wave is one that is closer to the nucleus with less energy or one that is farther from the nucleus with more energy; 2. The "shape" of the orbital, spherical or otherwise; 3. The "inclination" of the orbital, determining the magnetic moment of the orbital around the z-axis. 4. The "spin" of the electron. The collective name for these properties is the quantum state of the electron. The quantum state can be described by giving a number to each of these properties; these are known as the electron's quantum numbers. The quantum state of the electron is described by its wavefunction. The Pauli exclusion principle demands that no two electrons within an atom may have the same values of all four numbers.

17

Basic Concepts of Quantum Mechanics

18

The first property describing the orbital is the principal quantum number, n, which is the same as in Bohr's model. n denotes the energy level of each orbital. The possible values for n are integers: The shapes of the first five atomic orbitals: 1s, 2s, 2px,2py, and 2pz. The colours show the phase of the wavefunction.

The next quantum number, the azimuthal quantum number, denoted l, describes the shape of the orbital. The shape is a consequence of the angular momentum of the orbital. The angular momentum represents the resistance of a spinning object to speeding up or slowing down under the influence of external force. The azimuthal quantum number represents the orbital angular momentum of an electron around its nucleus. The possible values for l are integers from 0 to n − 1:

The shape of each orbital has its own letter as well. The first shape is denoted by the letter s (a mnemonic being "sphere"). The next shape is denoted by the letter p and has the form of a dumbbell. The other orbitals have more complicated shapes (see atomic orbital), and are denoted by the letters d, f, and g. The third quantum number, the magnetic quantum number, describes the magnetic moment of the electron, and is denoted by ml (or simply m). The possible values for ml are integers from −l to l: The magnetic quantum number measures the component of the angular momentum in a particular direction. The choice of direction is arbitrary, conventionally the z-direction is chosen. The fourth quantum number, the spin quantum number (pertaining to the "orientation" of the electron's spin) is denoted ms, with values +1⁄2 or −1⁄2. The chemist Linus Pauling wrote, by way of example: In the case of a helium atom with two electrons in the 1s orbital, the Pauli Exclusion Principle requires that the two electrons differ in the value of one quantum number. Their values of n, l, and ml are the same; moreover, they have the same spin, s = 1⁄2. Accordingly they must differ in the value of ms, which can have the value of +1⁄2 for one electron and −1⁄2 for the other."[36] It is the underlying structure and symmetry of atomic orbitals, and the way that electrons fill them, that determines the organisation of the periodic table and the structure and strength of chemical bonds between atoms.

Basic Concepts of Quantum Mechanics

19

Dirac wave equation In 1928, Paul Dirac extended the Pauli equation, which described spinning electrons, to account for special relativity. The result was a theory that dealt properly with events, such as the speed at which an electron orbits the nucleus, occurring at a substantial fraction of the speed of light. By using the simplest electromagnetic interaction, Dirac was able to predict the value of the magnetic moment associated with the electron's spin, and found the experimentally observed value, which was too large to be that of a spinning charged sphere governed by classical physics. He was able to solve for the spectral lines of the hydrogen atom, and to reproduce from physical first principles Sommerfeld's successful formula for the fine structure of the hydrogen spectrum. Dirac's equations sometimes yielded a negative value for energy, for which he proposed a novel solution: he posited the existence of an antielectron and of a dynamical vacuum. This led to the many-particle quantum field theory.

Paul Dirac (1902 - 1984)

Quantum entanglement The Pauli exclusion principle says that two electrons in one system cannot be in the same state. Nature leaves open the possibility, however, that two electrons can have both states "superimposed" over each of them. Recall that the wave functions that Superposition of two quantum characteristics, and two resolution possibilities. emerge simultaneously from the double slits arrive at the detection screen in a state of superposition. Nothing is certain until the superimposed waveforms "collapse," At that instant an electron shows up somewhere in accordance with the probabilities that are the squares of the amplitudes of the two superimposed waveforms. The situation there is already very abstract. A concrete way of thinking about entangled photons, photons in which two contrary states are superimposed on each of them in the same event, is as follows: Imagine that the superposition of a state that can be mentally labeled as blue and another state that can be mentally labeled as red will then appear (in imagination, of course) as a purple state. Two photons are produced as the result of the same atomic event. Perhaps they are produced by the excitation of a crystal that characteristically absorbs a photon of a certain frequency and emits two photons of half the original frequency. So the two photons come out "purple." If the experimenter now performs some experiment that will determine whether one of the photons is either blue or red, then that experiment changes the photon involved from one having a superposition of "blue" and "red" characteristics to a photon that has only one of those characteristics. The problem that Einstein had with such an imagined situation was that if one of these photons had been kept bouncing between mirrors in a laboratory on earth, and the other one had traveled halfway to the nearest star, when its twin was made to reveal itself as either blue or red, that meant that the distant photon now had to lose its "purple" status too. So whenever it might be investigated after its twin had been measured, it would necessarily show up in the opposite state to whatever its twin had revealed.

Basic Concepts of Quantum Mechanics

20

In trying to show that quantum mechanics was not a complete theory, Einstein started with the theory's prediction that two or more particles that have interacted in the past can appear strongly correlated when their various properties are later measured. He sought to explain this seeming interaction in a classical way, through their common past, and preferably not by some "spooky action at a distance." The argument is worked out in a famous paper, Einstein, Podolsky, and Rosen (1935; abbreviated EPR), setting out what is now called the EPR paradox. Assuming what is now usually called local realism, EPR attempted to show from quantum theory that a particle has both position and momentum simultaneously, while according to the Copenhagen interpretation, only one of those two properties actually exists and only at the moment that it is being measured. EPR concluded that quantum theory is incomplete in that it refuses to consider physical properties which objectively exist in nature. (Einstein, Podolsky, & Rosen 1935 is currently Einstein's most cited publication in physics journals.) In the same year, Erwin Schrödinger used the word "entanglement" and declared: "I would not call that one but rather the characteristic trait of quantum mechanics." [38] The question of whether entanglement is a real condition is still in dispute.[39] The Bell inequalities are the most powerful challenge to Einstein's claims.

Quantum field theory The idea of quantum field theory began in the late 1920s with British physicist Paul Dirac, when he attempted to quantise the electromagnetic field — a procedure for constructing a quantum theory starting from a classical theory. A field in physics is "a region or space in which a given effect (such as magnetism) exists."[40] Other effects that manifest themselves as fields are gravitation and static electricity.[41] In 2008, physicist Richard Hammond wrote that Sometimes we distinguish between quantum mechanics (QM) and quantum field theory (QFT). QM refers to a system in which the number of particles is fixed, and the fields (such as the electromechanical field) are continuous classical entities. QFT . . . goes a step further and allows for the creation and annihilation of particles . . . . He added, however, that quantum mechanics is often used to refer to "the entire notion of quantum view."[42]:108 In 1931, Dirac proposed the existence of particles that later became known as anti-matter.[43] Dirac shared the Nobel Prize in physics for 1933 with Schrödinger, "for the discovery of new productive forms of atomic theory."[44]

Quantum electrodynamics

This sculpture in Bristol, England — a series of clustering cones — presents the idea of small worlds that Paul Dirac studied to reach his discovery of anti-matter.

Quantum electrodynamics (QED) is the name of the quantum theory of the electromagnetic force. Understanding QED begins with understanding electromagnetism. Electromagnetism can be called "electrodynamics" because it is a dynamic interaction between electrical and magnetic forces. Electromagnetism begins with the electric charge. Electric charges are the sources of, and create, electric fields. An electric field is a field which exerts a force on any particles that carry electric charges, at any point in space. This includes the electron, proton, and even quarks, among others. As a force is exerted, electric charges move, a current flows and a magnetic field is produced. The magnetic field, in turn causes electric current (moving electrons). The interacting electric and magnetic field is called an electromagnetic field.

Basic Concepts of Quantum Mechanics The physical description of interacting charged particles, electrical currents, electrical fields, and magnetic fields is called electromagnetism. In 1928 Paul Dirac produced a relativistic quantum theory of electromagnetism. This was the progenitor to modern quantum electrodynamics, in that it had essential ingredients of the modern theory. However, the problem of unsolvable infinities developed in this relativistic quantum theory. Years later, renormalization solved this problem. Initially viewed as a suspect, provisional procedure by some of its originators, renormalization eventually was embraced as an important and self-consistent tool in QED and other fields of physics. Also, in the late 1940s Feynman's diagrams depicted all possible interactions pertaining to a given event. The diagrams showed that the electromagnetic force is the interactions of photons between interacting particles. An example of a prediction of quantum electrodynamics which has been verified experimentally is the Lamb shift. This refers to an effect whereby the quantum nature of the electromagnetic field causes the energy levels in an atom or ion to deviate slightly from what they would otherwise be. As a result, spectral lines may shift or split. In the 1960s physicists realized that QED broke down at extremely high energies. From this inconsistency the Standard Model of particle physics was discovered, which remedied the higher energy breakdown in theory. The Standard Model unifies the electromagnetic and weak interactions into one theory. This is called the electroweak theory.

Interpretations The physical measurements, equations, and predictions pertinent to quantum mechanics are all consistent and hold a very high level of confirmation. However, the question of what these abstract models say about the underlying nature of the real world has received competing answers.

Applications Applications of quantum mechanics include the laser, the transistor, the electron microscope, and magnetic resonance imaging. A special class of quantum mechanical applications is related to macroscopic quantum phenomena such as superfluid helium and superconductors. The study of semiconductors led to the invention of the diode and the transistor, which are indispensable for modern electronics. In even the simple light switch, quantum tunnelling is absolutely vital, as otherwise the electrons in the electric current could not penetrate the potential barrier made up of a layer of oxide. Flash memory chips found in USB drives also use quantum tunnelling, to erase their memory cells.[45]

Notes [1] Quantum Mechanics from [[National Public Radio (http:/ / www. pbs. org/ trasnsistor/ science/ info/ quantum. html)]] [2] Classical physics also does not accurately describe the universe on the largest scales or at speeds close to that of light. An accurate description requires general relativity. [3] Feynman, Richard P. (1988). QED : the strange theory of light and matter (1st Princeton pbk., seventh printing with corrections. ed.). Princeton, N.J.: Princeton University Press. pp. 10. ISBN 978-0691024172. [4] Alan Macdonald, "Spooky action at a distance: The puzzle of entanglement in quantum theory," page 5 of 7, downloaded 13 June 2012 from http:/ / faculty. luther. edu/ ~macdonal/ [5] This result was published (in German) as Planck, Max (1901). "Ueber das Gesetz der Energieverteilung im Normalspectrum" (http:/ / www. physik. uni-augsburg. de/ annalen/ history/ historic-papers/ 1901_309_553-563. pdf). Ann. Phys. 309 (3): 553–63. Bibcode 1901AnP...309..553P. doi:10.1002/andp.19013090310. . English translation: " On the Law of Distribution of Energy in the Normal Spectrum (http:/ / dbhs. wvusd. k12. ca. us/ webdocs/ Chem-History/ Planck-1901/ Planck-1901. html)". [6] The word "quantum" comes from the Latin word for "how much" (as does "quantity"). Something which is "quantized," like the energy of Planck's harmonic oscillators, can only take specific values. For example, in most countries money is effectively quantized, with the "quantum of money" being the lowest-value coin in circulation. "Mechanics" is the branch of science that deals with the action of forces on objects, so "quantum mechanics" is the part of mechanics that deals with objects for which particular properties are quantized.

21

Basic Concepts of Quantum Mechanics [7] Francis Weston Sears (1958). Mechanics, Wave Motion, and Heat (http:/ / books. google. com/ books?hl=en& q="Mechanics,+ Wave+ Motion,+ and+ Heat"+ "where+ n+ =+ 1,"& btnG=Search+ Books). Addison-Wesley. p. 537. . [8] "The Nobel Prize in Physics 1918" (http:/ / nobelprize. org/ nobel_prizes/ physics/ laureates/ 1918/ ). The Nobel Foundation. . Retrieved 2009-08-01. [9] Kragh, Helge (1 December 2000). "Max Planck: the reluctant revolutionary" (http:/ / physicsworld. com/ cws/ article/ print/ 373). PhysicsWorld.com. [10] Einstein, Albert (1905). "Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt" (http:/ / www. zbp. univie. ac. at/ dokumente/ einstein1. pdf). Annalen der Physik 17 (6): 132–148. Bibcode 1905AnP...322..132E. doi:10.1002/andp.19053220607. ., translated into English as On a Heuristic Viewpoint Concerning the Production and Transformation of Light (http:/ / lorentz. phl. jhu. edu/ AnnusMirabilis/ AeReserveArticles/ eins_lq. pdf). The term "photon" was introduced in 1926. [11] Taylor, J. R.; Zafiratos, C. D.; Dubson, M. A. (2004). Modern Physics for Scientists and Engineers. Prentice Hall. pp. 127–9. ISBN 0-13-589789-0. [12] Stephen Hawking, The Universe in a Nutshell, Bantam, 2001. [13] Actually there can be intensity-dependent effects, but at intensities achievable with non-laser sources these effects are unobservable. [14] Dicke and Wittke, Introduction to Quantum Mechanics, p. 12 [15] Einstein's photoelectric effect equation can be derived and explained without requiring the concept of "photons". That is, the electromagnetic radiation can be treated as a classical electromagnetic wave, as long as the electrons in the material are treated by the laws of quantum mechanics. The results are quantitatively correct for thermal light sources (the sun, incandescent lamps, etc) both for the rate of electron emission as well as their angular distribution. For more on this point, see NTRS.NASA.gov (http:/ / ntrs. nasa. gov/ archive/ nasa/ casi. ntrs. nasa. gov/ 19680009569_1968009569. pdf) [16] The classical model of the atom is called the planetary model, or sometimes the Rutherford model after Ernest Rutherford who proposed it in 1911, based on the Geiger-Marsden gold foil experiment which first demonstrated the existence of the nucleus. [17] Taylor, J. R.; Zafiratos, C. D.; Dubson, M. A. (2004). Modern Physics for Scientists and Engineers. Prentice Hall. pp. 147–8. ISBN 0-13-589789-0. [18] McEvoy, J. P.; Zarate, O. (2004). Introducing Quantum Theory. Totem Books. pp. 70–89, especially p. 89. ISBN 1-84046-577-8. [19] World Book Encyclopedia, page 6, 2007. [20] Dicke and Wittke, Introduction to Quantum Mechanics, p. 10f. [21] In this case, the energy of the electron is the sum of its kinetic and potential energies. The electron has kinetic energy by virtue of its actual motion around the nucleus, and potential energy because of its electromagnetic interaction with the nucleus. [22] The model can be easily modified to account of the emission spectrum of any system consisting of a nucleus and a single electron (that is, ions such as He+ or O7+ which contain only one electron). [23] J. P. McEvoy and Oscar Zarate (2004). Introducing Quantum Theory. Totem Books. p. 110f. ISBN 1-84046-577-8. [24] Aezel, Amir D., Entanglrment, p. 51f. (Penguin, 2003) ISBN 0-452-28457 [25] J. P. McEvoy and Oscar Zarate (2004). Introducing Quantum Theory. Totem Books. p. 114. ISBN 1-84046-577-8. [26] A.S. Eddington, The Nature of the Physical World, the course of Gifford Lectures that Eddington delivered in the University of Edinburgh in January to March 1927, Kessinger Publishing, 2005, p. 201. (http:/ / books. google. com/ books?id=PGOTKcxSqMUC& pg=PA201& lpg=PA201& dq=We+ can+ scarcely+ describe+ such+ an+ entity+ as+ a+ wave+ or+ as+ a+ particle;+ perhaps+ as+ a+ compromise+ we+ had+ better+ call+ it+ a+ `wavicle& source=bl& ots=K0IfGzaXli& sig=zgrQiBJbHRLuUzVBT-yy8jZhC1Y& hl=en& ei=i8g1SpOHC4PgtgOu_4jVDg& sa=X& oi=book_result& ct=result& resnum=1) [27] Banesh Hoffman, The Strange Story of the Quantum, Dover, 1959 [28] "Schrodinger Equation (Physics)," Encyclopædia Britannica (http:/ / www. britannica. com/ EBchecked/ topic/ 528298/ Schrodinger-equation) [29] Erwin Schrödinger, "The Present Situation in Quantum Mechanics," p. 9. "This translation was originally published in Proceedings of the American Philosophical Society, 124, 323-38. [And then appeared as Section I.11 of Part I of Quantum Theory and Measurement (J.A. Wheeler and W.H. Zurek, eds., Princeton university Press, New Jersey 1983). This paper can be downloaded from http:/ / www. tu-harburg. de/ rzt/ rzt/ it/ QM/ cat. html. " [30] For a somewhat more sophisticated look at how Heisenberg transitioned from the old quantum theory and classical physics to the new quantum mechanics, see Heisenberg's entryway to matrix mechanics. [31] W. Moore, Schrödinger: Life and Thought, Cambridge University Press (1989), p. 222. [32] Heisenberg's Nobel Prize citation (http:/ / nobelprize. org/ nobel_prizes/ physics/ laureates/ 1932/ ) [33] Heisenberg first published his work on the uncertainty principle in the leading German physics journal Zeitschrift für Physik: Heisenberg, W. (1927). "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik". Z. Phys. 43 (3–4): 172–198. Bibcode 1927ZPhy...43..172H. doi:10.1007/BF01397280. [34] Nobel Prize in Physics presentation speech, 1932 (http:/ / nobelprize. org/ nobel_prizes/ physics/ laureates/ 1932/ press. html) [35] "Uncertainty principle," Encyclopædia Britannica (http:/ / www. britannica. com/ EBchecked/ topic/ 614029/ uncertainty-principle) [36] Linus Pauling, The Nature of the Chemical Bond, p. 47 [37] "Orbital (chemistry and physics)," Encyclopædia Britannica (http:/ / www. britannica. com/ EBchecked/ topic/ 431159/ orbital) [38] E. Schrödinger, Proceedings of the Cambridge Philosophical Society, 31 (1935), p. 555says: "When two systems, of which we know the states by their respective representation, enter into a temporary physical interaction due to known forces between them and when after a time

22

Basic Concepts of Quantum Mechanics of mutual influence the systems separate again, then they can no longer be described as before, viz., by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics." [39] "Quantum Nonlocality and the Possibility of Superluminal Effects", John G. Cramer, npl.washington.edu (http:/ / www. npl. washington. edu/ npl/ int_rep/ qm_nl. html) [40] "Mechanics," Merriam-Webster Online Dictionary (http:/ / www. merriam-webster. com/ dictionary/ field) [41] "Field," Encyclopædia Britannica (http:/ / www. britannica. com/ EBchecked/ topic/ 206162/ field) [42] Richard Hammond, The Unknown Universe, New Page Books, 2008. ISBN 978-1-60163-003-2 [43] The Physical World website (http:/ / www. physicalworld. org/ restless_universe/ html/ ru_dira. html) [44] "The Nobel Prize in Physics 1933" (http:/ / nobelprize. org/ nobel_prizes/ physics/ laureates/ 1933/ ). The Nobel Foundation. . Retrieved 2007-11-24. [45] Durrani, Z. A. K.; Ahmed, H. (2008). Vijay Kumar. ed. Nanosilicon. Elsevier. p. 345. ISBN 978-0-08-044528-1.

References • Bernstein, Jeremy (2005). "Max Born and the quantum theory". American Journal of Physics 73 (11). • Beller, Mara (2001). Quantum Dialogue: The Making of a Revolution. University of Chicago Press. • Bohr, Niels (1958). Atomic Physics and Human Knowledge. John Wiley & Sons. ASIN B00005VGVF. ISBN 0-486-47928-5. OCLC 530611. • de Broglie, Louis (1953). The Revolution in Physics. Noonday Press. LCCN 53010401. • Einstein, Albert (1934). Essays in Science. Philosophical Library. ISBN 0-486-47011-3. LCCN 55003947. • Feigl, Herbert; Brodbeck, May (1953). Readings in the Philosophy of Science. Appleton-Century-Crofts. ISBN 0-390-30488-3. LCCN 53006438. • Feynman, Richard P. (1949). "Space-Time Approach to Quantum Electrodynamics" (http://www.physics. princeton.edu/~mcdonald/examples/QED/feynman_pr_76_769_49.pdf). Physical Review 76 (6): 769–789. Bibcode 1949PhRv...76..769F. doi:10.1103/PhysRev.76.769. • Fowler, Michael (1999). The Bohr Atom. University of Virginia. • Heisenberg, Werner (1958). Physics and Philosophy. Harper and Brothers. ISBN 0-06-130549-9. LCCN 99010404. • Lakshmibala, S. (2004). "Heisenberg, Matrix Mechanics and the Uncertainty Principle". Resonance, Journal of Science Education 9 (8). • Liboff, Richard L. (1992). Introductory Quantum Mechanics (2nd ed.). • Lindsay, Robert Bruce; Margenau, Henry (1957). Foundations of Physics. Dover. ISBN 0-918024-18-8. LCCN 57014416. • McEvoy, J. P.; Zarate, Oscar. Introducing Quantum Theory. ISBN 1-874166-37-4. • Müller-Kirsten, H. J. W. (2012). Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral (2nd ed.). World Scientific. ISBN 978-981-4397-74-2. • Nave, Carl Rod (2005). "Quantum Physics" (http://hyperphysics.phy-astr.gsu.edu/hbase/quacon. html#quacon). HyperPhysics. Georgia State University. • Peat, F. David (2002). From Certainty to Uncertainty: The Story of Science and Ideas in the Twenty-First Century. Joseph Henry Press. • Reichenbach, Hans (1944). Philosophic Foundations of Quantum Mechanics. University of California Press. ISBN 0-486-40459-5. LCCN a44004471. • Schlipp, Paul Arthur (1949). Albert Einstein: Philosopher-Scientist. Tudor Publishing Company. LCCN 50005340. • Scientific American Reader, 1953. • Sears, Francis Weston (1949). Optics (3rd ed.). Addison-Wesley. ISBN 0-19-504601-3. LCCN 51001018. • Shimony, A. (1983). "(title not given in citation)". Foundations of Quantum Mechanics in the Light of New Technology (S. Kamefuchi et al., eds.). Tokyo: Japan Physical Society. pp. 225.; cited in: Popescu, Sandu; Daniel Rohrlich (1996). "Action and Passion at a Distance: An Essay in Honor of Professor Abner Shimony". arXiv:quant-ph/9605004 [quant-ph].

23

Basic Concepts of Quantum Mechanics • Tavel, Morton; Tavel, Judith (illustrations) (2002). Contemporary physics and the limits of knowledge (http:// books.google.com/?id=SELS0HbIhjYC&pg=PA200&dq=Wave+function+collapse). Rutgers University Press. ISBN 978-0-8135-3077-2. • Van Vleck, J. H.,1928, "The Correspondence Principle in the Statistical Interpretation of Quantum Mechanics," Proc. Nat. Acad. Sci. 14: 179. • Wheeler, John Archibald; Feynman, Richard P. (1949). "Classical Electrodynamics in Terms of Direct Interparticle Action". Reviews of Modern Physics 21 (3): 425–433. Bibcode 1949RvMP...21..425W. doi:10.1103/RevModPhys.21.425. • Wieman, Carl; Perkins, Katherine (2005). "Transforming Physics Education". Physics Today. • Westmoreland; Benjamin Schumacher (1998). "Quantum Entanglement and the Nonexistence of Superluminal Signals". arXiv:quant-ph/9801014 [quant-ph]. • Bronner, Patrick; Strunz, Andreas; Silberhorn, Christine; Meyn, Jan-Peter (2009). "Demonstrating quantum random with single photons". European Journal of Physics 30 (5): 1189–1200. Bibcode 2009EJPh...30.1189B. doi:10.1088/0143-0807/30/5/026.

Further reading The following titles, all by working physicists, attempt to communicate quantum theory to lay people, using a minimum of technical apparatus. • Malin, Shimon (2012). Nature Loves to Hide: Quantum Physics and the Nature of Reality, a Western Perspective (Revised ed.). World Scientific. ISBN 978-981-4324-57-1. • Jim Al-Khalili (2003) Quantum: A Guide for the Perplexed. Weidenfield & Nicholson. • Richard Feynman (1985) QED: The Strange Theory of Light and Matter. Princeton University Press. ISBN 0-691-08388-6 • Ford, Kenneth (2005) The Quantum World. Harvard Univ. Press. Includes elementary particle physics. • Ghirardi, GianCarlo (2004) Sneaking a Look at God's Cards, Gerald Malsbary, trans. Princeton Univ. Press. The most technical of the works cited here. Passages using algebra, trigonometry, and bra-ket notation can be passed over on a first reading. • Tony Hey and Walters, Patrick (2003) The New Quantum Universe. Cambridge Univ. Press. Includes much about the technologies quantum theory has made possible. • Vladimir G. Ivancevic, Tijana T. Ivancevic (2008) Quantum leap: from Dirac and Feynman, across the universe, to human body and mind. World Scientific Publishing Company. Provides an intuitive introduction in non-mathematical terms and an introduction in comparatively basic mathematical terms. • N. David Mermin (1990) “Spooky actions at a distance: mysteries of the QT” in his Boojums all the way through. Cambridge Univ. Press: 110–176. The author is a rare physicist who tries to communicate to philosophers and humanists. • Roland Omnes (1999) Understanding Quantum Mechanics. Princeton Univ. Press. • Victor Stenger (2000) Timeless Reality: Symmetry, Simplicity, and Multiple Universes. Buffalo NY: Prometheus Books. Chpts. 5–8. • Martinus Veltman (2003) Facts and Mysteries in Elementary Particle Physics. World Scientific Publishing Company. • Brian Cox and Jeff Forshaw (2011) The Quantum Universe. Allen Lane. • A website with good introduction to Quantum mechanics can be found here. (http://www.chem1.com/acad/ webtext/atoms/atpt-4.html)

24

Basic Concepts of Quantum Mechanics

External links • Takada, Kenjiro, Emeritus professor at Kyushu University, " Microscopic World – Introduction to Quantum Mechanics. (http://www2.kutl.kyushu-u.ac.jp/seminar/MicroWorld1_E/MicroWorld_1_E.html)" • Quantum Theory. (http://www.encyclopedia.com/doc/1E1-quantumt.html) • Quantum Mechanics. (http://www.aip.org/history/heisenberg/p07.htm) • The spooky quantum (http://www.imamu.edu.sa/Scientific_selections/abstracts/Physics/THE SPOOKY QUANTUM.pdf) • Everything you wanted to know about the quantum world. (http://www.newscientist.com/channel/ fundamentals/quantum-world) From the New Scientist. • This Quantum World. (http://thisquantumworld.com/ht/index.php) • The Quantum Exchange (http://www.compadre.org/quantum) (tutorials and open source learning software). • Theoretical Physics wiki (http://theoreticalphysics.wetpaint.com) • " Uncertainty Principle, (http://www.thebigview.com/spacetime/index.html)" a recording of Werner Heisenberg's voice. • Single and double slit interference (http://class.phys.psu.edu/251Labs/10_Interference_&_Diffraction/ Single_and_Double-Slit_Interference.pdf) • Time-Evolution of a Wavepacket in a Square Well (http://demonstrations.wolfram.com/ TimeEvolutionOfAWavepacketInASquareWell/) An animated demonstration of a wave packet dispersion over time. • Experiments with single photons (http://www.didaktik.physik.uni-erlangen.de/quantumlab/english/) An introduction into quantum physics with interactive experiments • Hitachi video recording of double-slit experiment done with electrons. You can see the interference pattern build up over time. (http://www.youtube.com/watch?v=oxknfn97vFE)

25

Introduction to Quantum Mechanics

26

Introduction to Quantum Mechanics Quantum mechanics is the body of scientific principles that explains the behavior of matter and its interactions with energy on the scale of atoms and subatomic particles and how these phenomena could be related to everyday life (see: Schrodinger's cat). Classical physics explains matter and energy at the macroscopic level of the scale familiar to human experience, including the behavior of astronomical bodies. It remains the key to measurement for much of modern science and technology; but at the end of the 19th Century observers discovered phenomena in both the large (macro) and the small (micro) worlds that classical physics could not explain.[1] Coming to terms with these limitations led to the development of quantum mechanics, a major revolution in physics. This article describes how physicists discovered the limitations of classical physics and developed the main concepts of the quantum theory that replaced them in the early decades of the 20th century.[2] These concepts are described in roughly the order they were first discovered; for a more complete history of the subject, see History of quantum mechanics. Some aspects of quantum mechanics can seem counter-intuitive or even paradoxical, because they describe behavior quite different than that seen at larger length scales, where classical physics is an excellent approximation. In the words of Richard Feynman, quantum mechanics deals with "nature as She is — absurd."[3] Many types of energy, such as photons (discrete units of light), behave in some respects like particles and in other respects like waves. Radiators of photons (such as neon lights) have emission spectra that are discontinuous, in that only certain frequencies of light are present. Quantum mechanics predicts the energies, the colours, and the spectral intensities of all forms of electromagnetic radiation. Quantum mechanics ordains that the more closely one pins down one measure (such as the position of a particle), the less precise another measurement pertaining to the same particle (such as its momentum) must become. Put another way, measuring position first and then measuring momentum does not have the same outcome as measuring momentum first and then measuring position; the act of measuring the first property necessarily introduces additional energy into the micro-system being studied, thereby perturbing that system.

From above and from left to right:Max Planck, Albert Einstein,Niels Bohr, Louis de Broglie,Max Born, Paul Dirac,Werner Heisenberg, Wolfgang Pauli,Erwin Schrödinger, Richard Feynman.

Even more disconcerting, pairs of particles can be created as "entangled twins." As is described in more detail in the article on Quantum entanglement, entangled particles seem to exhibit what Einstein called "spooky action at a distance," matches between states that classical physics would insist must be random even when distance and the speed of light ensure that no physical causation could account for these correlations.[4]

Introduction to Quantum Mechanics

27

The first quantum theory: Max Planck and black body radiation

Hot metalwork from a blacksmith. The yellow-orange glow is the visible part of the thermal radiation emitted due to the high temperature. Everything else in the picture is glowing with thermal radiation as well, but less brightly and at longer wavelengths than the human eye can detect. A far-infrared camera can observe this radiation.

Thermal radiation is electromagnetic radiation emitted from the surface of an object due to the object's temperature. If an object is heated sufficiently, it starts to emit light at the red end of the spectrum — it is red hot. Heating it further causes the colour to change from red to yellow to blue to white, as light at shorter wavelengths (higher frequencies) begins to be emitted. It turns out that a perfect emitter is also a perfect absorber. When it is cold, such an object looks perfectly black, because it absorbs all the light that falls on it and emits none. Consequently, an ideal thermal emitter is known as a black body, and the radiation it emits is called black body radiation.

In the late 19th century, thermal radiation had been fairly well-characterized experimentally. How the wavelength at which the radiation is strongest changes with temperature is given by Wien's displacement law, and the overall power emitted per unit area is given by the Stefan–Boltzmann law. However, classical physics was unable to explain the relationship between temperatures and predominant frequencies of radiation. In fact, at short wavelengths, classical physics predicted that energy will be emitted by a hot body at an infinite rate. This result, which is clearly wrong, is known as the ultraviolet catastrophe. Physicists were searching for a single theory that explained why they got the experimental results that they did. The first model that was able to explain the full spectrum of thermal radiation was put forward by Max Planck in 1900.[5] He modeled the thermal radiation as being in equilibrium, using a set of harmonic oscillators. To reproduce the experimental results he had to assume that each oscillator produced an integer number of units of energy at its single characteristic frequency, rather than being able to emit any arbitrary amount of energy. In other words, the energy of each oscillator was "quantized."[6] The quantum of energy for each oscillator, according to Planck, was proportional to the frequency of the oscillator; the constant of proportionality is now known as the Planck constant. The Planck constant, usually written as h, has the value 663 × 10−34 J s, and so the energy E of an oscillator of frequency f is given by

Correct values (green) contrasted against the classical values (Rayleigh-Jeans law, red and Wien approximation, blue).

[7]

Planck's law was the first quantum theory in physics, and Planck won the Nobel Prize in 1918 "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta."[8] At the time, however, Planck's view was that quantization was purely a mathematical trick, rather than (as we now know) a fundamental change in our understanding of the world.[9]

Introduction to Quantum Mechanics

28

Photons: the quantisation of light In 1905, Albert Einstein took an extra step. He suggested that quantisation was not just a mathematical trick: the energy in a beam of light occurs in individual packets, which are now called photons.[10] The energy of a single photon is given by its frequency multiplied by Planck's constant:

For centuries, scientists had debated between two possible theories of light: was it a wave or did it instead comprise a stream of tiny particles? By the 19th century, the debate was generally considered to have been settled in favour of the wave theory, as it was able to explain observed effects such as refraction, diffraction and polarization. James Clerk Maxwell had shown that electricity, magnetism and light are all manifestations of the same phenomenon: the Einstein's portrait by Harm electromagnetic field. Maxwell's equations, which are the complete set of laws of Kamerlingh Onnes at the University of Leiden in 1920 classical electromagnetism, describe light as waves: a combination of oscillating electric and magnetic fields. Because of the preponderance of evidence in favour of the wave theory, Einstein's ideas were met initially with great scepticism. Eventually, however, the photon model became favoured; one of the most significant pieces of evidence in its favour was its ability to explain several puzzling properties of the photoelectric effect, described in the following section. Nonetheless, the wave analogy remained indispensable for helping to understand other characteristics of light, such as diffraction.

The photoelectric effect In 1887 Heinrich Hertz observed that light can eject electrons from metal.[11] In 1902 Philipp Lenard discovered that the maximum possible energy of an ejected electron is related to the frequency of the light, not to its intensity; if the frequency is too low, no electrons are ejected regardless of the intensity. The lowest frequency of light that causes electrons to be emitted, called the threshold frequency, is different for every metal. This observation is at odds with classical electromagnetism, which predicts that the electron's energy should be proportional to the intensity of the radiation.[12]:24

Light (red arrows, left) is shone upon a metal. If the light is of sufficient frequency (i.e. sufficient energy), electrons are ejected (blue arrows, right).

Einstein explained the effect by postulating that a beam of light is a stream of particles (photons), and that if the beam is of frequency f then each photon has an energy equal to hf.[11] An electron is likely to be struck only by a single photon, which imparts at most an energy hf to the electron.[11] Therefore, the intensity of the beam has no effect;[13] only its frequency determines the maximum energy that can be imparted to the electron.[11] To explain the threshold effect, Einstein argued that it takes a certain amount of energy, called the work function, denoted by φ, to remove an electron from the metal.[11] This amount of energy is different for each metal. If the energy of the photon is less than the work function then it does not carry sufficient energy to remove the electron from the metal. The threshold frequency, f0, is the frequency of a photon whose energy is equal to the work function: If f is greater than f0, the energy hf is enough to remove an electron. The ejected electron has a kinetic energy EK which is, at most, equal to the photon's energy minus the energy needed to dislodge the electron from the metal:

Introduction to Quantum Mechanics Einstein's description of light as being composed of particles extended Planck's notion of quantised energy: a single photon of a given frequency f delivers an invariant amount of energy hf. In other words, individual photons can deliver more or less energy, but only depending on their frequencies. However, although the photon is a particle it was still being described as having the wave-like property of frequency. Once again, the particle account of light was being "compromised".[14][15] The relationship between the frequency of electromagnetic radiation and the energy of each individual photon is why ultraviolet light can cause sunburn, but visible or infrared light cannot. A photon of ultraviolet light will deliver a high amount of energy—enough to contribute to cellular damage such as occurs in a sunburn. A photon of infrared light will deliver a lower amount of energy—only enough to warm one's skin. So an infrared lamp can warm a large surface, perhaps large enough to keep people comfortable in a cold room, but it cannot give anyone a sunburn. If each individual photon had identical energy, it would not be correct to talk of a "high energy" photon. Light of high frequency could carry more energy only because of flooding a surface with more photons arriving per second. Light of low frequency could carry more energy only for the same reason. If it were true that all photons carry the same energy, then if you doubled the rate of photon delivery, you would double the number of energy units arriving each second. Einstein rejected that wave-dependent classical approach in favour of a particle-based analysis where the energy of the particle must be absolute and varies with frequency in discrete steps (i.e. is quantised). All photons of the same frequency have identical energy, and all photons of different frequencies have proportionally different energies. In nature, single photons are rarely encountered. The sun emits photons continuously at all electromagnetic frequencies, so they appear to propagate as a continuous wave, not as discrete units. The emission sources available to Hertz and Lennard in the 19th century shared that characteristic. A sun that radiates red light, or a piece of iron in a forge that glows red, may both be said to contain a great deal of energy. It might be surmised that adding continuously to the total energy of some radiating body would make it radiate red light, orange light, yellow light, green light, blue light, violet light, and so on in that order. But that is not so for otherwise larger suns and larger pieces of iron in a forge would glow with colours more toward the violet end of the spectrum. To change the color of such a radiating body it is necessary to change its temperature, and increasing its temperature changes the quanta of energy that are available to excite individual atoms to higher levels and permit them to emit photons of higher frequencies. The total energy emitted per unit of time by a sun or by a piece of iron in a forge depends on both the number of photons emitted per unit of time and also on the amount of energy carried by each of the photons involved. In other words, the characteristic frequency of a radiating body is dependent on its temperature. When physicists were looking only at beams of light containing huge numbers of individual and virtually indistinguishable photons it was difficult to understand the importance of the energy levels of individual photons. So when physicists first discovered devices exhibiting the photoelectric effect, the effect that makes the light meters of modern cameras work, they initially expected that a higher intensity of light would produce a higher voltage from the photoelectric device. They discovered that strong beams of light toward the red end of the spectrum might produce no electrical potential at all, and that weak beams of light toward the violet end of the spectrum would produce higher and higher voltages. Einstein's idea that individual units of light may contain different amounts of energy depending on their frequency made it possible to explain the experimental results that hitherto had seemed quite counter-intuitive. Although the energy imparted by photons is invariant at any given frequency, the initial energy-state of the electrons in a photoelectric device prior to absorption of light is not necessarily uniform. Therefore anomalous results may occur in the case of individual electrons. An electron that was already excited above the equilibrium level of the photoelectric device might be ejected when it absorbed uncharacteristically low frequency illumination. Statistically, however, the characteristic behavior of a photoelectric device will reflect the behavior of the vast majority of its electrons, which will be at their equilibrium level. This point is helpful in comprehending the distinction between the study of individual particles in quantum dynamics and the study of massed particles in classical physics.

29

Introduction to Quantum Mechanics

30

The quantisation of matter: the Bohr model of the atom By the dawn of the 20th century, it was known that atoms comprise a diffuse cloud of negatively-charged electrons surrounding a small, dense, positively-charged nucleus. This understanding suggested a model in which the electrons circle around the nucleus like planets orbiting a sun.[16] However, it was also known that the atom in this model would be unstable: according to classical theory orbiting electrons are undergoing centripetal acceleration, and should therefore give off electromagnetic radiation, the loss of energy also causing them to spiral toward the nucleus, colliding with it in a fraction of a second. A second, related, puzzle was the emission spectrum of atoms. When a gas is heated, it gives off light only at discrete frequencies. For example, the visible light given off by hydrogen consists of four different colours, as shown in the picture below. By contrast, white light consists of a continuous emission across the whole range of visible frequencies.

Emission spectrum of hydrogen. When excited, hydrogen gas gives off light in four distinct colours (spectral lines) in the visible spectrum, as well as a number of lines in the infra-red and ultra-violet.

In 1885 the Swiss mathematician Johann Balmer discovered that each wavelength λ (lambda) in the visible spectrum of hydrogen is related to some integer n by the equation

where B is a constant which Balmer determined to be equal to 364.56 nm. Thus Balmer's constant was the basis of a system of discrete, i.e. quantised, integers. In 1888 Johannes Rydberg generalized and greatly increased the explanatory utility of Balmer's formula. He predicted that λ is related to two integers n and m according to what is now known as the Rydberg formula:[17]

where R is the Rydberg constant, equal to 0.0110 nm−1, and n must be greater than m. Rydberg's formula accounts for the four visible wavelengths of hydrogen by setting m = 2 and n = 3, 4, 5, 6. It also predicts additional wavelengths in the emission spectrum: for m = 1 and for n > 1, the emission spectrum should contain certain ultraviolet wavelengths, and for m = 3 and n > 3, it should also contain certain infrared wavelengths. Experimental observation of these wavelengths came two decades later: in 1908 Louis Paschen found some of the predicted infrared wavelengths, and in 1914 Theodore Lyman found some of the predicted ultraviolet wavelengths.[17]

Introduction to Quantum Mechanics

31

Bohr's model In 1913 Niels Bohr proposed a new model of the atom that included quantized electron orbits.[18] In Bohr's model, electrons could inhabit only certain orbits around the atomic nucleus. When an atom emitted (or absorbed) energy, the electron did not move in a continuous trajectory from one orbit around the nucleus to another, as might be expected classically. Instead, the electron would jump instantaneously from one orbit to another, giving off the emitted light in the form of a photon.[19] The possible energies of photons given off by each element were determined by the differences in energy between the orbits, and so the emission spectrum for each element would contain a number of lines.[20] Bohr theorised that the angular momentum, L, of an electron is quantised:

The Bohr model of the atom, showing an electron quantum jumping to ground state n = 1.

where n is an integer and h is the Planck constant. Starting from this assumption, Coulomb's law and the equations of circular motion show that an electron with n units of angular momentum will orbit a proton at a distance r given by , where ke is the Coulomb constant, m is the mass of an electron, and e is the charge on an electron. For simplicity this is written as

where a0, called the Bohr radius, is equal to 0.0529 nm. The Bohr radius is the radius of the smallest allowed orbit. The energy of the electron[21] can also be calculated, and is given by . Thus Bohr's assumption that angular momentum is quantised means that an electron can only inhabit certain orbits around the nucleus, and that it can have only certain energies. A consequence of these constraints is that the electron will not crash into the nucleus: it cannot continuously emit energy, and it cannot come closer to the nucleus than a0 (the Bohr radius). An electron loses energy by jumping instantaneously from its original orbit to a lower orbit; the extra energy is emitted in the form of a photon. Conversely, an electron that absorbs a photon gains energy, hence it jumps to an orbit that is farther from the nucleus. Each photon from glowing atomic hydrogen is due to an electron moving from a higher orbit, with radius rn, to a lower orbit, rm. The energy Eγ of this photon is the difference in the energies En and Em of the electron:

Since Planck's equation shows that the photon's energy is related to its wavelength by Eγ = hc/λ, the wavelengths of light that can be emitted are given by

This equation has the same form as the Rydberg formula, and predicts that the constant R should be given by

Introduction to Quantum Mechanics

32

Therefore the Bohr model of the atom can predict the emission spectrum of hydrogen in terms of fundamental constants.[22] However, it was not able to make accurate predictions for multi-electron atoms, or to explain why some spectral lines are brighter than others.

Wave-particle duality In 1924, Louis de Broglie proposed the idea that just as light has both wave-like and particle-like properties, matter also has wave-like properties.[23] The wavelength, λ , associated with a particle is related to its momentum, p through the Planck constant h :[24][25]

The relationship, called the de Broglie hypothesis, holds for all types of matter. Thus all matter exhibits properties of both particles and waves. Three years later, the wave-like nature of electrons was demonstrated by showing that a beam of electrons could exhibit diffraction, just like a beam of light. At the University of Aberdeen, George Thomson passed a beam of electrons through a thin metal film and observed the predicted diffraction patterns. At Bell Labs, Davisson and Germer guided their beam through a crystalline grid. Similar wave-like phenomena were later shown for atoms and even small molecules. De Broglie was awarded the Nobel Prize for Physics in 1929 for his hypothesis; Thomson and Davisson shared the Nobel Prize for Physics in 1937 for their experimental work. The concept of wave-particle duality says that neither the classical concept of "particle" nor of "wave" can fully describe the behavior of quantum-scale objects, either photons or matter. Indeed, astrophysicist A.S. Eddington proposed in 1927 that "We can scarcely describe such an entity as a wave or as a particle; perhaps as a compromise we had better call it a 'wavicle' ".[26] (This term was later popularised by mathematician Banesh Hoffmann.)[27]:172 Wave-particle duality is an example of the principle of complementarity in quantum physics. An elegant example of wave-particle duality, the double slit experiment, is discussed in the section below. De Broglie's treatment of quantum events served as a jumping off point for Schrödinger when he set about to construct a wave equation to describe quantum theoretical events.

The double-slit experiment

Light from one slit interferes with light from the other, producing an interference pattern (the 3 fringes shown at the right).

In the double-slit experiment as originally performed by Thomas Young and Augustin Fresnel in 1827, a beam of light is directed through two narrow, closely spaced slits, producing an interference pattern of light and dark bands on a screen. If one of the slits is covered up, one might naively expect that the intensity of the fringes due to interference would be halved everywhere. In fact, a much simpler pattern is seen, a simple diffraction pattern. Closing one slit results in a much simpler pattern diametrically opposite the open slit. Exactly the same behaviour can be demonstrated in water waves, and so the double-slit experiment was seen as a demonstration of the wave nature of light.

Introduction to Quantum Mechanics

33

The double-slit experiment has also been performed using electrons, atoms, and even molecules, and the same type of interference pattern is seen. Thus it has been demonstrated that all matter possesses both particle and wave characteristics. Even if the source intensity is turned down so that only one particle (e.g. photon or electron) is passing through the apparatus at a time, the same interference pattern develops over time. The quantum particle acts as a wave when passing through the double slits, but as a particle when it is detected. This is a typical feature of quantum complementarity: a quantum particle will act as a wave when we do an experiment to measure its wave-like properties, and like a particle when we do an experiment to measure its particle-like properties. Where on the detector screen any individual particle shows up will be the result of an entirely random process.

The diffraction pattern produced when light is shone through one slit (top) and the interference pattern produced by two slits (bottom). The interference pattern from two slits is much more complex, demonstrating the wave-like propagation of light.

Application to the Bohr model De Broglie expanded the Bohr model of the atom by showing that an electron in orbit around a nucleus could be thought of as having wave-like properties. In particular, an electron will be observed only in situations that permit a standing wave around a nucleus. An example of a standing wave is a violin string, which is fixed at both ends and can be made to vibrate. The waves created by a stringed instrument appear to oscillate in place, moving from crest to trough in an up-and-down motion. The wavelength of a standing wave is related to the length of the vibrating object and the boundary conditions. For example, because the violin string is fixed at both ends, it can carry standing waves of wavelengths 2l/n, where l is the length and n is a positive integer. De Broglie suggested that the allowed electron orbits were those for which the circumference of the orbit would be an integer number of wavelengths.

Development of modern quantum mechanics In 1925, building on de Broglie's hypothesis, Erwin Schrödinger developed the equation that describes the behaviour of a quantum mechanical wave. The equation, called the Schrödinger equation after its creator, is central to quantum mechanics, defines the permitted stationary states of a quantum system, and describes how the quantum state of a physical system changes in time.[28] In the paper that introduced Schrödinger's cat, he says that the psi-function featured in his equation provides the "means for predicting probability of measurement results," and that it therefore provides "future expectation[s] , somewhat as laid down in a catalog."[29] Schrödinger was able to calculate the energy levels of hydrogen by treating a hydrogen atom's electron as a classical wave, moving in a well of electrical potential created by the proton. This calculation accurately reproduced the energy levels of the Bohr model.

Erwin Schrödinger, about 1933, age 46

At a somewhat earlier time, Werner Heisenberg was trying to find an explanation for the intensities of the different lines in the hydrogen emission spectrum. By means of a series of mathematical analogies, Heisenberg wrote out the quantum mechanical analogue for the classical computation of intensities. Shortly afterwards, Heisenberg's colleague Max Born realised that Heisenberg's method of calculating the

Introduction to Quantum Mechanics

34

probabilities for transitions between the different energy levels could best be expressed by using the mathematical concept of matrices.[30] In May 1926, Schrödinger proved that Heisenberg's matrix mechanics and his own wave mechanics made the same predictions about the properties and behaviour of the electron; mathematically, the two theories were identical. Yet the two men disagreed on the interpretation of their mutual theory. For instance, Heisenberg saw no problem in the theoretical prediction of instantaneous transitions of electrons between orbits in an atom, but Schrödinger hoped that a theory based on continuous wave-like properties could avoid what he called (in the words of Wilhelm Wien[31]) "this nonsense about quantum jumps."

Copenhagen interpretation Bohr, Heisenberg and others tried to explain what these experimental results and mathematical models really mean. Their description, known as the Copenhagen interpretation of quantum mechanics, aimed to describe the nature of reality that was being probed by the measurements and described by the mathematical formulations of quantum mechanics. The main principles of the Copenhagen interpretation are: 1. A system is completely described by a wave function,

. (Heisenberg)

2. How changes over time is given by the Schrödinger equation. 3. The description of nature is essentially probabilistic. The probability of an event — for example, where on the screen a particle will show up in the two slit experiment — is related to the square of the amplitude of its wave function. (Born rule, due to Max Born, which gives a physical meaning to the wavefunction in the Copenhagen interpretation: the probability amplitude) 4. It is not possible to know the values of all of the properties of the system at the same time; those properties that are not known with precision must be described by probabilities. (Heisenberg's uncertainty principle) 5. Matter, like energy, exhibits a wave-particle duality. An experiment can demonstrate the particle-like properties of matter, or its wave-like properties; but not both at the same time. (Complementarity principle due to Bohr) 6. Measuring devices are essentially classical devices, and measure classical properties such as position and momentum. 7. The quantum mechanical description of large systems should closely approximate the classical description. (Correspondence principle of Bohr and Heisenberg) Various consequences of these principles are discussed in more detail in the following subsections.

Introduction to Quantum Mechanics

35

Uncertainty principle Suppose that we want to measure the position and speed of an object — for example a car going through a radar speed trap. Naively, we assume that the car has a definite position and speed at a particular moment in time, and how accurately we can measure these values depends on the quality of our measuring equipment — if we improve the precision of our measuring equipment, we will get a result that is closer to the true value. In particular, we would assume that how precisely we measure the speed of the car does not affect the measurement of its position, and vice versa. In 1927, Heisenberg proved that these assumptions are not correct.[33] Quantum mechanics shows that certain pairs of physical properties, like position and speed, cannot both be known to arbitrary precision: the more precisely one property is known, the less precisely the other can be known. This statement is known as the uncertainty principle. The uncertainty principle isn't a statement about the accuracy of our measuring equipment, but about the nature of the system itself — our naive assumption that the car had a definite position and speed was incorrect. On a scale of cars and people, these uncertainties are too small to notice, but when dealing with atoms and electrons they become critical.[34]

Werner Heisenberg at the age of 26. Heisenberg won the Nobel Prize in Physics in 1932 for the work that he [32] did at around this time.

Heisenberg gave, as an illustration, the measurement of the position and momentum of an electron using a photon of light. In measuring the electron's position, the higher the frequency of the photon the more accurate is the measurement of the position of the impact, but the greater is the disturbance of the electron, which absorbs a random amount of energy, rendering the measurement obtained of its momentum increasingly uncertain (momentum is velocity multiplied by mass), for one is necessarily measuring its post-impact disturbed momentum, from the collision products, not its original momentum. With a photon of lower frequency the disturbance - hence uncertainty - in the momentum is less, but so is the accuracy of the measurement of the position of the impact.[35] The uncertainty principle shows mathematically that the product of the uncertainty in the position and momentum of a particle (momentum is velocity multiplied by mass) could never be less than a certain value, and that this value is related to Planck's constant.

Wave function collapse Wave function collapse is a forced term for whatever happened when it becomes appropriate to replace the description of an uncertain state of a system by a description of the system in a definite state. Explanations for the nature of the process of becoming certain are controversial. At any time before a photon "shows up" on a detection screen it can only be described by a set of probabilities for where it might show up. When it does show up, for instance in the CCD of an electronic camera, the time and the space where it interacted with the device are known within very tight limits. However, the photon has disappeared, and the wave function has disappeared with it. In its place some physical change in the detection screen has appeared, e.g., an exposed spot in a sheet of photographic film, or a change in electric potential in some cell of a CCD.

Introduction to Quantum Mechanics

Eigenstates and eigenvalues For a more detailed introduction to this subject, see: Introduction to eigenstates Because of the uncertainty principle, statements about both the position and momentum of particles can only assign a probability that the position or momentum will have some numerical value. Therefore it is necessary to formulate clearly the difference between the state of something that is indeterminate, such as an electron in a probability cloud, and the state of something having a definite value. When an object can definitely be "pinned-down" in some respect, it is said to possess an eigenstate.

The Pauli exclusion principle In 1924, Wolfgang Pauli proposed a new quantum degree of freedom (or quantum number), with two possible values, to resolve inconsistencies between observed molecular spectra and the predictions of quantum mechanics. In particular, the spectrum of atomic hydrogen had a doublet, or pair of lines differing by a small amount, where only one line was expected. Pauli formulated his exclusion principle, stating that "There cannot exist an atom in such a quantum state that two electrons within [it] have the same set of quantum numbers."[36] A year later, Uhlenbeck and Goudsmit identified Pauli's new degree of freedom with a property called spin. The idea, originating with Ralph Kronig, was that electrons behave as if they rotate, or "spin", about an axis. Spin would account for the missing magnetic moment, and allow two electrons in the same orbital to occupy distinct quantum states if they "spun" in opposite directions, thus satisfying the exclusion principle. The quantum number represented the sense (positive or negative) of spin.

Application to the hydrogen atom Bohr's model of the atom was essentially two-dimensional — an electron orbiting in a plane around its nuclear "sun." However, the uncertainty principle states that an electron cannot be viewed as having an exact location at any given time. In the modern theory the orbit has been replaced by an atomic orbital, a "cloud" of possible locations. It is often depicted as a three-dimensional region within which there is a 95 percent probability of finding the electron.[37] Schrödinger was able to calculate the energy levels of hydrogen by treating a hydrogen atom's electron as a wave, represented by the "wave function" Ψ, in a electric potential well, V, created by the proton. The solutions to Schrödinger's equation are distributions of probabilities for electron positions and locations. Orbitals have a range of different shapes in three dimensions. The energies of the different orbitals can be calculated, and they accurately reproduce the energy levels of the Bohr model. Within Schrödinger's picture, each electron has four properties: 1. An "orbital" designation, indicating whether the particle wave is one that is closer to the nucleus with less energy or one that is farther from the nucleus with more energy; 2. The "shape" of the orbital, spherical or otherwise; 3. The "inclination" of the orbital, determining the magnetic moment of the orbital around the z-axis. 4. The "spin" of the electron. The collective name for these properties is the quantum state of the electron. The quantum state can be described by giving a number to each of these properties; these are known as the electron's quantum numbers. The quantum state of the electron is described by its wavefunction. The Pauli exclusion principle demands that no two electrons within an atom may have the same values of all four numbers.

36

Introduction to Quantum Mechanics

37

The first property describing the orbital is the principal quantum number, n, which is the same as in Bohr's model. n denotes the energy level of each orbital. The possible values for n are integers: The shapes of the first five atomic orbitals: 1s, 2s, 2px,2py, and 2pz. The colours show the phase of the wavefunction.

The next quantum number, the azimuthal quantum number, denoted l, describes the shape of the orbital. The shape is a consequence of the angular momentum of the orbital. The angular momentum represents the resistance of a spinning object to speeding up or slowing down under the influence of external force. The azimuthal quantum number represents the orbital angular momentum of an electron around its nucleus. The possible values for l are integers from 0 to n − 1:

The shape of each orbital has its own letter as well. The first shape is denoted by the letter s (a mnemonic being "sphere"). The next shape is denoted by the letter p and has the form of a dumbbell. The other orbitals have more complicated shapes (see atomic orbital), and are denoted by the letters d, f, and g. The third quantum number, the magnetic quantum number, describes the magnetic moment of the electron, and is denoted by ml (or simply m). The possible values for ml are integers from −l to l: The magnetic quantum number measures the component of the angular momentum in a particular direction. The choice of direction is arbitrary, conventionally the z-direction is chosen. The fourth quantum number, the spin quantum number (pertaining to the "orientation" of the electron's spin) is denoted ms, with values +1⁄2 or −1⁄2. The chemist Linus Pauling wrote, by way of example: In the case of a helium atom with two electrons in the 1s orbital, the Pauli Exclusion Principle requires that the two electrons differ in the value of one quantum number. Their values of n, l, and ml are the same; moreover, they have the same spin, s = 1⁄2. Accordingly they must differ in the value of ms, which can have the value of +1⁄2 for one electron and −1⁄2 for the other."[36] It is the underlying structure and symmetry of atomic orbitals, and the way that electrons fill them, that determines the organisation of the periodic table and the structure and strength of chemical bonds between atoms.

Introduction to Quantum Mechanics

38

Dirac wave equation In 1928, Paul Dirac extended the Pauli equation, which described spinning electrons, to account for special relativity. The result was a theory that dealt properly with events, such as the speed at which an electron orbits the nucleus, occurring at a substantial fraction of the speed of light. By using the simplest electromagnetic interaction, Dirac was able to predict the value of the magnetic moment associated with the electron's spin, and found the experimentally observed value, which was too large to be that of a spinning charged sphere governed by classical physics. He was able to solve for the spectral lines of the hydrogen atom, and to reproduce from physical first principles Sommerfeld's successful formula for the fine structure of the hydrogen spectrum. Dirac's equations sometimes yielded a negative value for energy, for which he proposed a novel solution: he posited the existence of an antielectron and of a dynamical vacuum. This led to the many-particle quantum field theory.

Paul Dirac (1902 - 1984)

Quantum entanglement The Pauli exclusion principle says that two electrons in one system cannot be in the same state. Nature leaves open the possibility, however, that two electrons can have both states "superimposed" over each of them. Recall that the wave functions that Superposition of two quantum characteristics, and two resolution possibilities. emerge simultaneously from the double slits arrive at the detection screen in a state of superposition. Nothing is certain until the superimposed waveforms "collapse," At that instant an electron shows up somewhere in accordance with the probabilities that are the squares of the amplitudes of the two superimposed waveforms. The situation there is already very abstract. A concrete way of thinking about entangled photons, photons in which two contrary states are superimposed on each of them in the same event, is as follows: Imagine that the superposition of a state that can be mentally labeled as blue and another state that can be mentally labeled as red will then appear (in imagination, of course) as a purple state. Two photons are produced as the result of the same atomic event. Perhaps they are produced by the excitation of a crystal that characteristically absorbs a photon of a certain frequency and emits two photons of half the original frequency. So the two photons come out "purple." If the experimenter now performs some experiment that will determine whether one of the photons is either blue or red, then that experiment changes the photon involved from one having a superposition of "blue" and "red" characteristics to a photon that has only one of those characteristics. The problem that Einstein had with such an imagined situation was that if one of these photons had been kept bouncing between mirrors in a laboratory on earth, and the other one had traveled halfway to the nearest star, when its twin was made to reveal itself as either blue or red, that meant that the distant photon now had to lose its "purple" status too. So whenever it might be investigated after its twin had been measured, it would necessarily show up in the opposite state to whatever its twin had revealed.

Introduction to Quantum Mechanics

39

In trying to show that quantum mechanics was not a complete theory, Einstein started with the theory's prediction that two or more particles that have interacted in the past can appear strongly correlated when their various properties are later measured. He sought to explain this seeming interaction in a classical way, through their common past, and preferably not by some "spooky action at a distance." The argument is worked out in a famous paper, Einstein, Podolsky, and Rosen (1935; abbreviated EPR), setting out what is now called the EPR paradox. Assuming what is now usually called local realism, EPR attempted to show from quantum theory that a particle has both position and momentum simultaneously, while according to the Copenhagen interpretation, only one of those two properties actually exists and only at the moment that it is being measured. EPR concluded that quantum theory is incomplete in that it refuses to consider physical properties which objectively exist in nature. (Einstein, Podolsky, & Rosen 1935 is currently Einstein's most cited publication in physics journals.) In the same year, Erwin Schrödinger used the word "entanglement" and declared: "I would not call that one but rather the characteristic trait of quantum mechanics." [38] The question of whether entanglement is a real condition is still in dispute.[39] The Bell inequalities are the most powerful challenge to Einstein's claims.

Quantum field theory The idea of quantum field theory began in the late 1920s with British physicist Paul Dirac, when he attempted to quantise the electromagnetic field — a procedure for constructing a quantum theory starting from a classical theory. A field in physics is "a region or space in which a given effect (such as magnetism) exists."[40] Other effects that manifest themselves as fields are gravitation and static electricity.[41] In 2008, physicist Richard Hammond wrote that Sometimes we distinguish between quantum mechanics (QM) and quantum field theory (QFT). QM refers to a system in which the number of particles is fixed, and the fields (such as the electromechanical field) are continuous classical entities. QFT . . . goes a step further and allows for the creation and annihilation of particles . . . . He added, however, that quantum mechanics is often used to refer to "the entire notion of quantum view."[42]:108 In 1931, Dirac proposed the existence of particles that later became known as anti-matter.[43] Dirac shared the Nobel Prize in physics for 1933 with Schrödinger, "for the discovery of new productive forms of atomic theory."[44]

Quantum electrodynamics

This sculpture in Bristol, England — a series of clustering cones — presents the idea of small worlds that Paul Dirac studied to reach his discovery of anti-matter.

Quantum electrodynamics (QED) is the name of the quantum theory of the electromagnetic force. Understanding QED begins with understanding electromagnetism. Electromagnetism can be called "electrodynamics" because it is a dynamic interaction between electrical and magnetic forces. Electromagnetism begins with the electric charge. Electric charges are the sources of, and create, electric fields. An electric field is a field which exerts a force on any particles that carry electric charges, at any point in space. This includes the electron, proton, and even quarks, among others. As a force is exerted, electric charges move, a current flows and a magnetic field is produced. The magnetic field, in turn causes electric current (moving electrons). The interacting electric and magnetic field is called an electromagnetic field.

Introduction to Quantum Mechanics The physical description of interacting charged particles, electrical currents, electrical fields, and magnetic fields is called electromagnetism. In 1928 Paul Dirac produced a relativistic quantum theory of electromagnetism. This was the progenitor to modern quantum electrodynamics, in that it had essential ingredients of the modern theory. However, the problem of unsolvable infinities developed in this relativistic quantum theory. Years later, renormalization solved this problem. Initially viewed as a suspect, provisional procedure by some of its originators, renormalization eventually was embraced as an important and self-consistent tool in QED and other fields of physics. Also, in the late 1940s Feynman's diagrams depicted all possible interactions pertaining to a given event. The diagrams showed that the electromagnetic force is the interactions of photons between interacting particles. An example of a prediction of quantum electrodynamics which has been verified experimentally is the Lamb shift. This refers to an effect whereby the quantum nature of the electromagnetic field causes the energy levels in an atom or ion to deviate slightly from what they would otherwise be. As a result, spectral lines may shift or split. In the 1960s physicists realized that QED broke down at extremely high energies. From this inconsistency the Standard Model of particle physics was discovered, which remedied the higher energy breakdown in theory. The Standard Model unifies the electromagnetic and weak interactions into one theory. This is called the electroweak theory.

Interpretations The physical measurements, equations, and predictions pertinent to quantum mechanics are all consistent and hold a very high level of confirmation. However, the question of what these abstract models say about the underlying nature of the real world has received competing answers.

Applications Applications of quantum mechanics include the laser, the transistor, the electron microscope, and magnetic resonance imaging. A special class of quantum mechanical applications is related to macroscopic quantum phenomena such as superfluid helium and superconductors. The study of semiconductors led to the invention of the diode and the transistor, which are indispensable for modern electronics. In even the simple light switch, quantum tunnelling is absolutely vital, as otherwise the electrons in the electric current could not penetrate the potential barrier made up of a layer of oxide. Flash memory chips found in USB drives also use quantum tunnelling, to erase their memory cells.[45]

Notes [1] Quantum Mechanics from [[National Public Radio (http:/ / www. pbs. org/ trasnsistor/ science/ info/ quantum. html)]] [2] Classical physics also does not accurately describe the universe on the largest scales or at speeds close to that of light. An accurate description requires general relativity. [3] Feynman, Richard P. (1988). QED : the strange theory of light and matter (1st Princeton pbk., seventh printing with corrections. ed.). Princeton, N.J.: Princeton University Press. pp. 10. ISBN 978-0691024172. [4] Alan Macdonald, "Spooky action at a distance: The puzzle of entanglement in quantum theory," page 5 of 7, downloaded 13 June 2012 from http:/ / faculty. luther. edu/ ~macdonal/ [5] This result was published (in German) as Planck, Max (1901). "Ueber das Gesetz der Energieverteilung im Normalspectrum" (http:/ / www. physik. uni-augsburg. de/ annalen/ history/ historic-papers/ 1901_309_553-563. pdf). Ann. Phys. 309 (3): 553–63. Bibcode 1901AnP...309..553P. doi:10.1002/andp.19013090310. . English translation: " On the Law of Distribution of Energy in the Normal Spectrum (http:/ / dbhs. wvusd. k12. ca. us/ webdocs/ Chem-History/ Planck-1901/ Planck-1901. html)". [6] The word "quantum" comes from the Latin word for "how much" (as does "quantity"). Something which is "quantized," like the energy of Planck's harmonic oscillators, can only take specific values. For example, in most countries money is effectively quantized, with the "quantum of money" being the lowest-value coin in circulation. "Mechanics" is the branch of science that deals with the action of forces on objects, so "quantum mechanics" is the part of mechanics that deals with objects for which particular properties are quantized.

40

Introduction to Quantum Mechanics [7] Francis Weston Sears (1958). Mechanics, Wave Motion, and Heat (http:/ / books. google. com/ books?hl=en& q="Mechanics,+ Wave+ Motion,+ and+ Heat"+ "where+ n+ =+ 1,"& btnG=Search+ Books). Addison-Wesley. p. 537. . [8] "The Nobel Prize in Physics 1918" (http:/ / nobelprize. org/ nobel_prizes/ physics/ laureates/ 1918/ ). The Nobel Foundation. . Retrieved 2009-08-01. [9] Kragh, Helge (1 December 2000). "Max Planck: the reluctant revolutionary" (http:/ / physicsworld. com/ cws/ article/ print/ 373). PhysicsWorld.com. [10] Einstein, Albert (1905). "Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt" (http:/ / www. zbp. univie. ac. at/ dokumente/ einstein1. pdf). Annalen der Physik 17 (6): 132–148. Bibcode 1905AnP...322..132E. doi:10.1002/andp.19053220607. ., translated into English as On a Heuristic Viewpoint Concerning the Production and Transformation of Light (http:/ / lorentz. phl. jhu. edu/ AnnusMirabilis/ AeReserveArticles/ eins_lq. pdf). The term "photon" was introduced in 1926. [11] Taylor, J. R.; Zafiratos, C. D.; Dubson, M. A. (2004). Modern Physics for Scientists and Engineers. Prentice Hall. pp. 127–9. ISBN 0-13-589789-0. [12] Stephen Hawking, The Universe in a Nutshell, Bantam, 2001. [13] Actually there can be intensity-dependent effects, but at intensities achievable with non-laser sources these effects are unobservable. [14] Dicke and Wittke, Introduction to Quantum Mechanics, p. 12 [15] Einstein's photoelectric effect equation can be derived and explained without requiring the concept of "photons". That is, the electromagnetic radiation can be treated as a classical electromagnetic wave, as long as the electrons in the material are treated by the laws of quantum mechanics. The results are quantitatively correct for thermal light sources (the sun, incandescent lamps, etc) both for the rate of electron emission as well as their angular distribution. For more on this point, see NTRS.NASA.gov (http:/ / ntrs. nasa. gov/ archive/ nasa/ casi. ntrs. nasa. gov/ 19680009569_1968009569. pdf) [16] The classical model of the atom is called the planetary model, or sometimes the Rutherford model after Ernest Rutherford who proposed it in 1911, based on the Geiger-Marsden gold foil experiment which first demonstrated the existence of the nucleus. [17] Taylor, J. R.; Zafiratos, C. D.; Dubson, M. A. (2004). Modern Physics for Scientists and Engineers. Prentice Hall. pp. 147–8. ISBN 0-13-589789-0. [18] McEvoy, J. P.; Zarate, O. (2004). Introducing Quantum Theory. Totem Books. pp. 70–89, especially p. 89. ISBN 1-84046-577-8. [19] World Book Encyclopedia, page 6, 2007. [20] Dicke and Wittke, Introduction to Quantum Mechanics, p. 10f. [21] In this case, the energy of the electron is the sum of its kinetic and potential energies. The electron has kinetic energy by virtue of its actual motion around the nucleus, and potential energy because of its electromagnetic interaction with the nucleus. [22] The model can be easily modified to account of the emission spectrum of any system consisting of a nucleus and a single electron (that is, ions such as He+ or O7+ which contain only one electron). [23] J. P. McEvoy and Oscar Zarate (2004). Introducing Quantum Theory. Totem Books. p. 110f. ISBN 1-84046-577-8. [24] Aezel, Amir D., Entanglrment, p. 51f. (Penguin, 2003) ISBN 0-452-28457 [25] J. P. McEvoy and Oscar Zarate (2004). Introducing Quantum Theory. Totem Books. p. 114. ISBN 1-84046-577-8. [26] A.S. Eddington, The Nature of the Physical World, the course of Gifford Lectures that Eddington delivered in the University of Edinburgh in January to March 1927, Kessinger Publishing, 2005, p. 201. (http:/ / books. google. com/ books?id=PGOTKcxSqMUC& pg=PA201& lpg=PA201& dq=We+ can+ scarcely+ describe+ such+ an+ entity+ as+ a+ wave+ or+ as+ a+ particle;+ perhaps+ as+ a+ compromise+ we+ had+ better+ call+ it+ a+ `wavicle& source=bl& ots=K0IfGzaXli& sig=zgrQiBJbHRLuUzVBT-yy8jZhC1Y& hl=en& ei=i8g1SpOHC4PgtgOu_4jVDg& sa=X& oi=book_result& ct=result& resnum=1) [27] Banesh Hoffman, The Strange Story of the Quantum, Dover, 1959 [28] "Schrodinger Equation (Physics)," Encyclopædia Britannica (http:/ / www. britannica. com/ EBchecked/ topic/ 528298/ Schrodinger-equation) [29] Erwin Schrödinger, "The Present Situation in Quantum Mechanics," p. 9. "This translation was originally published in Proceedings of the American Philosophical Society, 124, 323-38. [And then appeared as Section I.11 of Part I of Quantum Theory and Measurement (J.A. Wheeler and W.H. Zurek, eds., Princeton university Press, New Jersey 1983). This paper can be downloaded from http:/ / www. tu-harburg. de/ rzt/ rzt/ it/ QM/ cat. html. " [30] For a somewhat more sophisticated look at how Heisenberg transitioned from the old quantum theory and classical physics to the new quantum mechanics, see Heisenberg's entryway to matrix mechanics. [31] W. Moore, Schrödinger: Life and Thought, Cambridge University Press (1989), p. 222. [32] Heisenberg's Nobel Prize citation (http:/ / nobelprize. org/ nobel_prizes/ physics/ laureates/ 1932/ ) [33] Heisenberg first published his work on the uncertainty principle in the leading German physics journal Zeitschrift für Physik: Heisenberg, W. (1927). "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik". Z. Phys. 43 (3–4): 172–198. Bibcode 1927ZPhy...43..172H. doi:10.1007/BF01397280. [34] Nobel Prize in Physics presentation speech, 1932 (http:/ / nobelprize. org/ nobel_prizes/ physics/ laureates/ 1932/ press. html) [35] "Uncertainty principle," Encyclopædia Britannica (http:/ / www. britannica. com/ EBchecked/ topic/ 614029/ uncertainty-principle) [36] Linus Pauling, The Nature of the Chemical Bond, p. 47 [37] "Orbital (chemistry and physics)," Encyclopædia Britannica (http:/ / www. britannica. com/ EBchecked/ topic/ 431159/ orbital) [38] E. Schrödinger, Proceedings of the Cambridge Philosophical Society, 31 (1935), p. 555says: "When two systems, of which we know the states by their respective representation, enter into a temporary physical interaction due to known forces between them and when after a time

41

Introduction to Quantum Mechanics of mutual influence the systems separate again, then they can no longer be described as before, viz., by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics." [39] "Quantum Nonlocality and the Possibility of Superluminal Effects", John G. Cramer, npl.washington.edu (http:/ / www. npl. washington. edu/ npl/ int_rep/ qm_nl. html) [40] "Mechanics," Merriam-Webster Online Dictionary (http:/ / www. merriam-webster. com/ dictionary/ field) [41] "Field," Encyclopædia Britannica (http:/ / www. britannica. com/ EBchecked/ topic/ 206162/ field) [42] Richard Hammond, The Unknown Universe, New Page Books, 2008. ISBN 978-1-60163-003-2 [43] The Physical World website (http:/ / www. physicalworld. org/ restless_universe/ html/ ru_dira. html) [44] "The Nobel Prize in Physics 1933" (http:/ / nobelprize. org/ nobel_prizes/ physics/ laureates/ 1933/ ). The Nobel Foundation. . Retrieved 2007-11-24. [45] Durrani, Z. A. K.; Ahmed, H. (2008). Vijay Kumar. ed. Nanosilicon. Elsevier. p. 345. ISBN 978-0-08-044528-1.

References • Bernstein, Jeremy (2005). "Max Born and the quantum theory". American Journal of Physics 73 (11). • Beller, Mara (2001). Quantum Dialogue: The Making of a Revolution. University of Chicago Press. • Bohr, Niels (1958). Atomic Physics and Human Knowledge. John Wiley & Sons. ASIN B00005VGVF. ISBN 0-486-47928-5. OCLC 530611. • de Broglie, Louis (1953). The Revolution in Physics. Noonday Press. LCCN 53010401. • Einstein, Albert (1934). Essays in Science. Philosophical Library. ISBN 0-486-47011-3. LCCN 55003947. • Feigl, Herbert; Brodbeck, May (1953). Readings in the Philosophy of Science. Appleton-Century-Crofts. ISBN 0-390-30488-3. LCCN 53006438. • Feynman, Richard P. (1949). "Space-Time Approach to Quantum Electrodynamics" (http://www.physics. princeton.edu/~mcdonald/examples/QED/feynman_pr_76_769_49.pdf). Physical Review 76 (6): 769–789. Bibcode 1949PhRv...76..769F. doi:10.1103/PhysRev.76.769. • Fowler, Michael (1999). The Bohr Atom. University of Virginia. • Heisenberg, Werner (1958). Physics and Philosophy. Harper and Brothers. ISBN 0-06-130549-9. LCCN 99010404. • Lakshmibala, S. (2004). "Heisenberg, Matrix Mechanics and the Uncertainty Principle". Resonance, Journal of Science Education 9 (8). • Liboff, Richard L. (1992). Introductory Quantum Mechanics (2nd ed.). • Lindsay, Robert Bruce; Margenau, Henry (1957). Foundations of Physics. Dover. ISBN 0-918024-18-8. LCCN 57014416. • McEvoy, J. P.; Zarate, Oscar. Introducing Quantum Theory. ISBN 1-874166-37-4. • Müller-Kirsten, H. J. W. (2012). Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral (2nd ed.). World Scientific. ISBN 978-981-4397-74-2. • Nave, Carl Rod (2005). "Quantum Physics" (http://hyperphysics.phy-astr.gsu.edu/hbase/quacon. html#quacon). HyperPhysics. Georgia State University. • Peat, F. David (2002). From Certainty to Uncertainty: The Story of Science and Ideas in the Twenty-First Century. Joseph Henry Press. • Reichenbach, Hans (1944). Philosophic Foundations of Quantum Mechanics. University of California Press. ISBN 0-486-40459-5. LCCN a44004471. • Schlipp, Paul Arthur (1949). Albert Einstein: Philosopher-Scientist. Tudor Publishing Company. LCCN 50005340. • Scientific American Reader, 1953. • Sears, Francis Weston (1949). Optics (3rd ed.). Addison-Wesley. ISBN 0-19-504601-3. LCCN 51001018. • Shimony, A. (1983). "(title not given in citation)". Foundations of Quantum Mechanics in the Light of New Technology (S. Kamefuchi et al., eds.). Tokyo: Japan Physical Society. pp. 225.; cited in: Popescu, Sandu; Daniel Rohrlich (1996). "Action and Passion at a Distance: An Essay in Honor of Professor Abner Shimony". arXiv:quant-ph/9605004 [quant-ph].

42

Introduction to Quantum Mechanics • Tavel, Morton; Tavel, Judith (illustrations) (2002). Contemporary physics and the limits of knowledge (http:// books.google.com/?id=SELS0HbIhjYC&pg=PA200&dq=Wave+function+collapse). Rutgers University Press. ISBN 978-0-8135-3077-2. • Van Vleck, J. H.,1928, "The Correspondence Principle in the Statistical Interpretation of Quantum Mechanics," Proc. Nat. Acad. Sci. 14: 179. • Wheeler, John Archibald; Feynman, Richard P. (1949). "Classical Electrodynamics in Terms of Direct Interparticle Action". Reviews of Modern Physics 21 (3): 425–433. Bibcode 1949RvMP...21..425W. doi:10.1103/RevModPhys.21.425. • Wieman, Carl; Perkins, Katherine (2005). "Transforming Physics Education". Physics Today. • Westmoreland; Benjamin Schumacher (1998). "Quantum Entanglement and the Nonexistence of Superluminal Signals". arXiv:quant-ph/9801014 [quant-ph]. • Bronner, Patrick; Strunz, Andreas; Silberhorn, Christine; Meyn, Jan-Peter (2009). "Demonstrating quantum random with single photons". European Journal of Physics 30 (5): 1189–1200. Bibcode 2009EJPh...30.1189B. doi:10.1088/0143-0807/30/5/026.

Further reading The following titles, all by working physicists, attempt to communicate quantum theory to lay people, using a minimum of technical apparatus. • Malin, Shimon (2012). Nature Loves to Hide: Quantum Physics and the Nature of Reality, a Western Perspective (Revised ed.). World Scientific. ISBN 978-981-4324-57-1. • Jim Al-Khalili (2003) Quantum: A Guide for the Perplexed. Weidenfield & Nicholson. • Richard Feynman (1985) QED: The Strange Theory of Light and Matter. Princeton University Press. ISBN 0-691-08388-6 • Ford, Kenneth (2005) The Quantum World. Harvard Univ. Press. Includes elementary particle physics. • Ghirardi, GianCarlo (2004) Sneaking a Look at God's Cards, Gerald Malsbary, trans. Princeton Univ. Press. The most technical of the works cited here. Passages using algebra, trigonometry, and bra-ket notation can be passed over on a first reading. • Tony Hey and Walters, Patrick (2003) The New Quantum Universe. Cambridge Univ. Press. Includes much about the technologies quantum theory has made possible. • Vladimir G. Ivancevic, Tijana T. Ivancevic (2008) Quantum leap: from Dirac and Feynman, across the universe, to human body and mind. World Scientific Publishing Company. Provides an intuitive introduction in non-mathematical terms and an introduction in comparatively basic mathematical terms. • N. David Mermin (1990) “Spooky actions at a distance: mysteries of the QT” in his Boojums all the way through. Cambridge Univ. Press: 110–176. The author is a rare physicist who tries to communicate to philosophers and humanists. • Roland Omnes (1999) Understanding Quantum Mechanics. Princeton Univ. Press. • Victor Stenger (2000) Timeless Reality: Symmetry, Simplicity, and Multiple Universes. Buffalo NY: Prometheus Books. Chpts. 5–8. • Martinus Veltman (2003) Facts and Mysteries in Elementary Particle Physics. World Scientific Publishing Company. • Brian Cox and Jeff Forshaw (2011) The Quantum Universe. Allen Lane. • A website with good introduction to Quantum mechanics can be found here. (http://www.chem1.com/acad/ webtext/atoms/atpt-4.html)

43

Introduction to Quantum Mechanics

External links • Takada, Kenjiro, Emeritus professor at Kyushu University, " Microscopic World – Introduction to Quantum Mechanics. (http://www2.kutl.kyushu-u.ac.jp/seminar/MicroWorld1_E/MicroWorld_1_E.html)" • Quantum Theory. (http://www.encyclopedia.com/doc/1E1-quantumt.html) • Quantum Mechanics. (http://www.aip.org/history/heisenberg/p07.htm) • The spooky quantum (http://www.imamu.edu.sa/Scientific_selections/abstracts/Physics/THE SPOOKY QUANTUM.pdf) • Everything you wanted to know about the quantum world. (http://www.newscientist.com/channel/ fundamentals/quantum-world) From the New Scientist. • This Quantum World. (http://thisquantumworld.com/ht/index.php) • The Quantum Exchange (http://www.compadre.org/quantum) (tutorials and open source learning software). • Theoretical Physics wiki (http://theoreticalphysics.wetpaint.com) • " Uncertainty Principle, (http://www.thebigview.com/spacetime/index.html)" a recording of Werner Heisenberg's voice. • Single and double slit interference (http://class.phys.psu.edu/251Labs/10_Interference_&_Diffraction/ Single_and_Double-Slit_Interference.pdf) • Time-Evolution of a Wavepacket in a Square Well (http://demonstrations.wolfram.com/ TimeEvolutionOfAWavepacketInASquareWell/) An animated demonstration of a wave packet dispersion over time. • Experiments with single photons (http://www.didaktik.physik.uni-erlangen.de/quantumlab/english/) An introduction into quantum physics with interactive experiments • Hitachi video recording of double-slit experiment done with electrons. You can see the interference pattern build up over time. (http://www.youtube.com/watch?v=oxknfn97vFE)

44

45

2. The Quantum Theories Old Quantum Theory The old quantum theory is a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was a collection of heuristic prescriptions which are now understood to be the first quantum corrections to classical mechanics.[1] The Bohr model was the focus of study, and Arnold Sommerfeld[2] made a crucial contribution by quantizing the z-component of the angular momentum, which in the old quantum era was inappropriately called space quantization (Richtungsquantelung). This allowed the orbits of the electron to be ellipses instead of circles, and introduced the concept of quantum degeneracy. The theory would have correctly explained the Zeeman effect, except for the issue of electron spin. The main tool was Bohr–Sommerfeld quantization, a procedure for selecting out certain discrete set of states of a classical integrable motion as allowed states. These are like the allowed orbits of the Bohr model of the atom; the system can only be in one of these states and not in any states in between. The theory did not extend to chaotic motions, because it required a full multiply periodic trajectory of the classical system for all time in order to pose the quantum conditions.

Basic principles The basic idea of the old quantum theory is that the motion in an atomic system is quantized, or discrete. The system obeys classical mechanics except that not every motion is allowed, only those motions which obey the old quantum condition:

where the

are the momenta of the system and the

are the corresponding coordinates. The quantum numbers

are integers and the integral is taken over one period of the motion at constant energy (as described by the Hamiltonian). The integral is an area in phase space, which is a quantity called the action and is quantized in units of Planck's constant. For this reason, Planck's constant was often called the quantum of action. In order for the old quantum condition to make sense, the classical motion must be separable, meaning that there are separate coordinates in terms of which the motion is periodic. The periods of the different motions do not have to be the same, they can even be incommensurate, but there must be a set of coordinates where the motion decomposes in a multi-periodic way. The motivation for the old quantum condition was the correspondence principle, complemented by the physical observation that the quantities which are quantized must be adiabatic invariants. Given Planck's quantization rule for the harmonic oscillator, either condition determines the correct classical quantity to quantize in a general system up to an additive constant.

Old Quantum Theory

46

Examples Harmonic oscillator The simplest system in the old quantum theory is the harmonic oscillator, whose Hamiltonian is:

The level sets of H are the orbits, and the quantum condition is that the area enclosed by an orbit in phase space is an integer. It follows that the energy is quantized according to the Planck rule:

a result which was known well before, and used to formulate the old quantum condition. Please note that this result differs by

from the results found with the help of quantum mechanics. This constant is neglected in the

derivation of the old quantum theory, and its value can not be determined using it. The thermal properties of a quantized oscillator may be found by averaging the energy in each of the discrete states assuming that they are occupied with a Boltzmann weight:

kT is Boltzmann constant times the absolute temperature, which is the temperature as measured in more natural units of energy. The quantity is more fundamental in thermodynamics than the temperature, because it is the thermodynamic potential associated to the energy. From this expression, it is easy to see that for large values of

, for very low temperatures, the average energy U in

the Harmonic oscillator approaches zero very quickly, exponentially fast. The reason is that kT is the typical energy of random motion at temperature T, and when this is smaller than , there is not enough energy to give the oscillator even one quantum of energy. So the oscillator stays in its ground state, storing next to no energy at all. This means that at very cold temperatures, the change in energy with respect to beta, or equivalently the change in energy with respect to temperature, is also exponentially small. The change in energy with respect to temperature is the specific heat, so the specific heat is exponentially small at low temperatures, going to zero like

At small values of

, at high temperatures, the average energy U is equal to

. This reproduces the

equipartition theorem of classical thermodynamics: every harmonic oscillator at temperature T has energy kT on average. This means that the specific heat of an oscillator is constant in classical mechanics and equal to k. For a collection of atoms connected by springs, a reasonable model of a solid, the total specific heat is equal to the total number of oscillators times k. There are overall three oscillators for each atom, corresponding to the three possible directions of independent oscillations in three dimensions. So the specific heat of a classical solid is always 3k per atom, or in chemistry units, 3R per mole of atoms. Monatomic solids at room temperatures have approximately the same specific heat of 3k per atom, but at low temperatures they don't. The specific heat is smaller at colder temperatures, and it goes to zero at absolute zero. This is true for all material systems, and this observation is called the third law of thermodynamics. Classical mechanics cannot explain the third law, because in classical mechanics the specific heat is independent of the temperature. This contradiction between classical mechanics and the specific heat of cold materials was noted by James Clerk Maxwell in the 19th century, and remained a deep puzzle for those who advocated an atomic theory of matter. Einstein resolved this problem in 1906 by proposing that atomic motion is quantized. This was the first application of quantum theory to mechanical systems. A short while later, Debye gave a quantitative theory of solid specific heats in terms of quantized oscillators with various frequencies (see Einstein solid and Debye model).

Old Quantum Theory

47

One-dimensional potential One-dimensional problems are easy to solve. At any energy E, the value of the momentum p is found from the conservation equation:

which is integrated over all values of q between the classical turning points, the places where the momentum vanishes. The integral is easiest for a particle in a box of length L, where the quantum condition is:

which gives the allowed momenta:

and the energy levels

Another easy case to solve with the old quantum theory is a linear potential on the positive halfline, the constant confining force F binding a particle to an impenetrable wall. This case is much more difficult in the full quantum mechanical treatment, and unlike the other examples, the semiclassical answer here is not exact but approximate, becoming more accurate at large quantum numbers.

so that the quantum condition is:

Which determines the energy levels.

Rotator Another simple system is the rotator. A rotator consists of a mass M at the end of a massless rigid rod of length R and in two dimensions has the Lagrangian:

which determines that the angular momentum J conjugate to

, the polar angle,

. The old quantum

condition requires that J multiplied by the period of

is an integer multiple of Planck's constant:

the angular momentum to be an integer multiple of was enough to determine the energy levels.

. In the Bohr model, this restriction imposed on circular orbits

In three dimensions, a rigid rotator can be described by two angles — and , where is the inclination relative to an arbitrarily chosen z-axis while is the rotator angle in the projection to the x–y plane. The kinetic energy is again the only contribution to the Lagrangian:

And the conjugate momenta are

and

. The equation of motion for

is trivial:

is a constant:

Old Quantum Theory which is the z-component of the angular momentum. The quantum condition demands that the integral of the constant as varies from 0 to is an integer multiple of h:

And m is called the magnetic quantum number, because the z component of the angular momentum is the magnetic moment of the rotator along the z direction in the case where the particle at the end of the rotator is charged. Since the three-dimensional rotator is rotating about an axis, the total angular momentum should be restricted in the same way as the two-dimensional rotator. The two quantum conditions restrict the total angular momentum and the z-component of the angular momentum to be the integers l,m. This condition is reproduced in modern quantum mechanics, but in the era of the old quantum theory it led to a paradox: how can the orientation of the angular momentum relative to the arbitrarily chosen z-axis be quantized? This seems to pick out a direction in space. This phenomenon, the quantization of angular momentum about an axis, was given the name space quantization, because it seemed incompatible with rotational invariance. In modern quantum mechanics, the angular momentum is quantized the same way, but the discrete states of definite angular momentum in any one orientation are quantum superpositions of the states in other orientations, so that the process of quantization does not pick out a preferred axis. For this reason, the name "space quantization" fell out of favor, and the same phenomenon is now called the quantization of angular momentum.

Hydrogen atom The angular part of the Hydrogen atom is just the rotator, and gives the quantum numbers l and m. The only remaining variable is the radial coordinate, which executes a periodic one-dimensional potential motion, which can be solved. For a fixed value of the total angular momentum L, the Hamiltonian for a classical Kepler problem is (the unit of mass and unit of energy redefined to absorb two constants):

Fixing the energy to be (a negative) constant and solving for the radial momentum p, the quantum condition integral is:

which is elementary, and gives a new quantum number k which determines the energy in combination with l. The energy is:

and it only depends on the sum of k and l, which is the principal quantum number n. Since k is positive, the allowed values of l for any given n are no bigger than n. The energies reproduce those in the Bohr model, except with the correct quantum mechanical multiplicities, with some ambiguity at the extreme values. The semiclassical hydrogen atom is called the Sommerfeld model, and its orbits are ellipses of various sizes at discrete inclinations. The Sommerfeld model predicted that the magnetic moment of an atom measured along an axis will only take on discrete values, a result which seems to contradict rotational invariance but which was confirmed by the Stern–Gerlach experiment. Bohr–Sommerfeld theory is a part of the development of quantum mechanics and describes the possibility of atomic energy levels being split by a magnetic field.

48

Old Quantum Theory

49

Relativistic orbit Arnold Sommerfeld derived the relativistic solution of atomic energy levels.[3] We will start this derivation with the relativistic equation for energy in the electric potential

After substitution

we get

For momentum

,

and their ratio

the equation of motion is (see Binet

equation)

with solution

The angular shift of periapsis per revolution is given by

With the quantum conditions

and

we will obtain energies

where

is the fine-structure constant. This solution is same as the solution of the Dirac equation.[4]

De Broglie waves In 1905, Einstein noted that the entropy of the quantized electromagnetic field oscillators in a box is, for short wavelength, equal to the entropy of a gas of point particles in the same box. The number of point particles is equal to the number of quanta. Einstein concluded that the quanta could be treated as if they were localizable objects (see[5] page 139/140), particles of light, and named them photons. Einstein's theoretical argument was based on thermodynamics, on counting the number of states, and so was not completely convincing. Nevertheless, he concluded that light had attributes of both waves and particles, more precisely that an electromagnetic standing wave with frequency with the quantized energy:

Old Quantum Theory

50

should be thought of as consisting of n photons each with an energy photons were related to the wave.

. Einstein could not describe how the

The photons have momentum as well as energy, and the momentum had to be

where

is the wavenumber of

the electromagnetic wave. This is required by relativity, because the momentum and energy form a four-vector, as do the frequency and wave-number. In 1924, as a PhD candidate, Louis de Broglie proposed a new interpretation of the quantum condition. He suggested that all matter, electrons as well as photons, are described by waves obeying the relations.

or, expressed in terms of wavelength

instead,

He then noted that the quantum condition:

counts the change in phase for the wave as it travels along the classical orbit, and requires that it be an integer multiple of . Expressed in wavelengths, the number of wavelengths along a classical orbit must be an integer. This is the condition for constructive interference, and it explained the reason for quantized orbits—the matter waves make standing waves only at discrete frequencies, at discrete energies. For example, for a particle confined in a box, a standing wave must fit an integer number of wavelengths between twice the distance between the walls. The condition becomes:

so that the quantized momenta are:

reproducing the old quantum energy levels. This development was given a more mathematical form by Einstein, who noted that the phase function for the waves: in a mechanical system should be identified with the solution to the Hamilton–Jacobi equation, an equation which even Hamilton considered to be the short-wavelength limit of wave mechanics. These ideas led to the development of the Schrödinger equation.

Kramers transition matrix The old quantum theory was formulated only for special mechanical systems which could be separated into action angle variables which were periodic. It did not deal with the emission and absorption of radiation. Nevertheless, Hendrik Kramers was able to find heuristics for describing how emission and absorption should be calculated. Kramers suggested that the orbits of a quantum system should be Fourier analyzed, decomposed into harmonics at multiples of the orbit frequency:

The index n describes the quantum numbers of the orbit, it would be n–l–m in the Sommerfeld model. The frequency is the angular frequency of the orbit while k is an index for the Fourier mode. Bohr had suggested that the k-th harmonic of the classical motion correspond to the transition from level n to level n−k. Kramers proposed that the transition between states were analogous to classical emission of radiation, which happens at frequencies at multiples of the orbit frequencies. The rate of emission of radiation is proportional to , as it would be in classical mechanics. The description was approximate, since the Fourier components did

Old Quantum Theory not have frequencies that exactly match the energy spacings between levels. This idea led to the development of matrix mechanics.

Limitations of the old quantum theory The old quantum theory had some limitations:[6] • The old quantum theory provides no means to calculate the intensities of the spectral lines; • It fails when applied to atoms with more than one electron. That is, it cannot be applied to many-body systems; • It fails to explain the anomalous Zeeman effect (that is, where the spin of the electron cannot be neglected). It was later understood that the old quantum theory is in fact the semi-classical approximation (also called quasi-classical) to the Schrödinger equation[7] which has limited applicability.

History The old quantum theory was sparked by the work of Max Planck on the emission and absorption of light, and began in earnest after the work of Albert Einstein on the specific heats of solids. Einstein, followed by Debye, applied quantum principles to the motion of atoms, explaining the specific heat anomaly. In 1913, Niels Bohr identified the correspondence principle and used it to formulate a model of the hydrogen atom which explained the line spectrum. In the next few years Arnold Sommerfeld extended the quantum rule to arbitrary integrable systems making use of the principle of adiabatic invariance of the quantum numbers introduced by Lorentz and Einstein. Sommerfeld's model was much closer to the modern quantum mechanical picture than Bohr's. Throughout the 1910s and well into the 1920s, many problems were attacked using the old quantum theory with mixed results. Molecular rotation and vibration spectra were understood and the electron's spin was discovered, leading to the confusion of half-integer quantum numbers. Max Planck introduced the zero point energy and Arnold Sommerfeld semiclassically quantized the relativistic hydrogen atom. Hendrik Kramers explained the Stark effect. Bose and Einstein gave the correct quantum statistics for photons. Kramers gave a prescription for calculating transition probabilities between quantum states in terms of Fourier components of the motion, ideas which were extended in collaboration with Werner Heisenberg to a semiclassical matrix-like description of atomic transition probabilities. Heisenberg went on to reformulate all of quantum theory in terms of a version of these transition matrices, creating matrix mechanics. In 1924, Louis de Broglie introduced the wave theory of matter, which was extended to a semiclassical equation for matter waves by Albert Einstein a short time later. In 1926 Erwin Schrödinger found a completely quantum mechanical wave-equation, which reproduced all the successes of the old quantum theory without ambiguities and inconsistencies. Schrödinger's wave mechanics developed separately from matrix mechanics until Schrödinger and others proved that the two methods predicted the same experimental consequences. Paul Dirac later proved in 1926 that both methods can be obtained from a more general method called transformation theory. Matrix mechanics and wave mechanics put an end to the era of the old-quantum theory.

51

Old Quantum Theory

References [1] [2] [3] [4]

ter Haar, D. (1967). The Old Quantum Theory. Pergamon Press. pp. 206. ISBN 0-08-012101-2. Sommerfeld, Arnold (1919). Atombau und Spektrallinien'. Braunschweig: Friedrich Vieweg und Sohn. ISBN 3-87144-484-7. Arnold Sommerfeld (1924). Atombau und Spektrallinien. Braunschweig. ISBN 3-87144-484-7. Ya I Granovski (2004). "Sommerfeld formula and Dirac's theory" (http:/ / www. iop. org/ EJ/ article/ 1063-7869/ 47/ 5/ L06/ PHU_47_5_L06. pdf). Physics ± Uspekhi 47 (5): 523–524. . [5] Einstein, Albert (1905). "Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt" (http:/ / www. physik. uni-augsburg. de/ annalen/ history/ einstein-papers/ 1905_17_132-148. pdf). Annalen der Physik 17 (6): 132–148. Bibcode 1905AnP...322..132E. doi:10.1002/andp.19053220607. . Retrieved 2008-02-18. [6] Chaddha, G.S. (2006). Quantum Mechanics (http:/ / books. google. com/ books?id=Bzj2JcPeAHAC). New Dehli: New Age international. p. 8-9. ISBN 81-224-1465-6. ., Extract of page 9 (http:/ / books. google. com/ books?id=Bzj2JcPeAHAC& pg=PA9)}} [7] L.D. Landau, E.M. Lifshitz (1977). Quantum Mechanics: Non-Relativistic Theory. Vol. 3 (3rd ed.). Pergamon Press. ISBN 978-0-08-020940-1.

Further reading • Thewlis, J., ed. (1962). Encyclopaedic Dictionary of Physics.

Quantum Mechanics after 1925 Quantum mechanics (QM – also known as quantum physics, or quantum theory) is a branch of physics dealing with physical phenomena at microscopic scales, where the action is on the order of the Planck constant. Quantum mechanics departs from classical mechanics primarily at the quantum realm of atomic and subatomic length scales. Quantum mechanics provides a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. In advanced topics of quantum mechanics, some of these behaviors are macroscopic and only emerge at extreme (i.e., very low or very high) energies or temperatures. The name quantum mechanics derives from the observation that some physical quantities can change only in discrete amounts (Latin quanta), and not in a continuous (cf. analog) way. For example, the angular momentum of an electron bound to an atom or molecule is quantized.[1] In the context of quantum mechanics, the wave–particle duality of energy and matter and the uncertainty principle provide a unified view of the behavior of photons, electrons, and other atomic-scale objects. The mathematical formulations of quantum mechanics are abstract. A mathematical function called the wavefunction provides information about the probability amplitude of position, momentum, and other physical properties of a particle. Mathematical manipulations of the wavefunction usually involve the bra-ket notation, which requires an understanding of complex numbers and linear functionals. The wavefunction treats the object as a quantum harmonic oscillator, and the mathematics is akin to that describing acoustic resonance. Many of the results of quantum mechanics are not easily visualized in terms of classical mechanics—for instance, the ground state in a quantum mechanical model is a non-zero energy state that is the lowest permitted energy state of a system, as opposed to a more "traditional" system that is thought of as simply being at rest, with zero kinetic energy. Instead of a traditional static, unchanging zero state, quantum mechanics allows for far more dynamic, chaotic possibilities, according to John Wheeler. The earliest versions of quantum mechanics were formulated in the first decade of the 20th century. At around the same time, the atomic theory and the corpuscular theory of light (as updated by Einstein) first came to be widely accepted as scientific fact; these latter theories can be viewed as quantum theories of matter and electromagnetic radiation, respectively. Early quantum theory was significantly reformulated in the mid-1920s by Werner Heisenberg, Max Born and Pascual Jordan, who created matrix mechanics; Louis de Broglie and Erwin Schrodinger (Wave Mechanics); and Wolfgang Pauli and Satyendra Nath Bose (statistics of subatomic particles). And the Copenhagen interpretation of Niels Bohr became widely accepted. By 1930, quantum mechanics had been further

52

Quantum Mechanics after 1925 unified and formalized by the work of David Hilbert, Paul Dirac and John von Neumann,[2] with a greater emphasis placed on measurement in quantum mechanics, the statistical nature of our knowledge of reality, and philosophical speculation about the role of the observer. Quantum mechanics has since branched out into almost every aspect of 20th century physics and other disciplines, such as quantum chemistry, quantum electronics, quantum optics, and quantum information science. Much 19th century physics has been re-evaluated as the "classical limit" of quantum mechanics, and its more advanced developments in terms of quantum field theory, string theory, and speculative quantum gravity theories.

History The first study of quantum mechanics goes back to the 17th and 18th centuries when scientists such as Robert Hooke, Christian Huygens and Leonhard Euler proposed a wave theory of light based on experimental observations.[3] In 1803, Thomas Young, an English polymath, performed the famous double-slit experiment that he later described in a paper entitled "On the nature of light and colours". This experiment played a major role in the general acceptance of the wave theory of light. In 1838 with the discovery of cathode rays by Michael Faraday, these studies were followed by the 1859 statement of the black-body radiation problem by Gustav Kirchhoff, the 1877 suggestion by Ludwig Boltzmann that the energy states of a physical system can be discrete, and the 1900 quantum hypothesis of Max Planck.[4] Planck's hypothesis that energy is radiated and absorbed in discrete "quanta" (or "energy elements") precisely matched the observed patterns of black-body radiation. In 1896, Wilhelm Wien empirically determined a distribution law of black-body radiation, later named Wien's law after him. However, it was only valid at high frequencies, and underestimated the radiancy at low frequencies. Later Max Planck corrected the theory and proposed what is now called Planck's law, which led to the development of quantum mechanics. The first studies of quantum phenomena in nature were by the work of several scientists as Arthur Compton, C.V. Raman, Pieter Zeeman (each one of them has a quantum effect named after their works), Albert Einstein and Robert A. Millikan (both studied the Photoelectric effect). At the same time Niels Bohr developed his theory of the atomic structure later confirmed with experiments by Henry Moseley. In 1913, Peter Debye extended Niels Bohr's theory of atomic structure, introducing elliptical orbits, a concept also introduced by Arnold Sommerfeld[5] . This phase is known as Old quantum theory. According to Planck, each energy element E is proportional to its frequency ν:

53

Quantum Mechanics after 1925

54

where h is Planck's constant. Planck (cautiously) insisted that this was simply an aspect of the processes of absorption and emission of radiation and had nothing to do with the physical reality of the radiation itself.[6] However, in 1905 Albert Einstein interpreted Planck's quantum hypothesis realistically and used it to explain the photoelectric effect, in which shining light on certain materials can eject electrons from the material.

Planck is considered the father of the Quantum Theory

The foundations of quantum mechanics were established during the first half of the 20th century by Max Planck, Niels Bohr, Werner Heisenberg, Louis de Broglie, Arthur Compton, Albert Einstein, Erwin Schrödinger, Max Born, John von Neumann, Paul Dirac, Enrico Fermi, Wolfgang Pauli, Max Von Laue, Freeman Dyson, David Hilbert, Wilhelm Wien, Satyendra Nath Bose, Arnold Sommerfeld and others. In the mid-1920s, developments in quantum mechanics led to its becoming the standard formulation for atomic physics. In the summer of 1925, Bohr and Heisenberg published results that The 1927 Solvay Conference in Brussels. closed the "Old Quantum Theory". Out of deference to their particle-like behavior in certain processes and measurements, light quanta came to be called photons (1926). From Einstein's simple postulation was born a flurry of debating, theorizing, and testing. Thus the entire field of quantum physics emerged, leading to its wider acceptance at the Fifth Solvay Conference in 1927. The other exemplar that led to quantum mechanics was the study of electromagnetic waves, such as visible light. When it was found in 1900 by Max Planck that the energy of waves could be described as consisting of small packets or "quanta", Albert Einstein further developed this idea to show that an electromagnetic wave such as light could be described as a particle (later called the photon) with a discrete quantum of energy that was dependent on its frequency.[7] This led to a theory of unity between subatomic particles and electromagnetic waves, called wave–particle duality, in which particles and waves were neither one nor the other, but had certain properties of both. While quantum mechanics traditionally described the world of the very small, it is also needed to explain certain recently investigated macroscopic systems such as superconductors and superfluids. The word quantum derives from the Latin, meaning "how great" or "how much".[8] In quantum mechanics, it refers to a discrete unit that quantum theory assigns to certain physical quantities, such as the energy of an atom at rest (see Figure 1). The discovery that particles are discrete packets of energy with wave-like properties led to the branch of physics dealing with atomic and sub-atomic systems which is today called quantum mechanics. It is the underlying mathematical framework of many fields of physics and chemistry, including condensed matter physics, solid-state

Quantum Mechanics after 1925 physics, atomic physics, molecular physics, computational physics, computational chemistry, quantum chemistry, particle physics, nuclear chemistry, and nuclear physics.[9] Some fundamental aspects of the theory are still actively studied.[10] Quantum mechanics is essential to understanding the behavior of systems at atomic length scales and smaller. For example, if classical mechanics truly governed the workings of an atom, electrons would rapidly travel toward, and collide with, the nucleus, making stable atoms impossible. However, in the natural world electrons normally remain in an uncertain, non-deterministic, "smeared", probabilistic wave–particle wavefunction orbital path around (or through) the nucleus, defying classical electromagnetism.[11] Quantum mechanics was initially developed to provide a better explanation of the atom, especially the differences in the spectra of light emitted by different isotopes of the same element. The quantum theory of the atom was developed as an explanation for the electron remaining in its orbit, which could not be explained by Newton's laws of motion and Maxwell's laws of (classical) electromagnetism. Broadly speaking, quantum mechanics incorporates four classes of phenomena for which classical physics cannot account: • The quantization of certain physical properties • Wave–particle duality • The Uncertainty principle • Quantum entanglement.

Mathematical formulations In the mathematically rigorous formulation of quantum mechanics developed by Paul Dirac[12] David Hilbert,[13] and John von Neumann,[14] the possible states of a quantum mechanical system are represented by unit vectors (called "state vectors"). Formally, these reside in a complex separable Hilbert space - variously called the "state space" or the "associated Hilbert space" of the system - that is well defined up to a complex number of norm 1 (the phase factor). In other words, the possible states are points in the projective space of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system - for example, the state space for position and momentum states is the space of square-integrable functions, while the state space for the spin of a single proton is just the product of two complex planes. Each observable is represented by a maximally Hermitian (precisely: by a self-adjoint) linear operator acting on the state space. Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. If the operator's spectrum is discrete, the observable can only attain those discrete eigenvalues. In the formalism of quantum mechanics, the state of a system at a given time is described by a complex wave function, also referred to as state vector in a complex vector space.[15] This abstract mathematical object allows for the calculation of probabilities of outcomes of concrete experiments. For example, it allows one to compute the probability of finding an electron in a particular region around the nucleus at a particular time. Contrary to classical mechanics, one can never make simultaneous predictions of conjugate variables, such as position and momentum, with accuracy. For instance, electrons may be considered (to a certain probability) to be located somewhere within a given region of space, but with their exact positions unknown. Contours of constant probability, often referred to as "clouds", may be drawn around the nucleus of an atom to conceptualize where the electron might be located with the most probability. Heisenberg's uncertainty principle quantifies the inability to precisely locate the particle given its conjugate momentum.[16] According to one interpretation, as the result of a measurement the wave function containing the probability information for a system collapses from a given initial state to a particular eigenstate. The possible results of a measurement are the eigenvalues of the operator representing the observable — which explains the choice of Hermitian operators, for which all the eigenvalues are real. The probability distribution of an observable in a given

55

Quantum Mechanics after 1925 state can be found by computing the spectral decomposition of the corresponding operator. Heisenberg's uncertainty principle is represented by the statement that the operators corresponding to certain observables do not commute. The probabilistic nature of quantum mechanics thus stems from the act of measurement. This is one of the most difficult aspects of quantum systems to understand. It was the central topic in the famous Bohr-Einstein debates, in which the two scientists attempted to clarify these fundamental principles by way of thought experiments. In the decades after the formulation of quantum mechanics, the question of what constitutes a "measurement" has been extensively studied. Newer interpretations of quantum mechanics have been formulated that do away with the concept of "wavefunction collapse" (see, for example, the relative state interpretation). The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wavefunctions become entangled, so that the original quantum system ceases to exist as an independent entity. For details, see the article on measurement in quantum mechanics.[17] Generally, quantum mechanics does not assign definite values. Instead, it makes a prediction using a probability distribution; that is, it describes the probability of obtaining the possible outcomes from measuring an observable. Often these results are skewed by many causes, such as dense probability clouds. Probability clouds are approximate, but better than the Bohr model, whereby electron location is given by a probability function, the wave function eigenvalue, such that the probability is the squared modulus of the complex amplitude, or quantum state nuclear attraction.[18][19] Naturally, these probabilities will depend on the quantum state at the "instant" of the measurement. Hence, uncertainty is involved in the value. There are, however, certain states that are associated with a definite value of a particular observable. These are known as eigenstates of the observable ("eigen" can be translated from German as meaning "inherent" or "characteristic").[20] In the everyday world, it is natural and intuitive to think of everything (every observable) as being in an eigenstate. Everything appears to have a definite position, a definite momentum, a definite energy, and a definite time of occurrence. However, quantum mechanics does not pinpoint the exact values of a particle's position and momentum (since they are conjugate pairs) or its energy and time (since they too are conjugate pairs); rather, it only provides a range of probabilities of where that particle might be given its momentum and momentum probability. Therefore, it is helpful to use different words to describe states having uncertain values and states having definite values (eigenstates). Usually, a system will not be in an eigenstate of the observable (particle) we are interested in. However, if one measures the observable, the wavefunction will instantaneously be an eigenstate (or "generalized" eigenstate) of that observable. This process is known as wavefunction collapse, a controversial and much-debated process[21] that involves expanding the system under study to include the measurement device. If one knows the corresponding wave function at the instant before the measurement, one will be able to compute the probability of the wavefunction collapsing into each of the possible eigenstates. For example, the free particle in the previous example will usually have a wavefunction that is a wave packet centered around some mean position x0 (neither an eigenstate of position nor of momentum). When one measures the position of the particle, it is impossible to predict with certainty the result.[17] It is probable, but not certain, that it will be near x0, where the amplitude of the wave function is large. After the measurement is performed, having obtained some result x, the wave function collapses into a position eigenstate centered at x.[22] The time evolution of a quantum state is described by the Schrödinger equation, in which the Hamiltonian (the operator corresponding to the total energy of the system) generates the time evolution. The time evolution of wave functions is deterministic in the sense that - given a wavefunction at an initial time - it makes a definite prediction of what the wavefunction will be at any later time.[23] During a measurement, on the other hand, the change of the initial wavefunction into another, later wavefunction is not deterministic, it is unpredictable (i.e. random). A time-evolution simulation can be seen here.[24][25] Wave functions change as time progresses. The Schrödinger equation describes how wavefunctions change in time, playing a role similar to Newton's second law in classical mechanics. The Schrödinger equation, applied to the aforementioned example of the free particle, predicts that the center of a wave packet will move through space at a

56

Quantum Mechanics after 1925 constant velocity (like a classical particle with no forces acting on it). However, the wave packet will also spread out as time progresses, which means that the position becomes more uncertain with time. This also has the effect of turning a position eigenstate (which can be thought of as an infinitely sharp wave packet) into a broadened wave packet that no longer represents a (definite, certain) position eigenstate.[26] Some wave functions produce probability distributions that are constant, or independent of time - such as when in a stationary state of constant energy, time vanishes in the absolute square of the wave function. Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus, whereas in quantum mechanics it is described by a static, spherically symmetric wavefunction surrounding the nucleus (Fig. 1) (note, however, that only the lowest angular momentum states, labeled s, are spherically symmetric).[27] The Schrödinger equation acts on the entire probability amplitude, not merely its absolute Fig. 1: Probability densities corresponding to the wavefunctions of an value. Whereas the absolute value of the electron in a hydrogen atom possessing definite energy levels (increasing from the top of the image to the bottom: n = 1, 2, 3, ...) and angular probability amplitude encodes information about momenta (increasing across from left to right: s, p, d, ...). Brighter areas probabilities, its phase encodes information about correspond to higher probability density in a position measurement. the interference between quantum states. This Wavefunctions like these are directly comparable to Chladni's figures of gives rise to the "wave-like" behavior of quantum acoustic modes of vibration in classical physics, and are indeed modes of oscillation as well, possessing a sharp energy and, thus, a definite states. As it turns out, analytic solutions of the frequency. The angular momentum and energy are quantized, and take only Schrödinger equation are only available for a very discrete values like those shown (as is the case for resonant frequencies in small number of relatively simple model acoustics) Hamiltonians, of which the quantum harmonic oscillator, the particle in a box, the hydrogen molecular ion, and the hydrogen atom are the most important representatives. Even the helium atom - which contains just one more electron than does the hydrogen atom - has defied all attempts at a fully analytic treatment. There exist several techniques for generating approximate solutions, however. In the important method known as perturbation theory, one uses the analytic result for a simple quantum mechanical model to generate a result for a more complicated model that is related to the simpler model by (for one example) the addition of a weak potential energy. Another method is the "semi-classical equation of motion" approach, which applies to systems for which quantum mechanics produces only weak (small) deviations from classical behavior. These deviations can then be computed based on the classical motion. This approach is particularly important in the field of quantum chaos.

57

Quantum Mechanics after 1925

Mathematically equivalent formulations of quantum mechanics There are numerous mathematically equivalent formulations of quantum mechanics. One of the oldest and most commonly used formulations is the "transformation theory" proposed by the late Cambridge theoretical physicist Paul Dirac, which unifies and generalizes the two earliest formulations of quantum mechanics - matrix mechanics (invented by Werner Heisenberg)[28] and wave mechanics (invented by Erwin Schrödinger).[29] Especially since Werner Heisenberg was awarded the Nobel Prize in Physics in 1932 for the creation of quantum mechanics, the role of Max Born in the development of QM has become somewhat confused and overlooked. A 2005 biography of Born details his role as the creator of the matrix formulation of quantum mechanics. This fact was recognized in a paper that Heisenberg himself published in 1940 honoring Max Planck.[30] and In the matrix formulation, the instantaneous state of a quantum system encodes the probabilities of its measurable properties, or "observables". Examples of observables include energy, position, momentum, and angular momentum. Observables can be either continuous (e.g., the position of a particle) or discrete (e.g., the energy of an electron bound to a hydrogen atom).[31] An alternative formulation of quantum mechanics is Feynman's path integral formulation, in which a quantum-mechanical amplitude is considered as a sum over all possible histories between the initial and final states. This is the quantum-mechanical counterpart of the action principle in classical mechanics.

Interactions with other scientific theories The rules of quantum mechanics are fundamental. They assert that the state space of a system is a Hilbert space, and that observables of that system are Hermitian operators acting on that space—although they do not tell us which Hilbert space or which operators. These can be chosen appropriately in order to obtain a quantitative description of a quantum system. An important guide for making these choices is the correspondence principle, which states that the predictions of quantum mechanics reduce to those of classical mechanics when a system moves to higher energies or—equivalently—larger quantum numbers, i.e. whereas a single particle exhibits a degree of randomness, in systems incorporating millions of particles averaging takes over and, at the high energy limit, the statistical probability of random behaviour approaches zero. In other words, classical mechanics is simply a quantum mechanics of large systems. This "high energy" limit is known as the classical or correspondence limit. One can even start from an established classical model of a particular system, then attempt to guess the underlying quantum model that would give rise to the classical model in the correspondence limit. When quantum mechanics was originally formulated, it was applied to models whose correspondence limit was non-relativistic classical mechanics. For instance, the well-known model of the quantum harmonic oscillator uses an explicitly non-relativistic expression for the kinetic energy of the oscillator, and is thus a quantum version of the classical harmonic oscillator. Early attempts to merge quantum mechanics with special relativity involved the replacement of the Schrödinger equation with a covariant equation such as the Klein-Gordon equation or the Dirac equation. While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of quantum field theory, which applies quantization to a field (rather than a fixed set of particles). The first complete quantum field theory, quantum electrodynamics, provides a fully quantum description of the electromagnetic interaction. The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one that has been employed since the inception of quantum mechanics, is to treat charged particles as quantum mechanical objects being acted on by a classical electromagnetic field. For example, the elementary quantum model of the hydrogen atom describes the electric field of the hydrogen atom using a classical Coulomb potential. This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by charged particles.

58

Quantum Mechanics after 1925 Quantum field theories for the strong nuclear force and the weak nuclear force have also been developed. The quantum field theory of the strong nuclear force is called quantum chromodynamics, and describes the interactions of subnuclear particles such as quarks and gluons. The weak nuclear force and the electromagnetic force were unified, in their quantized forms, into a single quantum field theory (known as electroweak theory), by the physicists Abdus Salam, Sheldon Glashow and Steven Weinberg. These three men shared the Nobel Prize in Physics in 1979 for this work.[32] It has proven difficult to construct quantum models of gravity, the remaining fundamental force. Semi-classical approximations are workable, and have led to predictions such as Hawking radiation. However, the formulation of a complete theory of quantum gravity is hindered by apparent incompatibilities between general relativity (the most accurate theory of gravity currently known) and some of the fundamental assumptions of quantum theory. The resolution of these incompatibilities is an area of active research, and theories such as string theory are among the possible candidates for a future theory of quantum gravity. Classical mechanics has also been extended into the complex domain, with complex classical mechanics exhibiting behaviors similar to quantum mechanics.[33]

Quantum mechanics and classical physics Predictions of quantum mechanics have been verified experimentally to an extremely high degree of accuracy. According to the correspondence principle between classical and quantum mechanics, all objects obey the laws of quantum mechanics, and classical mechanics is just an approximation for large systems of objects (or a statistical quantum mechanics of a large collection of particles). The laws of classical mechanics thus follow from the laws of quantum mechanics as a statistical average at the limit of large systems or large quantum numbers.[34] However, chaotic systems do not have good quantum numbers, and quantum chaos studies the relationship between classical and quantum descriptions in these systems. Quantum coherence is an essential difference between classical and quantum theories, and is illustrated by the Einstein-Podolsky-Rosen paradox. Quantum interference involves adding together probability amplitudes, whereas classical "waves" infer that there is an adding together of intensities. For microscopic bodies, the extension of the system is much smaller than the coherence length, which gives rise to long-range entanglement and other nonlocal phenomena that are characteristic of quantum systems.[35] Quantum coherence is not typically evident at macroscopic scales - although an exception to this rule can occur at extremely low temperatures (i.e. approaching absolute zero), when quantum behavior can manifest itself on more macroscopic scales (see macroscopic quantum phenomena, Bose-Einstein condensate, and Quantum machine). This is in accordance with the following observations: • Many macroscopic properties of a classical system are a direct consequence of the quantum behavior of its parts. For example, the stability of bulk matter (which consists of atoms and molecules which would quickly collapse under electric forces alone), the rigidity of solids, and the mechanical, thermal, chemical, optical and magnetic properties of matter are all results of the interaction of electric charges under the rules of quantum mechanics.[36] • While the seemingly "exotic" behavior of matter posited by quantum mechanics and relativity theory become more apparent when dealing with particles of extremely small size or velocities approaching the speed of light, the laws of classical Newtonian physics remain accurate in predicting the behavior of the vast majority of "large" objects (on the order of the size of large molecules or bigger) at velocities much smaller than the velocity of light.[37]

59

Quantum Mechanics after 1925

Relativity and quantum mechanics Main articles: Quantum gravity and Theory of everything Even with the defining postulates of both Einstein's theory of general relativity and quantum theory being indisputably supported by rigorous and repeated empirical evidence and while they do not directly contradict each other theoretically (at least with regard to their primary claims), they have proven extremely difficult to incorporate into one consistent, cohesive model.[38] Einstein himself is well known for rejecting some of the claims of quantum mechanics. While clearly contributing to the field, he did not accept many of the more "philosophical consequences and interpretations" of quantum mechanics, such as the lack of deterministic causality. He is famously quoted as saying, in response to this aspect, "My God does not play with dice". He also had difficulty with the assertion that a single subatomic particle can occupy numerous areas of space at one time. However, he was also the first to notice some of the apparently exotic consequences of entanglement, and used them to formulate the Einstein-Podolsky-Rosen paradox in the hope of showing that quantum mechanics had unacceptable implications. This was 1935, but in 1964 it was shown by John Bell (see Bell inequality) that - although Einstein was correct in identifying seemingly paradoxical implications of quantum mechanical nonlocality - these implications could be experimentally tested. Alain Aspect's initial experiments in 1982, and many subsequent experiments since, have definitively verified quantum entanglement. According to the paper of J. Bell and the Copenhagen interpretation - the common interpretation of quantum mechanics by physicists since 1927 - and contrary to Einstein's ideas, quantum mechanics was not, at the same time: • a "realistic" theory and • a local theory. The Einstein-Podolsky-Rosen paradox shows in any case that there exist experiments by which one can measure the state of one particle and instantaneously change the state of its entangled partner - although the two particles can be an arbitrary distance apart. However, this effect does not violate causality, since no transfer of information happens. Quantum entanglement forms the basis of quantum cryptography, which is used in high-security commercial applications in banking and government. Gravity is negligible in many areas of particle physics, so that unification between general relativity and quantum mechanics is not an urgent issue in those particular applications. However, the lack of a correct theory of quantum gravity is an important issue in cosmology and the search by physicists for an elegant "Theory of Everything" (TOE). Consequently, resolving the inconsistencies between both theories has been a major goal of 20th and 21st century physics. Many prominent physicists, including Stephen Hawking, have labored for many years in the attempt to discover a theory underlying everything. This TOE would combine not only the different models of subatomic physics, but also derive the four fundamental forces of nature - the strong force, electromagnetism, the weak force, and gravity - from a single force or phenomenon. While Stephen Hawking was initially a believer in the Theory of Everything, after considering Gödel's Incompleteness Theorem, he has concluded that one is not obtainable, and has stated so publicly in his lecture "Gödel and the End of Physics" (2002).[39]

Attempts at a unified field theory The quest to unify the fundamental forces through quantum mechanics is still ongoing. Quantum electrodynamics (or "quantum electromagnetism"), which is currently (in the perturbative regime at least) the most accurately tested physical theory,[40] (blog) has been successfully merged with the weak nuclear force into the electroweak force and work is currently being done to merge the electroweak and strong force into the electrostrong force. Current predictions state that at around 1014 GeV the three aforementioned forces are fused into a single unified field,[41] Beyond this "grand unification," it is speculated that it may be possible to merge gravity with the other three gauge symmetries, expected to occur at roughly 1019 GeV. However — and while special relativity is parsimoniously

60

Quantum Mechanics after 1925 incorporated into quantum electrodynamics — the expanded general relativity, currently the best theory describing the gravitation force, has not been fully incorporated into quantum theory. One of the leading authorities continuing the search for a coherent TOE is Edward Witten, a theoretical physicist who formulated the groundbreaking M-theory, which is an attempt at describing the supersymmetrical based string theory. M-theory posits that our apparent 4-dimensional spacetime is, in reality, actually an 11-dimensional spacetime containing 10 spatial dimensions and 1 time dimension, although 7 of the spatial dimensions are - at lower energies - completely "compactified" (or infinitely curved) and not readily amenable to measurement or probing. Other popular theory is Loop quantum gravity (LQG) a theory that describes the quantum properties of gravity. It is also a theory of quantum space and quantum time, because, as discovered with general relativity, the geometry of spacetime is a manifestation of gravity. LQG is an attempt to merge and adapt standard quantum mechanics and standard general relativity. The main output of the theory is a physical picture of space where space is granular. The granularity is a direct consequence of the quantization. It has the same nature of the granularity of the photons in the quantum theory of electromagnetism or the discrete levels of the energy of the atoms. But here it is space itself which is discrete. More precisely, space can be viewed as an extremely fine fabric or network "woven" of finite loops. These networks of loops are called spin networks. The evolution of a spin network over time, is called a spin foam. The predicted size of this structure is the Planck length, which is approximately 1.616×10−35 m. According to theory, there is no meaning to length shorter than this (cf. Planck scale energy). Therefore LQG predicts that not just matter, but also space itself, has an atomic structure. Loop quantum Gravity was first proposed by Carlo Rovelli.

Philosophical implications Since its inception, the many counter-intuitive aspects and results of quantum mechanics have provoked strong philosophical debates and many interpretations. Even fundamental issues, such as Max Born's basic rules concerning probability amplitudes and probability distributions took decades to be appreciated by society and many leading scientists. Indeed, the renowned physicist Richard Feynman once said, "I think I can safely say that nobody understands quantum mechanics."[42] The Copenhagen interpretation - due largely to the Danish theoretical physicist Niels Bohr - remains the quantum mechanical formalism that is currently most widely accepted amongst physicists, some 75 years after its enunciation. According to this interpretation, the probabilistic nature of quantum mechanics is not a temporary feature which will eventually be replaced by a deterministic theory, but instead must be considered a final renunciation of the classical idea of "causality". It is also believed therein that any well-defined application of the quantum mechanical formalism must always make reference to the experimental arrangement, due to the complementarity nature of evidence obtained under different experimental situations. Albert Einstein, himself one of the founders of quantum theory, disliked this loss of determinism in measurement. Einstein held that there should be a local hidden variable theory underlying quantum mechanics and, consequently, that the present theory was incomplete. He produced a series of objections to the theory, the most famous of which has become known as the Einstein-Podolsky-Rosen paradox. John Bell showed that this "EPR" paradox led to experimentally testable differences between quantum mechanics and local realistic theories. Experiments have been performed confirming the accuracy of quantum mechanics, thereby demonstrating that the physical world cannot be described by any local realistic theory.[43] The Bohr-Einstein debates provide a vibrant critique of the Copenhagen Interpretation from an epistemological point of view. The Everett many-worlds interpretation, formulated in 1956, holds that all the possibilities described by quantum theory simultaneously occur in a multiverse composed of mostly independent parallel universes.[44] This is not accomplished by introducing some "new axiom" to quantum mechanics, but on the contrary, by removing the axiom of the collapse of the wave packet. All of the possible consistent states of the measured system and the measuring apparatus (including the observer) are present in a real physical - not just formally mathematical, as in other interpretations - quantum superposition. Such a superposition of consistent state combinations of different systems is

61

Quantum Mechanics after 1925

62

called an entangled state. While the multiverse is deterministic, we perceive non-deterministic behavior governed by probabilities, because we can observe only the universe (i.e., the consistent state contribution to the aforementioned superposition) that we, as observers, inhabit. Everett's interpretation is perfectly consistent with John Bell's experiments and makes them intuitively understandable. However, according to the theory of quantum decoherence, these "parallel universes" will never be accessible to us. The inaccessibility can be understood as follows: once a measurement is done, the measured system becomes entangled with both the physicist who measured it and a huge number of other particles, some of which are photons flying away at the speed of light towards the other end of the universe. In order to prove that the wave function did not collapse, one would have to bring all these particles back and measure them again, together with the system that was originally measured. Not only is this completely impractical, but even if one could theoretically do this, it would destroy any evidence that the original measurement took place (to include the physicist's memory). In light of these Bell tests, Cramer (1986) formulated his Transactional interpretation.[45] Relational quantum mechanics appeared in the late 1990s as the modern derivative of the Copenhagen Interpretation.

Applications Quantum mechanics had enormous[46] success in explaining many of the features of our world. The individual behaviors of the subatomic particles that make up all forms of matter (electrons, protons, neutrons, photons, and others) can often only be satisfactorily described using quantum mechanics. Quantum mechanics has strongly influenced string theories, candidates for a Theory of Everything (see reductionism), and the multiverse hypotheses. Quantum mechanics is also critically important for understanding how individual atoms combine covalently to form molecules. The application of quantum mechanics to chemistry is known as quantum chemistry. Relativistic quantum mechanics can, in principle, mathematically describe most of chemistry. Quantum mechanics can also provide quantitative insight into ionic and covalent bonding processes by explicitly showing which molecules are energetically favorable to which others, and the magnitudes of the energies involved.[47] Furthermore, most of the calculations performed in modern computational chemistry rely on quantum mechanics. A great deal of modern technological inventions operate at a scale where quantum effects are significant. Examples include the laser, the transistor (and thus the microchip), the electron microscope, and magnetic resonance imaging (MRI). The study of semiconductors led to the invention of the diode and the transistor, which are indispensable parts of modern electronics systems and devices. Researchers are currently seeking robust methods of directly manipulating quantum states. Efforts are being made to more fully develop quantum cryptography, which will theoretically allow guaranteed secure transmission of information. A more distant goal is the development of

A working mechanism of a resonant tunneling diode device, based on the phenomenon of quantum tunneling through potential barriers

Quantum Mechanics after 1925 quantum computers, which are expected to perform certain computational tasks exponentially faster than classical computers. Another active research topic is quantum teleportation, which deals with techniques to transmit quantum information over arbitrary distances. Quantum tunneling is vital to the operation of many devices - even in the simple light switch, as otherwise the electrons in the electric current could not penetrate the potential barrier made up of a layer of oxide. Flash memory chips found in USB drives use quantum tunneling to erase their memory cells. While quantum mechanics primarily applies to the atomic regimes of matter and energy, some systems exhibit quantum mechanical effects on a large scale - superfluidity, the frictionless flow of a liquid at temperatures near absolute zero, is one well-known example. Quantum theory also provides accurate descriptions for many previously unexplained phenomena, such as black body radiation and the stability of the orbitals of electrons in atoms. It has also given insight into the workings of many different biological systems, including smell receptors and protein structures.[48] Recent work on photosynthesis has provided evidence that quantum correlations play an essential role in this basic fundamental process of the plant kingdom.[49] Even so, classical physics can often provide good approximations to results otherwise obtained by quantum physics, typically in circumstances with large numbers of particles or large quantum numbers.

Examples Free particle For example, consider a free particle. In quantum mechanics, there is wave-particle duality, so the properties of the particle can be described as the properties of a wave. Therefore, its quantum state can be represented as a wave of arbitrary shape and extending over space as a wave function. The position and momentum of the particle are observables. The Uncertainty Principle states that both the position and the momentum cannot simultaneously be measured with complete precision. However, one can measure the position (alone) of a moving free particle, creating an eigenstate of position with a wavefunction that is very large (a Dirac delta) at a particular position x, and zero everywhere else. If one performs a position measurement on such a wavefunction, the resultant x will be obtained with 100% probability (i.e., with full certainty, or complete precision). This is called an eigenstate of position—or, stated in mathematical terms, a generalized position eigenstate (eigendistribution). If the particle is in an eigenstate of position, then its momentum is completely unknown. On the other hand, if the particle is in an eigenstate of momentum, then its position is completely unknown.[50] In an eigenstate of momentum having a plane wave form, it can be shown that the wavelength is equal to h/p, where h is Planck's constant and p is the momentum of the eigenstate.[51]

63

Quantum Mechanics after 1925

64

3D confined electron wave functions for each eigenstate in a Quantum Dot. Here, rectangular and triangular-shaped quantum dots are shown. Energy states in rectangular dots are more ‘s-type’ and ‘p-type’. However, in a triangular dot, the wave functions are mixed due to confinement symmetry.

Step potential The potential in this case is given by:

The solutions are superpositions of left- and right-moving waves: ,

where the wave vectors are related to the energy via , and Scattering at a finite potential step of height V0, shown in green. The amplitudes and direction of left- and right-moving waves are indicated. Yellow is the incident wave, blue are reflected and transmitted waves, red does not occur. E > V0 for this figure.

and the coefficients A and B are determined from the boundary conditions and by imposing a continuous derivative on the solution.

Each term of the solution can be interpreted as an incident, reflected, or transmitted component of the wave, allowing the calculation of transmission and reflection coefficients. In contrast to classical mechanics, incident particles with energies higher than the size of the potential step are still partially reflected.

Quantum Mechanics after 1925

65

Rectangular potential barrier This is a model for the quantum tunneling effect, which has important applications to modern devices such as flash memory and the scanning tunneling microscope.

Particle in a box The particle in a one-dimensional potential energy box is the most simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy everywhere inside a certain region, and infinite potential energy everywhere outside' that region. For the one-dimensional case in the direction, the time-independent Schrödinger equation can be written as:[52]

Writing the differential operator 1-dimensional potential energy box (or infinite potential well)

the previous equation can be seen to be evocative of the classic kinetic energy analogue

with

as the energy for the state

, which in this case coincides with the kinetic energy of the particle.

The general solutions of the Schrödinger equation for the particle in a box are:

or, from Euler's formula,

The presence of the walls of the box determines the values of C, D, and k. At each wall (x = 0 and x = L), ψ = 0. Thus when x = 0,

and so D = 0. When x = L,

C cannot be zero, since this would conflict with the Born interpretation. Therefore, sin kL = 0, and so it must be that kL is an integer multiple of π. And additionally,

The quantization of energy levels follows from this constraint on k, since

Quantum Mechanics after 1925

66

Finite potential well This is the generalization of the infinite potential well problem to potential wells of finite depth.

Harmonic oscillator As in the classical case, the potential for the quantum harmonic oscillator is given by:

This problem can be solved either by solving the Schrödinger equation directly, which is not trivial, or by using the more elegant "ladder method", first proposed by Paul Dirac. The eigenstates are given by:

Some trajectories of a harmonic oscillator (i.e. a ball attached to a spring) in classical mechanics (A-B) and quantum mechanics (C-H). In quantum mechanics, the position of the ball is represented by a wave (called the wavefunction), with the real part shown in blue and the imaginary part shown in red. Some of the trajectories (such as C,D,E,and F) are standing waves (or "stationary states"). Each standing-wave frequency is proportional to a possible energy level of the oscillator. This "energy quantization" does not occur in classical physics, where the oscillator can have any energy.

where Hn are the Hermite polynomials:

and the corresponding energy levels are . This is another example which illustrates the quantization of energy for bound states.

Quantum Mechanics after 1925

Notes [1] The angular momentum of an unbound electron, in contrast, is not quantized. [2] van Hove, Leon (1958). "Von Neumann's contributions to quantum mechanics" (http:/ / www. ams. org/ journals/ bull/ 1958-64-03/ S0002-9904-1958-10206-2/ S0002-9904-1958-10206-2. pdf) (PDF). Bulletin of the American Mathematical Society 64: Part2:95–99. . [3] Max Born & Emil Wolf, Principles of Optics, 1999, Cambridge University Press [4] Mehra, J.; Rechenberg, H. (1982). The historical development of quantum theory. New York: Springer-Verlag. ISBN 0387906428. [5] http:/ / www. ias. ac. in/ resonance/ December2010/ p1056-1059. pdf [6] Kuhn, T. S. (1978). Black-body theory and the quantum discontinuity 1894-1912. Oxford: Clarendon Press. ISBN 0195023838. [7] Einstein, A. (1905). "Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt [On a heuristic point of view concerning the production and transformation of light]". Annalen der Physik 17 (6): 132–148. Bibcode 1905AnP...322..132E. doi:10.1002/andp.19053220607. Reprinted in The collected papers of Albert Einstein, John Stachel, editor, Princeton University Press, 1989, Vol. 2, pp. 149-166, in German; see also Einstein's early work on the quantum hypothesis, ibid. pp. 134-148. [8] "Quantum - Definition and More from the Free Merriam-Webster Dictionary" (http:/ / www. merriam-webster. com/ dictionary/ quantum). Merriam-webster.com. . Retrieved 2012-08-18. [9] http:/ / mooni. fccj. org/ ~ethall/ quantum/ quant. htm [10] Compare the list of conferences presented here (http:/ / ysfine. com/ ) [11] Oocities.com (http:/ / web. archive. org/ 20091026095410/ http:/ / geocities. com/ mik_malm/ quantmech. html) at the Wayback Machine (archived October 26, 2009) [12] P.A.M. Dirac, The Principles of Quantum Mechanics, Clarendon Press, Oxford, 1930. [13] D. Hilbert Lectures on Quantum Theory, 1915-1927 [14] J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, 1932 (English translation: Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955). [15] Greiner, Walter; Müller, Berndt (1994). Quantum Mechanics Symmetries, Second edition (http:/ / books. google. com/ books?id=gCfvWx6vuzUC& pg=PA52). Springer-Verlag. p. 52. ISBN 3-540-58080-8. ., [16] "Heisenberg - Quantum Mechanics, 1925-1927: The Uncertainty Relations" (http:/ / www. aip. org/ history/ heisenberg/ p08a. htm). Aip.org. . Retrieved 2012-08-18. [17] Greenstein, George; Zajonc, Arthur (2006). The Quantum Challenge: Modern Research on the Foundations of Quantum Mechanics, Second edition (http:/ / books. google. com/ books?id=5t0tm0FB1CsC& pg=PA215). Jones and Bartlett Publishers, Inc. p. 215. ISBN 0-7637-2470-X. ., [18] "[Abstract] Visualization of Uncertain Particle Movement" (http:/ / www. actapress. com/ PaperInfo. aspx?PaperID=25988& reason=500). Actapress.com. . Retrieved 2012-08-18. [19] Hirshleifer, Jack (2001). The Dark Side of the Force: Economic Foundations of Conflict Theory (http:/ / books. google. com/ books?id=W2J2IXgiZVgC& pg=PA265). Campbridge University Press. p. 265. ISBN 0-521-80412-4. ., [20] Dict.cc (http:/ / www. dict. cc/ german-english/ eigen. html) De.pons.eu (http:/ / de. pons. eu/ deutsch-englisch/ eigen) [21] "Topics: Wave-Function Collapse" (http:/ / www. phy. olemiss. edu/ ~luca/ Topics/ qm/ collapse. html). Phy.olemiss.edu. 2012-07-27. . Retrieved 2012-08-18. [22] "Collapse of the wave-function" (http:/ / farside. ph. utexas. edu/ teaching/ qmech/ lectures/ node28. html). Farside.ph.utexas.edu. . Retrieved 2012-08-18. [23] "Determinism and Naive Realism : philosophy" (http:/ / www. reddit. com/ r/ philosophy/ comments/ 8p2qv/ determinism_and_naive_realism/ ). Reddit.com. 2009-06-01. . Retrieved 2012-08-18. [24] Michael Trott. "Time-Evolution of a Wavepacket in a Square Well — Wolfram Demonstrations Project" (http:/ / demonstrations. wolfram. com/ TimeEvolutionOfAWavepacketInASquareWell/ ). Demonstrations.wolfram.com. . Retrieved 2010-10-15. [25] Michael Trott. "Time Evolution of a Wavepacket In a Square Well" (http:/ / demonstrations. wolfram. com/ TimeEvolutionOfAWavepacketInASquareWell/ ). Demonstrations.wolfram.com. . Retrieved 2010-10-15. [26] Mathews, Piravonu Mathews; Venkatesan, K. (1976). A Textbook of Quantum Mechanics (http:/ / books. google. com/ books?id=_qzs1DD3TcsC& pg=PA36). Tata McGraw-Hill. p. 36. ISBN 0-07-096510-2. ., [27] "Wave Functions and the Schrödinger Equation" (http:/ / physics. ukzn. ac. za/ ~petruccione/ Phys120/ Wave Functions and the Schrödinger Equation. pdf) (PDF). . Retrieved 2010-10-15. [28] "Quantum Physics: Werner Heisenberg Uncertainty Principle of Quantum Mechanics. Werner Heisenberg Biography" (http:/ / www. spaceandmotion. com/ physics-quantum-mechanics-werner-heisenberg. htm). Spaceandmotion.com. 1976-02-01. . Retrieved 2012-08-18. [29] http:/ / th-www. if. uj. edu. pl/ acta/ vol19/ pdf/ v19p0683. pdf [30] Nancy Thorndike Greenspan, "The End of the Certain World: The Life and Science of Max Born" (Basic Books, 2005), pp. 124-8 and 285-6. [31] http:/ / ocw. usu. edu/ physics/ classical-mechanics/ pdf_lectures/ 06. pdf [32] "The Nobel Prize in Physics 1979" (http:/ / nobelprize. org/ nobel_prizes/ physics/ laureates/ 1979/ index. html). Nobel Foundation. . Retrieved 2010-02-16. [33] Carl M. Bender, Daniel W. Hook, Karta Kooner (2009-12-31). "Complex Elliptic Pendulum". arXiv:1001.0131 [hep-th].

67

Quantum Mechanics after 1925 [34] "Quantum mechanics course iwhatisquantummechanics" (http:/ / www. scribd. com/ doc/ 5998949/ Quantum-mechanics-course-iwhatisquantummechanics). Scribd.com. 2008-09-14. . Retrieved 2012-08-18. [35] "Between classical and quantum�" (http:/ / philsci-archive. pitt. edu/ 2328/ 1/ handbook. pdf) (PDF). . Retrieved 2012-08-19. [36] "Atomic Properties" (http:/ / academic. brooklyn. cuny. edu/ physics/ sobel/ Nucphys/ atomprop. html). Academic.brooklyn.cuny.edu. . Retrieved 2012-08-18. [37] http:/ / assets. cambridge. org/ 97805218/ 29526/ excerpt/ 9780521829526_excerpt. pdf [38] "There is as yet no logically consistent and complete relativistic quantum field theory.", p. 4.  — V. B. Berestetskii, E. M. Lifshitz, L P Pitaevskii (1971). J. B. Sykes, J. S. Bell (translators). Relativistic Quantum Theory 4, part I. Course of Theoretical Physics (Landau and Lifshitz) ISBN 0-08-016025-5 [39] http:/ / www. damtp. cam. ac. uk/ strings02/ dirac/ hawking/  [40] "Life on the lattice: The most accurate theory we have" (http:/ / latticeqcd. blogspot. com/ 2005/ 06/ most-accurate-theory-we-have. html). Latticeqcd.blogspot.com. 2005-06-03. . Retrieved 2010-10-15. [41] Parker, B. (1993). Overcoming some of the problems. pp. 259–279. [42] The Character of Physical Law (1965) Ch. 6; also quoted in The New Quantum Universe (2003), by Tony Hey and Patrick Walters [43] "Action at a Distance in Quantum Mechanics (Stanford Encyclopedia of Philosophy)" (http:/ / plato. stanford. edu/ entries/ qm-action-distance/ ). Plato.stanford.edu. 2007-01-26. . Retrieved 2012-08-18. [44] "Everett's Relative-State Formulation of Quantum Mechanics (Stanford Encyclopedia of Philosophy)" (http:/ / plato. stanford. edu/ entries/ qm-everett/ ). Plato.stanford.edu. . Retrieved 2012-08-18. [45] The Transactional Interpretation of Quantum Mechanics (http:/ / www. npl. washington. edu/ npl/ int_rep/ tiqm/ TI_toc. html) by John Cramer. Reviews of Modern Physics 58, 647-688, July (1986) [46] See, for example, the Feynman Lectures on Physics for some of the technological applications which use quantum mechanics, e.g., transistors (vol III, pp. 14-11 ff), integrated circuits, which are follow-on technology in solid-state physics (vol II, pp. 8-6), and lasers (vol III, pp. 9-13). [47] Introduction to Quantum Mechanics with Applications to Chemistry - Linus Pauling, E. Bright Wilson (http:/ / books. google. com/ books?id=vdXU6SD4_UYC). Books.google.com. 1985-03-01. ISBN 9780486648712. . Retrieved 2012-08-18. [48] Anderson, Mark (2009-01-13). "Is Quantum Mechanics Controlling Your Thoughts? | Subatomic Particles" (http:/ / discovermagazine. com/ 2009/ feb/ 13-is-quantum-mechanics-controlling-your-thoughts/ article_view?b_start:int=1& -C). DISCOVER Magazine. . Retrieved 2012-08-18. [49] "Quantum mechanics boosts photosynthesis" (http:/ / physicsworld. com/ cws/ article/ news/ 41632). physicsworld.com. . Retrieved 2010-10-23. [50] Davies, P. C. W.; Betts, David S. (1984). Quantum Mechanics, Second edition (http:/ / books. google. com/ books?id=XRyHCrGNstoC& pg=PA79). Chapman and Hall. p. 79. ISBN 0-7487-4446-0. ., [51] Baofu, Peter (2007-12-31). The Future of Complexity: Conceiving a Better Way to Understand Order and Chaos (http:/ / books. google. com/ books?id=tKm-Ekwke_UC). Books.google.com. ISBN 9789812708991. . Retrieved 2012-08-18. [52] Derivation of particle in a box, chemistry.tidalswan.com (http:/ / chemistry. tidalswan. com/ index. php?title=Quantum_Mechanics)

References The following titles, all by working physicists, attempt to communicate quantum theory to lay people, using a minimum of technical apparatus. • Malin, Shimon (2012). Nature Loves to Hide: Quantum Physics and the Nature of Reality, a Western Perspective (Revised ed.). World Scientific. ISBN 978-981-4324-57-1. • Chester, Marvin (1987) Primer of Quantum Mechanics. John Wiley. ISBN 0-486-42878-8 • Richard Feynman, 1985. QED: The Strange Theory of Light and Matter, Princeton University Press. ISBN 0-691-08388-6. Four elementary lectures on quantum electrodynamics and quantum field theory, yet containing many insights for the expert. • Ghirardi, GianCarlo, 2004. Sneaking a Look at God's Cards, Gerald Malsbary, trans. Princeton Univ. Press. The most technical of the works cited here. Passages using algebra, trigonometry, and bra-ket notation can be passed over on a first reading. • N. David Mermin, 1990, "Spooky actions at a distance: mysteries of the QT" in his Boojums all the way through. Cambridge University Press: 110-76. • Victor Stenger, 2000. Timeless Reality: Symmetry, Simplicity, and Multiple Universes. Buffalo NY: Prometheus Books. Chpts. 5-8. Includes cosmological and philosophical considerations. More technical:

68

Quantum Mechanics after 1925 • Bryce DeWitt, R. Neill Graham, eds., 1973. The Many-Worlds Interpretation of Quantum Mechanics, Princeton Series in Physics, Princeton University Press. ISBN 0-691-08131-X • Dirac, P. A. M. (1930). The Principles of Quantum Mechanics. ISBN 0-19-852011-5. The beginning chapters make up a very clear and comprehensible introduction. • Hugh Everett, 1957, "Relative State Formulation of Quantum Mechanics," Reviews of Modern Physics 29: 454-62. • Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (1965). The Feynman Lectures on Physics. 1-3. Addison-Wesley. ISBN 0-7382-0008-5. • Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-111892-7. OCLC 40251748. A standard undergraduate text. • Max Jammer, 1966. The Conceptual Development of Quantum Mechanics. McGraw Hill. • Hagen Kleinert, 2004. Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 3rd ed. Singapore: World Scientific. Draft of 4th edition. (http://www.physik.fu-berlin.de/~kleinert/b5) • Gunther Ludwig, 1968. Wave Mechanics. London: Pergamon Press. ISBN 0-08-203204-1 • George Mackey (2004). The mathematical foundations of quantum mechanics. Dover Publications. ISBN 0-486-43517-2. • Albert Messiah, 1966. Quantum Mechanics (Vol. I), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons. Cf. chpt. IV, section III. • Omnès, Roland (1999). Understanding Quantum Mechanics. Princeton University Press. ISBN 0-691-00435-8. OCLC 39849482. • Scerri, Eric R., 2006. The Periodic Table: Its Story and Its Significance. Oxford University Press. Considers the extent to which chemistry and the periodic system have been reduced to quantum mechanics. ISBN 0-19-530573-6 • Transnational College of Lex (1996). What is Quantum Mechanics? A Physics Adventure. Language Research Foundation, Boston. ISBN 0-9643504-1-6. OCLC 34661512. • von Neumann, John (1955). Mathematical Foundations of Quantum Mechanics. Princeton University Press. ISBN 0-691-02893-1. • Hermann Weyl, 1950. The Theory of Groups and Quantum Mechanics, Dover Publications. • D. Greenberger, K. Hentschel, F. Weinert, eds., 2009. Compendium of quantum physics, Concepts, experiments, history and philosophy, Springer-Verlag, Berlin, Heidelberg.

Further reading • Bernstein, Jeremy (2009). Quantum Leaps (http://books.google.com/books?id=j0Me3brYOL0C& printsec=frontcover). Cambridge, Massachusetts: Belknap Press of Harvard University Press. ISBN 978-0-674-03541-6. • Müller-Kirsten, H. J. W. (2012). Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral (2nd ed.). World Scientific. ISBN 978-981-4397-74-2. • Bohm, David (1989). Quantum Theory. Dover Publications. ISBN 0-486-65969-0. • Eisberg, Robert; Resnick, Robert (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd ed.). Wiley. ISBN 0-471-87373-X. • Liboff, Richard L. (2002). Introductory Quantum Mechanics. Addison-Wesley. ISBN 0-8053-8714-5. • Merzbacher, Eugen (1998). Quantum Mechanics. Wiley, John & Sons, Inc. ISBN 0-471-88702-1. • Sakurai, J. J. (1994). Modern Quantum Mechanics. Addison Wesley. ISBN 0-201-53929-2. • Shankar, R. (1994). Principles of Quantum Mechanics. Springer. ISBN 0-306-44790-8. • Cox, Brian; Forshaw, Jeff (2011). The Quantum Universe: Everything That Can Happen Does Happen. Allen Lane. ISBN 1-84614-432-9.

69

Quantum Mechanics after 1925

External links • Quantum Cook Book (http://oyc.yale.edu/sites/default/files/notes_quantum_cookbook.pdf) by R. Shankar, Open Yale PHYS 201 material (4pp) • A foundation approach to quantum Theory that does not rely on wave-particle duality. (http://www.mesacc. edu/~kevinlg/i256/QM_basics.pdf) • The Modern Revolution in Physics (http://www.lightandmatter.com/lm/) - an online textbook. • J. O'Connor and E. F. Robertson: A history of quantum mechanics. (http://www-history.mcs.st-andrews.ac.uk/ history/HistTopics/The_Quantum_age_begins.html) • Introduction to Quantum Theory at Quantiki. (http://www.quantiki.org/wiki/index.php/ Introduction_to_Quantum_Theory) • Quantum Physics Made Relatively Simple (http://bethe.cornell.edu/): three video lectures by Hans Bethe • H is for h-bar. (http://www.nonlocal.com/hbar/) • Quantum Mechanics Books Collection (http://www.freebookcentre.net/Physics/Quantum-Mechanics-Books. html): Collection of free books Course material • Doron Cohen: Lecture notes in Quantum Mechanics (comprehensive, with advanced topics). (http://arxiv.org/ abs/quant-ph/0605180) • MIT OpenCourseWare: Chemistry (http://ocw.mit.edu/OcwWeb/Chemistry/index.htm). • MIT OpenCourseWare: Physics (http://ocw.mit.edu/OcwWeb/Physics/index.htm). See 8.04 (http://ocw. mit.edu/OcwWeb/Physics/8-04Spring-2006/CourseHome/index.htm) • Stanford Continuing Education PHY 25: Quantum Mechanics (http://www.youtube.com/stanford#g/c/ 84C10A9CB1D13841) by Leonard Susskind, see course description (http://continuingstudies.stanford.edu/ courses/course.php?cid=20072_PHY 25) Fall 2007 • 5½ Examples in Quantum Mechanics (http://www.physics.csbsju.edu/QM/) • Imperial College Quantum Mechanics Course. (http://www.imperial.ac.uk/quantuminformation/qi/tutorials) • Spark Notes - Quantum Physics. (http://www.sparknotes.com/testprep/books/sat2/physics/ chapter19section3.rhtml) • Quantum Physics Online : interactive introduction to quantum mechanics (RS applets). (http://www. quantum-physics.polytechnique.fr/) • Experiments to the foundations of quantum physics with single photons. (http://www.didaktik.physik. uni-erlangen.de/quantumlab/english/index.html) • AQME (http://www.nanohub.org/topics/AQME) : Advancing Quantum Mechanics for Engineers — by T.Barzso, D.Vasileska and G.Klimeck online learning resource with simulation tools on nanohub • Quantum Mechanics (http://www.lsr.ph.ic.ac.uk/~plenio/lecture.pdf) by Martin Plenio • Quantum Mechanics (http://farside.ph.utexas.edu/teaching/qm/389.pdf) by Richard Fitzpatrick • Online course on Quantum Transport (http://nanohub.org/resources/2039) FAQs • Many-worlds or relative-state interpretation. (http://www.hedweb.com/manworld.htm) • Measurement in Quantum mechanics. (http://www.mtnmath.com/faq/meas-qm.html) Media • PHYS 201: Fundamentals of Physics II (http://oyc.yale.edu/physics/phys-201#sessions) by Ramamurti Shankar, Open Yale Course • Lectures on Quantum Mechanics (http://www.youtube.com/view_play_list?p=84C10A9CB1D13841) by Leonard Susskind • Everything you wanted to know about the quantum world (http://www.newscientist.com/channel/ fundamentals/quantum-world) — archive of articles from New Scientist.

70

Quantum Mechanics after 1925 • Quantum Physics Research (http://www.sciencedaily.com/news/matter_energy/quantum_physics/) from Science Daily • Overbye, Dennis (December 27, 2005). "Quantum Trickery: Testing Einstein's Strangest Theory" (http://www. nytimes.com/2005/12/27/science/27eins.html?scp=1&sq=quantum trickery&st=cse). The New York Times. Retrieved April 12, 2010. • Audio: Astronomy Cast (http://www.astronomycast.com/physics/ep-138-quantum-mechanics/) Quantum Mechanics — June 2009. Fraser Cain interviews Pamela L. Gay. Philosophy • "Quantum Mechanics" (http://plato.stanford.edu/entries/qm) entry by Jenann Ismael in the Stanford Encyclopedia of Philosophy • "Measurement in Quantum Theory" (http://plato.stanford.edu/entries/qt-measurement) entry by Henry Krips in the Stanford Encyclopedia of Philosophy

71

72

3. The Interpretation of Quantum Mechanics Interpretations of Quantum Mechanics An interpretation of quantum mechanics is a set of statements which attempt to explain how quantum mechanics informs our understanding of nature. Although quantum mechanics has held up to rigorous and thorough experimental testing, many of these experiments are open to different interpretations. There exist a number of contending schools of thought, differing over whether quantum mechanics can be understood to be deterministic, which elements of quantum mechanics can be considered "real", and other matters. This question is of special interest to philosophers of physics, as physicists continue to show a strong interest in the subject. They usually consider an interpretation of quantum mechanics as an interpretation of the mathematical formalism of quantum mechanics, specifying the physical meaning of the mathematical entities of the theory.

Historical background The definition of terms used by researchers in quantum theory (such as wavefunctions and matrix mechanics) progressed through many stages. For instance, Schrödinger originally viewed the wavefunction associated with the electron as corresponding to the charge density of an object smeared out over an extended, possibly infinite, volume of space. Max Born interpreted it as simply corresponding to a probability distribution. There are two different interpretations of the wavefunction: 1. In one it corresponds to a material field; 2. in the other it corresponds to a probability distribution — specifically, the probability that the quantum of charge is located at any particular point within spatial dimensions. The Copenhagen interpretation was traditionally the most popular among physicists, next to a purely instrumentalist position that denies any need for explanation (a view expressed in David Mermin's famous quote "shut up and calculate", often misattributed to Richard Feynman.[1]) However, the many-worlds interpretation has been gaining acceptance;[2] a controversial poll mentioned in "The Physics of Immortality" (published in 1994), of 72 "leading cosmologists and other quantum field theorists" found that 58% supported the many-worlds interpretation, including Stephen Hawking and Nobel laureates Murray Gell-Mann and Richard Feynman.[3] Moreover, the instrumentalist position has been challenged by proposals for falsifiable experiments that might one day distinguish interpretations, e.g. by measuring an AI consciousness[4] or via quantum computing.[5]

The nature of interpretation More or less all interpretations of quantum mechanics share two qualities: 1. They are interpretations of a formalism — a set of equations and formulae for generating results and predictions — and 2. they are interpretations of a phenomenology, a set of observations, including both those obtained by empirical research, and more informal subjective ones (that humans invariably observe an unequivocal world is important in the interpretation of quantum mechanics). The qualities that vary between interpretations are: 1. the ontology which is concerned with what, if anything, the interpreted theory is "really about" and 2. the epistemology which is concerned in what is knowledge, how are we acquiring it and to what extent is it possible for a given subject or entity to be known.

Interpretations of Quantum Mechanics The same phenomenon may be given an ontological reading under one interpretation, and an epistemological one under another. For instance, indeterminism may be attributed to the real existence of a "maybe" in the universe (ontology) or to limitations of an observer's information and predictive abilities (epistemology). Interpretations may be broadly classed as leaning more towards ontology, i.e. realism, or towards anti-realism. Some approaches tend to avoid giving any interpretation of phenomena or formalism. These can be described as instrumentalist. Other approaches suggest modifications to the formalism, and are therefore, strictly speaking, alternative theories rather than interpretations. In some cases, for instance Bohmian mechanics, it is open to debate as to whether an approach is equivalent to the standard formalism.

Problems of interpretation The difficulties of interpretation reflect a number of points about the orthodox description of quantum mechanics, including: 1. The abstract, mathematical nature of that description. 2. The existence of what appear to be non-deterministic and irreversible processes. 3. The phenomenon of entanglement, and in particular the correlations between remote events that are not expected in classical theory. 4. The complementarity of the proffered descriptions of reality. 5. The role played by observers and the process of measurement. 6. The rapid rate at which quantum descriptions become more complicated as the size of a system increases. Firstly, the accepted mathematical structure of quantum mechanics is based on fairly abstract mathematics, such as Hilbert spaces and operators on those spaces. In classical mechanics and electromagnetism, on the other hand, properties of a point mass or properties of a field are described by real numbers or functions defined on two or three dimensional sets. These have direct, spatial meaning, and in these theories there seems to be less need to provide special interpretation for those numbers or functions. Furthermore, the process of measurement may play an essential role in quantum theory - a hotly contested point. The world around us seems to be in a specific state, but quantum mechanics describes it by wave functions that govern the probability of all values. In general, the wave-function assigns non-zero probabilities to all possible values of any given physical quantity, such as position. How, then, do we see a particle in a specific position when its wave function is spread across all space? In order to describe how specific outcomes arise from the probabilities, the direct interpretation introduced the concept of measurement. According to the theory, wave functions interact with each other and evolve in time in accordance with the laws of quantum mechanics until a measurement is performed, at which point the system takes on one of its possible values, with a probability that's governed by the wave-function. Measurement can interact with the system state in somewhat peculiar ways, as is illustrated by the double-slit experiment. Thus the mathematical formalism used to describe the time evolution of a non-relativistic system proposes two opposed kinds of transformation: • Reversible transformations described by unitary operators on the state space. These transformations are determined by solutions to the Schrödinger equation. • Non-reversible and unpredictable transformations described by mathematically more complicated transformations (see quantum operations). Examples include the transformations undergone by a system as a result of measurement. A solution to the problem of interpretation consists in providing some form of plausible picture, by resolving the second kind of transformation. This can be achieved by purely mathematical solutions, as offered by the many-worlds or the consistent histories interpretations.

73

Interpretations of Quantum Mechanics In addition to the unpredictable and irreversible character of measurement processes, there are other elements of quantum physics that distinguish it sharply from classical physics and which are not present in any classical theory. One of these is the phenomenon of entanglement, as illustrated in the EPR paradox, which seemingly violates principles of local causality.[6] Another obstruction to interpretation is the phenomenon of complementarity, which seems to violate basic principles of propositional logic. Complementarity says there is no logical picture (one obeying classical propositional logic) that can simultaneously describe and be used to reason about all properties of a quantum system S. This is often phrased by saying that there are "complementary" propositions A and B that can each describe S, but not at the same time. Examples of A and B are propositions using a wave description of S and a corpuscular description of S. The latter statement is one part of Niels Bohr's original formulation, which is often equated to the principle of complementarity itself. Complementarity does not usually imply that it is classical logic which is at fault (although Hilary Putnam did take that view in his paper "Is logic empirical?"). Rather, complementarity means that the composition of physical properties for S (such as position and momentum both having values within certain ranges), using propositional connectives, does not obey the rules of classical propositional logic (see also Quantum logic). As is now well-known (Omnès, 1999) the "origin of complementarity lies in the non-commutativity of [the] operators" that describe observables (i.e., particles) in quantum mechanics. Because the complexity of a quantum system is exponential in its number of degrees of freedom, it is difficult to overlap the quantum and classical descriptions to see how the classical approximations are being made.

Problematic status of interpretations As classical physics and non-mathematical language cannot match the precision of quantum mechanics mathematics, anything said outside the mathematical formulation is necessarily limited in accuracy. Also, the precise ontological status of each interpretation remains a matter of philosophical argument. In other words, if we interpret the formal structure X of quantum mechanics by means of a structure Y (via a mathematical equivalence of the two structures), what is the status of Y? This is the old question of saving the phenomena, in a new guise. Some physicists, for example Asher Peres and Chris Fuchs, argue that an interpretation is nothing more than a formal equivalence between sets of rules for operating on experimental data, thereby implying that the whole exercise of interpretation is unnecessary.

Instrumentalist interpretation Any modern scientific theory requires at the very least an instrumentalist description that relates the mathematical formalism to experimental practice and prediction. In the case of quantum mechanics, the most common instrumentalist description is an assertion of statistical regularity between state preparation processes and measurement processes. That is, if a measurement of a real-value quantity is performed many times, each time starting with the same initial conditions, the outcome is a well-defined probability distribution agreeing with the real numbers; moreover, quantum mechanics provides a computational instrument to determine statistical properties of this distribution, such as its expectation value. Calculations for measurements performed on a system S postulate a Hilbert space H over the complex numbers. When the system S is prepared in a pure state, it is associated with a vector in H. Measurable quantities are associated with Hermitian operators acting on H: these are referred to as observables. Repeated measurement of an observable A where S is prepared in state ψ yields a distribution of values. The expectation value of this distribution is given by the expression

74

Interpretations of Quantum Mechanics

75

This mathematical machinery gives a simple, direct way to compute a statistical property of the outcome of an experiment, once it is understood how to associate the initial state with a Hilbert space vector, and the measured quantity with an observable (that is, a specific Hermitian operator). As an example of such a computation, the probability of finding the system in a given state

is given by

computing the expectation value of a (rank-1) projection operator The probability is then the non-negative real number given by

By abuse of language, a bare instrumentalist description could be referred to as an interpretation, although this usage is somewhat misleading since instrumentalism explicitly avoids any explanatory role; that is, it does not attempt to answer the question why.

Summary of common interpretations of quantum mechanics Classification adopted by Einstein An interpretation (i.e. a semantic explanation of the formal mathematics of quantum mechanics) can be characterized by its treatment of certain matters addressed by Einstein, such as: • • • •

Realism Completeness Local realism Determinism

To explain these properties, we need to be more explicit about the kind of picture an interpretation provides. To that end we will regard an interpretation as a correspondence between the elements of the mathematical formalism M and the elements of an interpreting structure I, where: • The mathematical formalism M consists of the Hilbert space machinery of ket-vectors, self-adjoint operators acting on the space of ket-vectors, unitary time dependence of the ket-vectors, and measurement operations. In this context a measurement operation is a transformation which turns a ket-vector into a probability distribution (for a formalization of this concept see quantum operations). • The interpreting structure I includes states, transitions between states, measurement operations, and possibly information about spatial extension of these elements. A measurement operation refers to an operation which returns a value and might result in a system state change. Spatial information would be exhibited by states represented as functions on configuration space. The transitions may be non-deterministic or probabilistic or there may be infinitely many states. The crucial aspect of an interpretation is whether the elements of I are regarded as physically real. Hence the bare instrumentalist view of quantum mechanics outlined in the previous section is not an interpretation at all, for it makes no claims about elements of physical reality. The current usage of realism and completeness originated in the 1935 paper in which Einstein and others proposed the EPR paradox.[7] In that paper the authors proposed the concepts element of reality and the completeness of a physical theory. They characterised element of reality as a quantity whose value can be predicted with certainty before measuring or otherwise disturbing it, and defined a complete physical theory as one in which every element of physical reality is accounted for by the theory. In a semantic view of interpretation, an interpretation is complete if every element of the interpreting structure is present in the mathematics. Realism is also a property of each of the elements of the maths; an element is real if it corresponds to something in the interpreting structure. For example, in some interpretations of quantum mechanics (such as the many-worlds interpretation) the ket vector associated to the system state is said to correspond to an element of physical reality, while in other interpretations it is not.

Interpretations of Quantum Mechanics Determinism is a property characterizing state changes due to the passage of time, namely that the state at a future instant is a function of the state in the present (see time evolution). It may not always be clear whether a particular interpretation is deterministic or not, as there may not be a clear choice of a time parameter. Moreover, a given theory may have two interpretations, one of which is deterministic and the other not. Local realism has two aspects: • The value returned by a measurement corresponds to the value of some function in the state space. In other words, that value is an element of reality; • The effects of measurement have a propagation speed not exceeding some universal limit (e.g. the speed of light). In order for this to make sense, measurement operations in the interpreting structure must be localized. A precise formulation of local realism in terms of a local hidden variable theory was proposed by John Bell. Bell's theorem, combined with experimental testing, restricts the kinds of properties a quantum theory can have. For instance, Bell's theorem implies that quantum mechanics cannot satisfy both local realism and counterfactual definiteness.

The Copenhagen interpretation The Copenhagen interpretation is the "standard" interpretation of quantum mechanics formulated by Niels Bohr and Werner Heisenberg while collaborating in Copenhagen around 1927. Bohr and Heisenberg extended the probabilistic interpretation of the wavefunction proposed originally by Max Born. The Copenhagen interpretation rejects questions like "where was the particle before I measured its position?" as meaningless. The measurement process randomly picks out exactly one of the many possibilities allowed for by the state's wave function in a manner consistent with the well-defined probabilities that are assigned to each possible state. According to the interpretation, the interaction of an observer or apparatus that is external to the quantum system is the cause of wave function collapse, thus according to Heisenberg "reality is in the observations, not in the electron".[8]

Many worlds The many-worlds interpretation is an interpretation of quantum mechanics in which a universal wavefunction obeys the same deterministic, reversible laws at all times; in particular there is no (indeterministic and irreversible) wavefunction collapse associated with measurement. The phenomena associated with measurement are claimed to be explained by decoherence, which occurs when states interact with the environment producing entanglement, repeatedly splitting the universe into mutually unobservable alternate histories—distinct universes within a greater multiverse.

Consistent histories The consistent histories interpretation generalizes the conventional Copenhagen interpretation and attempts to provide a natural interpretation of quantum cosmology. The theory is based on a consistency criterion that allows the history of a system to be described so that the probabilities for each history obey the additive rules of classical probability. It is claimed to be consistent with the Schrödinger equation. According to this interpretation, the purpose of a quantum-mechanical theory is to predict the relative probabilities of various alternative histories (for example, of a particle).

76

Interpretations of Quantum Mechanics

Ensemble interpretation, or statistical interpretation The ensemble interpretation, also called the statistical interpretation, can be viewed as a minimalist interpretation. That is, it claims to make the fewest assumptions associated with the standard mathematics. It takes the statistical interpretation of Born to the fullest extent. The interpretation states that the wave function does not apply to an individual system – for example, a single particle – but is an abstract statistical quantity that only applies to an ensemble (a vast multitude) of similarly prepared systems or particles. Probably the most notable supporter of such an interpretation was Einstein: The attempt to conceive the quantum-theoretical description as the complete description of the individual systems leads to unnatural theoretical interpretations, which become immediately unnecessary if one accepts the interpretation that the description refers to ensembles of systems and not to individual systems. —Einstein in Albert Einstein: Philosopher-Scientist, ed. P.A. Schilpp (Harper & Row, New York) The most prominent current advocate of the ensemble interpretation is Leslie E. Ballentine, professor at Simon Fraser University, author of the graduate level text book Quantum Mechanics, A Modern Development. An experiment illustrating the ensemble interpretation is provided in Akira Tonomura's Video clip 1 .[9] It is evident from this double-slit experiment with an ensemble of individual electrons that, since the quantum mechanical wave function (absolutely squared) describes the completed interference pattern, it must describe an ensemble.

de Broglie–Bohm theory The de Broglie–Bohm theory of quantum mechanics is a theory by Louis de Broglie and extended later by David Bohm to include measurements. Particles, which always have positions, are guided by the wavefunction. The wavefunction evolves according to the Schrödinger wave equation, and the wavefunction never collapses. The theory takes place in a single space-time, is non-local, and is deterministic. The simultaneous determination of a particle's position and velocity is subject to the usual uncertainty principle constraint. The theory is considered to be a hidden variable theory, and by embracing non-locality it satisfies Bell's inequality. The measurement problem is resolved, since the particles have definite positions at all times.[10] Collapse is explained as phenomenological.[11]

Relational quantum mechanics The essential idea behind relational quantum mechanics, following the precedent of special relativity, is that different observers may give different accounts of the same series of events: for example, to one observer at a given point in time, a system may be in a single, "collapsed" eigenstate, while to another observer at the same time, it may be in a superposition of two or more states. Consequently, if quantum mechanics is to be a complete theory, relational quantum mechanics argues that the notion of "state" describes not the observed system itself, but the relationship, or correlation, between the system and its observer(s). The state vector of conventional quantum mechanics becomes a description of the correlation of some degrees of freedom in the observer, with respect to the observed system. However, it is held by relational quantum mechanics that this applies to all physical objects, whether or not they are conscious or macroscopic. Any "measurement event" is seen simply as an ordinary physical interaction, an establishment of the sort of correlation discussed above. Thus the physical content of the theory has to do not with objects themselves, but the relations between them.[12][13] An independent relational approach to quantum mechanics was developed in analogy with David Bohm's elucidation of special relativity,[14] in which a detection event is regarded as establishing a relationship between the quantized field and the detector. The inherent ambiguity associated with applying Heisenberg's uncertainty principle is subsequently avoided.[15]

77

Interpretations of Quantum Mechanics

Transactional interpretation The transactional interpretation of quantum mechanics (TIQM) by John G. Cramer is an interpretation of quantum mechanics inspired by the Wheeler–Feynman absorber theory.[16] It describes a quantum interaction in terms of a standing wave formed by the sum of a retarded (forward-in-time) and an advanced (backward-in-time) wave. The author argues that it avoids the philosophical problems with the Copenhagen interpretation and the role of the observer, and resolves various quantum paradoxes.

Stochastic mechanics An entirely classical derivation and interpretation of Schrödinger's wave equation by analogy with Brownian motion was suggested by Princeton University professor Edward Nelson in 1966.[17] Similar considerations had previously been published, for example by R. Fürth (1933), I. Fényes (1952), and Walter Weizel (1953), and are referenced in Nelson's paper. More recent work on the stochastic interpretation has been done by M. Pavon.[18] An alternative stochastic interpretation was developed by Roumen Tsekov.[19]

Objective collapse theories Objective collapse theories differ from the Copenhagen interpretation in regarding both the wavefunction and the process of collapse as ontologically objective. In objective theories, collapse occurs randomly ("spontaneous localization"), or when some physical threshold is reached, with observers having no special role. Thus, they are realistic, indeterministic, no-hidden-variables theories. The mechanism of collapse is not specified by standard quantum mechanics, which needs to be extended if this approach is correct, meaning that Objective Collapse is more of a theory than an interpretation. Examples include the Ghirardi-Rimini-Weber theory[20] and the Penrose interpretation.[21]

von Neumann/Wigner interpretation: consciousness causes the collapse In his treatise The Mathematical Foundations of Quantum Mechanics, John von Neumann deeply analyzed the so-called measurement problem. He concluded that the entire physical universe could be made subject to the Schrödinger equation (the universal wave function). Since something "outside the calculation" was needed to collapse the wave function, von Neumann concluded that the collapse was caused by the consciousness of the experimenter.[22] This point of view was prominently expanded on by Eugene Wigner, but he later abandoned this interpretation.[23][24] Variations of the von Neumann interpretation include: Subjective reduction research This principle, that consciousness causes the collapse, is the point of intersection between quantum mechanics and the mind/body problem; and researchers are working to detect conscious events correlated with physical events that, according to quantum theory, should involve a wave function collapse; but, thus far, results are inconclusive.[25][26] Participatory anthropic principle (PAP) John Archibald Wheeler's participatory anthropic principle says that consciousness plays some role in bringing the universe into existence.[27] Other physicists have elaborated their own variations of the von Neumann interpretation; including: • Henry P. Stapp (Mindful Universe: Quantum Mechanics and the Participating Observer) • Bruce Rosenblum and Fred Kuttner (Quantum Enigma: Physics Encounters Consciousness)

78

Interpretations of Quantum Mechanics

Many minds The many-minds interpretation of quantum mechanics extends the many-worlds interpretation by proposing that the distinction between worlds should be made at the level of the mind of an individual observer.

Quantum logic Quantum logic can be regarded as a kind of propositional logic suitable for understanding the apparent anomalies regarding quantum measurement, most notably those concerning composition of measurement operations of complementary variables. This research area and its name originated in the 1936 paper by Garrett Birkhoff and John von Neumann, who attempted to reconcile some of the apparent inconsistencies of classical boolean logic with the facts related to measurement and observation in quantum mechanics.

Quantum information theories Informational approaches subdivide into two kinds[28] • Information ontologies, such as J. A. Wheeler's "it from bit". These approaches have been described as a revival of immaterialism[29] • Interpretations where quantum mechanics is said to describe an observer's knowledge of the world, rather than the world itself. This approach has some similarity with Bohr's thinking.[30] Collapse (also known as reduction) is often interpreted as an observer acquiring information from a measurement, rather than as an objective event. These approaches have been appraised as similar to instrumentalism. The state is not an objective property of an individual system but is that information, obtained from a knowledge of how a system was prepared, which can be used for making predictions about future measurements. ...A quantum mechanical state being a summary of the observer’s information about an individual physical system changes both by dynamical laws, and whenever the observer acquires new information about the system through the process of measurement. The existence of two laws for the evolution of the state vector...becomes problematical only if it is believed that the state vector is an objective property of the system...The “reduction of the wavepacket” does take place in the consciousness of the observer, not because of any unique physical process which takes place there, but only because the state is a construct of the observer and not an objective property of the physical system[31]

Modal interpretations of quantum theory Modal interpretations of quantum mechanics were first conceived of in 1972 by B. van Fraassen, in his paper “A formal approach to the philosophy of science.” However, this term now is used to describe a larger set of models that grew out of this approach. The Stanford Encyclopedia of Philosophy describes several versions:[32] • The Copenhagen variant • Kochen-Dieks-Healey Interpretations • Motivating Early Modal Interpretations, based on the work of R. Clifton, M. Dickson and J. Bub.

Time-symmetric theories Several theories have been proposed which modify the equations of quantum mechanics to be symmetric with respect to time reversal.[33][34][35][36] This creates retrocausality: events in the future can affect ones in the past, exactly as events in the past can affect ones in the future. In these theories, a single measurement cannot fully determine the state of a system (making them a type of hidden variables theory), but given two measurements performed at different times, it is possible to calculate the exact state of the system at all intermediate times. The collapse of the wavefunction is therefore not a physical change to the system, just a change in our knowledge of it

79

Interpretations of Quantum Mechanics

80

due to the second measurement. Similarly, they explain entanglement as not being a true physical state but just an illusion created by ignoring retrocausality. The point where two particles appear to "become entangled" is simply a point where each particle is being influenced by events that occur to the other particle in the future.

Branching space-time theories BST theories resemble the many worlds interpretation; however, "the main difference is that the BST interpretation takes the branching of history to be feature of the topology of the set of events with their causal relationships... rather than a consequence of the separate evolution of different components of a state vector."[37] In MWI, it is the wave functions that branches, whereas in BST, the space-time topology itself branches. BST has applications to Bells theorem, quantum computation and quantum gravity. It also has some resemblance to hidden variable theories and the ensemble interpretation.: particles in BST have multiple well defined trajectories at the microscopic level. These can only be treated stochastically at a coarse grained level, in line with the ensemble interpretation.[37]

Other interpretations As well as the mainstream interpretations discussed above, a number of other interpretations have been proposed which have not made a significant scientific impact. These range from proposals by mainstream physicists to the more occult ideas of quantum mysticism.

Comparison of interpretations The most common interpretations are summarized in the table below. The values shown in the cells of the table are not without controversy, for the precise meanings of some of the concepts involved are unclear and, in fact, are themselves at the center of the controversy surrounding the given interpretation. No experimental evidence exists that distinguishes among these interpretations. To that extent, the physical theory stands, and is consistent with itself and with reality; difficulties arise only when one attempts to "interpret" the theory. Nevertheless, designing experiments which would test the various interpretations is the subject of active research. Most of these interpretations have variants. For example, it is difficult to get a precise definition of the Copenhagen interpretation as it was developed and argued about by many people. Interpretation

Author(s)

Ensemble interpretation

Max Born, 1926

Agnostic

No

Yes

Agnostic

Copenhagen interpretation

Niels Bohr, Werner Heisenberg, 1927

No

No1

Yes

de Broglie–Bohm theory

Louis de Broglie, 1927, David Bohm, 1952

Yes

Yes3

von Neumann interpretation

von Neumann, 1932, Wheeler, Wigner

No

Quantum logic

Garrett Birkhoff, 1936

Many-worlds interpretation

Hugh Everett, 1957

Popper's [38] interpretation

Deterministic? Wavefunction real?

[39]

Karl Popper, 1957

Unique history?

Hidden Collapsing variables? wavefunctions?

Observer role?

Local?

Counterfactual definiteness?

No

None

No

No

No

Yes2

Causal

No

No

Yes4

Yes

No

None

No

Yes

Yes

Yes

No

Yes

Causal

No

No

Agnostic

Agnostic

Yes5

No

No

Yes

Yes

No

No

No

None

Yes

No

No

Yes

Yes

Yes

No

None

Yes

Yes13

Interpretational6 Agnostic

No

Interpretations of Quantum Mechanics

81

Time-symmetric theories

William C. Davidon, 1976

Yes

Yes

Yes

Yes

No

No

Yes

No

Stochastic interpretation

Edward Nelson, 1966

No

No

Yes

No

No

None

No

No

Many-minds interpretation

H. Dieter Zeh, 1970

Yes

Yes

No

No

No

Interpretational7

Yes

No

Consistent histories

Robert B. Griffiths, 1984

Agnostic8

Agnostic8

No

No

No

Interpretational6

Yes

No

No

Yes

Yes

No

Yes

None

No

No

Objective Ghirardi–Rimini–Weber, collapse theories 1986, Penrose interpretation, 1989 Transactional interpretation

John G. Cramer, 1986

No

Yes

Yes

No

Yes9

None

No

Yes14

Relational interpretation

Carlo Rovelli, 1994

No

No

Agnostic10

No

Yes11

Intrinsic12

Yes

No

•

• • • • • • • • • • • •

•

1

  According to Bohr, the concept of a physical state independent of the conditions of its experimental observation does not have a well-defined meaning. According to Heisenberg the wavefunction represents a probability, but not an objective reality itself in space and time. 2   According to the Copenhagen interpretation, the wavefunction collapses when a measurement is performed. 3   Both particle AND guiding wavefunction are real. 4   Unique particle history, but multiple wave histories. 5   But quantum logic is more limited in applicability than Coherent Histories. 6   Quantum mechanics is regarded as a way of predicting observations, or a theory of measurement. 7   Observers separate the universal wavefunction into orthogonal sets of experiences. 8   If wavefunction is real then this becomes the many-worlds interpretation. If wavefunction less than real, but more than just information, then Zurek calls this the "existential interpretation". 9   In the TI the collapse of the state vector is interpreted as the completion of the transaction between emitter and absorber. 10   Comparing histories between systems in this interpretation has no well-defined meaning. 11   Any physical interaction is treated as a collapse event relative to the systems involved, not just macroscopic or conscious observers. 12   The state of the system is observer-dependent, i.e., the state is specific to the reference frame of the observer. 13   Caused by the fact that Popper holds both CFD and locality to be true, it is under dispute whether Popper's interpretation can really be considered an interpretation of Quantum Mechanics (which is what Popper claimed) or whether it must be considered a modification of Quantum Mechanics (which is what many Physicists claim), and, in case of the latter, if this modification has been empirically refuted or not. Popper exchanged many long letters with Einstein, Bell etc. about the issue. 14   The transactional interpretation is explicitly non-local.

Interpretations of Quantum Mechanics

Sources • Bub, J. and Clifton, R. 1996. “A uniqueness theorem for interpretations of quantum mechanics,” Studies in History and Philosophy of Modern Physics 27B: 181-219 • Rudolf Carnap, 1939, "The interpretation of physics," in Foundations of Logic and Mathematics of the International Encyclopedia of Unified Science. University of Chicago Press. • Dickson, M., 1994, "Wavefunction tails in the modal interpretation" in Hull, D., Forbes, M., and Burian, R., eds., Proceedings of the PSA 1" 366–76. East Lansing, Michigan: Philosophy of Science Association. • --------, and Clifton, R., 1998, "Lorentz-invariance in modal interpretations" in Dieks, D. and Vermaas, P., eds., The Modal Interpretation of Quantum Mechanics. Dordrecht: Kluwer Academic Publishers: 9–48. • Fuchs, Christopher, 2002, "Quantum Mechanics as Quantum Information (and only a little more)." arXiv:quant-ph/0205039 • -------- and A. Peres, 2000, "Quantum theory needs no ‘interpretation’," Physics Today. • Herbert, N., 1985. Quantum Reality: Beyond the New Physics. New York: Doubleday. ISBN 0-385-23569-0. • Hey, Anthony, and Walters, P., 2003. The New Quantum Universe, 2nd ed. Cambridge Univ. Press. ISBN 0-521-56457-3. • Roman Jackiw and D. Kleppner, 2000, "One Hundred Years of Quantum Physics," Science 289(5481): 893. • Max Jammer, 1966. The Conceptual Development of Quantum Mechanics. McGraw-Hill. • --------, 1974. The Philosophy of Quantum Mechanics. Wiley & Sons. • Al-Khalili, 2003. Quantum: A Guide for the Perplexed. London: Weidenfeld & Nicholson. • de Muynck, W. M., 2002. Foundations of quantum mechanics, an empiricist approach. Dordrecht: Kluwer Academic Publishers. ISBN 1-4020-0932-1.[40] • Roland Omnès, 1999. Understanding Quantum Mechanics. Princeton Univ. Press. • Karl Popper, 1963. Conjectures and Refutations. London: Routledge and Kegan Paul. The chapter "Three views Concerning Human Knowledge" addresses, among other things, instrumentalism in the physical sciences. • Hans Reichenbach, 1944. Philosophic Foundations of Quantum Mechanics. Univ. of California Press. • Max Tegmark and J. A. Wheeler, 2001, "100 Years of Quantum Mysteries," Scientific American 284: 68. • Bas van Fraassen, 1972, "A formal approach to the philosophy of science," in R. Colodny, ed., Paradigms and Paradoxes: The Philosophical Challenge of the Quantum Domain. Univ. of Pittsburgh Press: 303-66. • John A. Wheeler and Wojciech Hubert Zurek (eds), Quantum Theory and Measurement, Princeton: Princeton University Press, ISBN 0-691-08316-9, LoC QC174.125.Q38 1983.

References [1] For a discussion of the provenance of the phrase "shut up and calculate", see (http:/ / scitation. aip. org/ journals/ doc/ PHTOAD-ft/ vol_57/ iss_5/ 10_1. shtml) [2] Vaidman, L. (2002, March 24). Many-Worlds Interpretation of Quantum Mechanics. Retrieved March 19, 2010, from Stanford Encyclopedia of Philosophy: http:/ / plato. stanford. edu/ entries/ qm-manyworlds/ #Teg98 [3] "Who believes in many-worlds?" (http:/ / www. hedweb. com/ everett/ everett. htm#believes). Hedweb.com. . Retrieved 2011-01-24. [4] Quantum theory as a universal physical theory, by David Deutsch, International Journal of Theoretical Physics, Vol 24 #1 (1985) [5] Three connections between Everett's interpretation and experiment Quantum Concepts of Space and Time, by David Deutsch, Oxford University Press (1986) [6] La nouvelle cuisine, by John S. Bell, last article of Speakable and Unspeakable in Quantum Mechanics, second edition. [7] A. Einstein, B. Podolsky and N. Rosen, 1935, "Can quantum-mechanical description of physical reality be considered complete?" Phys. Rev. 47: 777. [8] http:/ / www. naturalthinker. net/ trl/ texts/ Heisenberg,Werner/ Heisenberg,%20Werner%20-%20Physics%20and%20philosophy. pdf [9] "An experiment illustrating the ensemble interpretation" (http:/ / www. hitachi. com/ rd/ research/ em/ doubleslit. html). Hitachi.com. . Retrieved 2011-01-24. [10] Why Bohm's Theory Solves the Measurement Problem by T. Maudlin, Philosophy of Science 62, pp. 479-483 (September, 1995). [11] Bohmian Mechanics as the Foundation of Quantum Mechanics by D. Durr, N. Zanghi, and S. Goldstein in Bohmian Mechanics and Quantum Theory: An Appraisal, edited by J.T. Cushing, A. Fine, and S. Goldstein, Boston Studies in the Philosophy of Science 184, 21-44 (Kluwer, 1996) 1997 arXiv:quant-ph/9511016

82

Interpretations of Quantum Mechanics [12] "Relational Quantum Mechanics (Stanford Encyclopedia of Philosophy)" (http:/ / plato. stanford. edu/ entries/ qm-relational/ ). Plato.stanford.edu. . Retrieved 2011-01-24. [13] For more information, see Carlo Rovelli (1996). "Relational Quantum Mechanics". International Journal of Theoretical Physics 35 (8): 1637. arXiv:quant-ph/9609002. Bibcode 1996IJTP...35.1637R. doi:10.1007/BF02302261. [14] David Bohm, The Special Theory of Relativity, Benjamin, New York, 1965 [15] (http:/ / www. quantum-relativity. org/ Quantum-Relativity. pdf). For a full account (http:/ / www. quantum-relativity. org/ Quantum_Optics_as_a_Relativistic_Theory_of_Light. pdf), see Q. Zheng and T. Kobayashi, 1996, "Quantum Optics as a Relativistic Theory of Light," Physics Essays 9: 447. Annual Report, Department of Physics, School of Science, University of Tokyo (1992) 240. [16] "Quantum Nocality - Cramer" (http:/ / www. npl. washington. edu/ npl/ int_rep/ qm_nl. html). Npl.washington.edu. . Retrieved 2011-01-24. [17] Nelson,E. (1966) Derivation of the Schrödinger Equation from Newtonian Mechanics, Phys. Rev. 150, 1079-1085 [18] M. Pavon, “Stochastic mechanics and the Feynman integral”, J. Math. Phys. 41, 6060-6078 (2000) [19] Roumen Tsekov (2009). "Bohmian Mechanics versus Madelung Quantum Hydrodynamics". arXiv:0904.0723 [quant-ph]. [20] "Frigg, R. GRW theory" (http:/ / www. romanfrigg. org/ writings/ GRW Theory. pdf) (PDF). . Retrieved 2011-01-24. [21] "Review of Penrose's Shadows of the Mind" (http:/ / www. thymos. com/ mind/ penrose. html). Thymos.com. . Retrieved 2011-01-24. [22] von Neumann, John. (1932/1955). Mathematical Foundations of Quantum Mechanics. Princeton: Princeton University Press. Translated by Robert T. Beyer. [23] [ Michael Esfeld, (1999), Essay Review: Wigner’s View of Physical Reality, published in Studies in History and Philosophy of Modern Physics, 30B, pp. 145–154, Elsevier Science Ltd.] [24] Zvi Schreiber (1995). "The Nine Lives of Schroedinger's Cat". arXiv:quant-ph/9501014 [quant-ph]. [25] Dick J. Bierman and Stephen Whitmarsh. (2006). Consciousness and Quantum Physics: Empirical Research on the Subjective Reduction of the State Vector. in Jack A. Tuszynski (Ed). The Emerging Physics of Consciousness. p. 27-48. [26] C. M. H. Nunn et. al. (1994). Collapse of a Quantum Field may Affect Brain Function. Journal of Consciousness Studies. 1(1):127-139. [27] "- The anthropic universe" (http:/ / www. abc. net. au/ rn/ scienceshow/ stories/ 2006/ 1572643. htm). Abc.net.au. 2006-02-18. . Retrieved 2011-01-24. [28] Information, Immaterialism, Instrumentalism: Old and New in Quantum Information. Christopher G. Timpson (http:/ / users. ox. ac. uk/ ~bras2317/ iii_2. pdf) [29] Timpson,Op. Cit.: "Let us call the thought that information might be the basic category from which all else flows informational immaterialism." [30] "Physics concerns what we can say about nature". (Niels Bohr, quoted in Petersen, A. (1963). The philosophy of Niels Bohr. Bulletin of the Atomic Scientists, 19(7):8–14.) [31] Hartle, J. B. (1968). Quantum mechanics of individual systems. Am. J. Phys., 36(8):704– 712. [32] "Modal Interpretations of Quantum Mechanics (Stanford Encyclopedia of Philosophy)" (http:/ / www. science. uva. nl/ ~seop/ entries/ qm-modal/ ). Science.uva.nl. . Retrieved 2011-01-24. [33] Davidon, W.C. "Quantum Physics of Single Systems." Il Nuovo Cimento, Volume 36B, pp. 34-40 (1976). [34] Aharonov, Y. and Vaidman, L. "On the Two-State Vector Reformulation of Quantum Mechanics." Physica Scripta, Volume T76, pp. 85-92 (1998). [35] Wharton, K. B. "Time-Symmetric Quantum Mechanics." Foundations of Physics, 37(1), pp. 159-168 (2007). [36] Wharton, K. B. "A Novel Interpretation of the Klein-Gordon Equation." Foundations of Physics, 40(3), pp. 313-332 (2010). [37] Sharlow, Mark; "What Branching Spacetime might do for Phyiscs" (http:/ / philsci-archive. pitt. edu/ 3781/ 1/ what_branching_spacetime_might_do. pdf) p.2 [38] Marie-Christine Combourieu: Karl R. Popper, 1992: About the EPR controversy. Foundations of Physics 22:10, 1303-1323 [39] Karl Popper: The Propensity Interpretation of the Calculus of Probability and of the Quantum Theory. Observation and Interpretation. Buttersworth Scientific Publications, Korner & Price (eds.) 1957. pp 65–70. [40] de Muynck, Willem M (2002). Foundations of quantum mechanics: an empiricist approach (http:/ / books. google. com/ ?id=k3rUe8XVjJUC& printsec=frontcover& dq=an+ empiricist+ approach#v=onepage& q=& f=false). Klower Academic Publishers. ISBN 1-4020-0932-1. . Retrieved 2011-01-24.

Further reading Almost all authors below are professional physicists. • David Z Albert, 1992. Quantum Mechanics and Experience. Harvard Univ. Press. ISBN 0-674-74112-9. • John S. Bell, 1987. Speakable and Unspeakable in Quantum Mechanics. Cambridge Univ. Press, ISBN 0-521-36869-3. The 2004 edition (ISBN 0-521-52338-9) includes two additional papers and an introduction by Alain Aspect. • Dmitrii Ivanovich Blokhintsev, 1968. The Philosophy of Quantum Mechanics. D. Reidel Publishing Company. ISBN 90-277-0105-9. • David Bohm, 1980. Wholeness and the Implicate Order. London: Routledge. ISBN 0-7100-0971-2.

83

Interpretations of Quantum Mechanics • Adan Cabello (15 November 2004). "Bibliographic guide to the foundations of quantum mechanics and quantum information". arXiv:quant-ph/0012089 [quant-ph]. • David Deutsch, 1997. The Fabric of Reality. London: Allen Lane. ISBN 0-14-027541-X; ISBN 0-7139-9061-9. Argues forcefully against instrumentalism. For general readers. • Bernard d'Espagnat, 1976. Conceptual Foundation of Quantum Mechanics, 2nd ed. Addison Wesley. ISBN 0-8133-4087-X. • --------, 1983. In Search of Reality. Springer. ISBN 0-387-11399-1. • --------, 2003. Veiled Reality: An Analysis of Quantum Mechanical Concepts. Westview Press. • --------, 2006. On Physics and Philosophy. Princeton Univ. Press. • Arthur Fine, 1986. The Shaky Game: Einstein Realism and the Quantum Theory. Science and its Conceptual Foundations. Univ. of Chicago Press. ISBN 0-226-24948-4. • Ghirardi, Giancarlo, 2004. Sneaking a Look at God’s Cards. Princeton Univ. Press. • Gregg Jaeger (2009) Entanglement, Information, and the Interpretation of Quantum Mechanics. (http://www. springer.com/physics/quantum+physics/book/978-3-540-92127-1) Springer. ISBN 978-3-540-92127-1. • N. David Mermin (1990) Boojums all the way through. (http://www.cambridge.org/catalogue/catalogue. asp?isbn=0521388805) Cambridge Univ. Press. ISBN 0-521-38880-5. • Roland Omnes, 1994. The Interpretation of Quantum Mechanics. Princeton Univ. Press. ISBN 0-691-03669-1. • --------, 1999. Understanding Quantum Mechanics. Princeton Univ. Press. • --------, 1999. Quantum Philosophy: Understanding and Interpreting Contemporary Science. Princeton Univ. Press. • Roger Penrose, 1989. The Emperor's New Mind. Oxford Univ. Press. ISBN 0-19-851973-7. Especially chpt. 6. • --------, 1994. Shadows of the Mind. Oxford Univ. Press. ISBN 0-19-853978-9. • --------, 2004. The Road to Reality. New York: Alfred A. Knopf. Argues that quantum theory is incomplete. • Styer, Daniel F. (March 2002). "Nine formulations of quantum mechanics". American Journal of Physics 70 (3): 288–297. doi:10.1119/1.1445404.

External links • Stanford Encyclopedia of Philosophy: • " Bohmian mechanics (http://plato.stanford.edu/entries/qm-bohm/)" by Sheldon Goldstein. • " Collapse Theories. (http://plato.stanford.edu/entries/qm-collapse/)" by Giancarlo Ghirardi. • " Copenhagen Interpretation of Quantum Mechanics (http://plato.stanford.edu/entries/qm-copenhagen/)" by Jan Faye. • " Everett's Relative State Formulation of Quantum Mechanics (http://plato.stanford.edu/entries/qm-everett/ )" by Jeffrey Barrett. • " Many-Worlds Interpretation of Quantum Mechanics (http://plato.stanford.edu/entries/qm-manyworlds/)" by Lev Vaidman. • " Modal Interpretation of Quantum Mechanics (http://plato.stanford.edu/entries/qm-modal/)" by Michael Dickson and Dennis Dieks. • " Quantum Entanglement and Information (http://plato.stanford.edu/entries/qt-entangle/)" by Jeffrey Bub. • " Quantum mechanics (http://plato.stanford.edu/entries/qm/)" by Jenann Ismael. • " Relational Quantum Mechanics (http://plato.stanford.edu/entries/qm-relational/)" by Federico Laudisa and Carlo Rovelli. • " The Role of Decoherence in Quantum Mechanics (http://plato.stanford.edu/entries/qm-decoherence/)" by Guido Bacciagaluppi. • Willem M. de Muynck, Broad overview (http://www.phys.tue.nl/ktn/Wim/muynck.htm#quantum) of the realist vs. empiricist interpretations, against oversimplified view of the measurement process.

84

Interpretations of Quantum Mechanics • Schreiber, Z., " The Nine Lives of Schrodinger's Cat. (http://arxiv.org/abs/quant-ph/9501014)" Overview of competing interpretations. • Interpretations of quantum mechanics on arxiv.org. (http://xstructure.inr.ac.ru/x-bin/subthemes3. py?level=2&index1=362483&skip=0) • The many worlds of quantum mechanics. (http://www.johnsankey.ca/qm.html) • Erich Joos' Decoherence Website. (http://www.decoherence.de/) • Quantum Mechanics for Philosophers. (http://home.sprynet.com/~owl1/qm.htm) Argues for the superiority of the Bohm interpretation. • Hidden Variables in Quantum Theory: The Hidden Cultural Variables of their Rejection. (http://www. miguel-montenegro.com/Hidden_cultural_variables.htm) • Numerous Many Worlds-related Topics and Articles. (http://www.station1.net/DouglasJones/many.htm) • Relational Approach to Quantum Physics. (http://www.quantum-relativity.org/) • Theory of incomplete measurements. (http://cc3d.free.fr/tim.pdf) Deriving quantum mechanics axioms from properties of acceptable measurements. • Alfred Neumaier's FAQ. (http://www.mat.univie.ac.at/~neum/physics-faq.txt) • Measurement in Quantum Mechanics FAQ. (http://www.mtnmath.com/faq/meas-qm.html)

The Copenhagen Interpretation The Copenhagen interpretation is one of the earliest and most commonly taught interpretations of quantum mechanics.[1] It holds that quantum mechanics does not yield a description of an objective reality but deals only with probabilities of observing, or measuring, various aspects of energy quanta, entities which fit neither the classical idea of particles nor the classical idea of waves. According to the interpretation, the act of measurement causes the set of probabilities to immediately and randomly assume only one of the possible values. This feature of the mathematics is known as wavefunction collapse. The essential concepts of the interpretation were devised by Niels Bohr, Werner Heisenberg and others in the years 1924–27.

Background Classical physics draws a distinction between particles and energy, holding that only the latter exhibit waveform characteristics, whereas quantum mechanics is based on the observation that matter has both wave and particle aspects and postulates that the state of every subatomic particle can be described by a wavefunction—a mathematical expression used to calculate the probability that the particle, if measured, will be in a given location or state of motion. In the early work of Max Planck, Albert Einstein and Niels Bohr, the existence of energy in discrete quantities had been postulated, in order to explain phenomena, such as the spectrum of black-body radiation, the photoelectric effect, and the stability and spectrum of atoms such as hydrogen, that had eluded explanation by, and even appeared to be in contradiction with, classical physics. Also, while elementary particles showed predictable properties in many experiments, they became highly unpredictable in certain contexts, for example, if one attempted to measure their individual trajectories through a simple physical apparatus. The Copenhagen interpretation is an attempt to explain the mathematical formulations of quantum mechanics and the corresponding experimental results. Early twentieth-century experiments on the physics of very small-scale phenomena led to the discovery of phenomena which could not be predicted on the basis of classical physics, and to the development of new models (theories) that described and predicted very accurately these micro-scale phenomena. These models could not easily be reconciled with the way objects are observed to behave on the macro scale of everyday life. The predictions they offered often appeared counter-intuitive and caused much consternation

85

The Copenhagen Interpretation

86

among the physicists—often including their discoverers.

Origin of the term Werner Heisenberg had been an assistant to Niels Bohr at his institute in Copenhagen during part of the 1920s, when they helped originate quantum mechanical theory. In 1929, Heisenberg gave a series of invited lectures at the University of Chicago explaining the new field of quantum mechanics. The lectures then served as the basis for his textbook, The Physical Principles of the Quantum Theory, published in 1930.[2] In the book's preface, Heisenberg wrote: On the whole the book contains nothing that is not to be found in previous publications, particularly in the investigations of Bohr. The purpose of the book seems to me to be fulfilled if it contributes somewhat to the diffusion of that 'Kopenhagener Geist der Quantentheorie' [i.e., Copenhagen spirit of quantum theory] if I may so express myself, which has directed the entire development of modern atomic physics. The term 'Copenhagen interpretation' suggests something more than just a spirit, such as some definite set of rules for interpreting the mathematical formalism of quantum mechanics, presumably dating back to the 1920s. However, no such text exists, apart from some informal popular lectures by Bohr and Heisenberg, which contradict each other on several important issues. It appears that the particular term, with its more definite sense, was coined by Heisenberg in the 1950s,[3] while criticizing alternate "interpretations" (e.g., David Bohm's[4]) that had been developed.[5] Lectures with the titles 'The Copenhagen Interpretation of Quantum Theory' and 'Criticisms and Counterproposals to the Copenhagen Interpretation', that Heisenberg delivered in 1955, are reprinted in the collection Physics and Philosophy.[6]

Principles Because it consists of the views developed by a number of scientists and philosophers during the second quarter of the 20th Century, there is no definitive statement of the Copenhagen interpretation.[7] Thus, various ideas have been associated with it; Asher Peres remarked that very different, sometimes opposite, views are presented as "the Copenhagen interpretation" by different authors.[8] Nonetheless, there are several basic principles that are generally accepted as being part of the interpretation: 1. A system is completely described by a wave function

2. 3. 4.

5. 6.

, representing the state of the system, which evolves

smoothly in time, except when a measurement is made, at which point it instantaneously collapses to an eigenstate of the observable measured. The description of nature is essentially probabilistic, with the probability of a given outcome of a measurement given by the square of the amplitude of the wave function. (The Born rule, after Max Born) It is not possible to know the value of all the properties of the system at the same time; those properties that are not known exactly must be described by probabilities. (Heisenberg's uncertainty principle) Matter exhibits a wave–particle duality. An experiment can show the particle-like properties of matter, or the wave-like properties; in some experiments both of these complementary viewpoints must be invoked to explain the results, according to the complementarity principle of Niels Bohr. Measuring devices are essentially classical devices, and measure only classical properties such as position and momentum. The quantum mechanical description of large systems will closely approximate the classical description. (This is the correspondence principle of Bohr and Heisenberg.)

The Copenhagen Interpretation

Meaning of the wave function The Copenhagen Interpretation denies that the wave function is anything more than a theoretical concept, or is at least non-committal about its being a discrete entity or a discernible component of some discrete entity. The subjective view, that the wave function is merely a mathematical tool for calculating the probabilities in a specific experiment, has some similarities to the Ensemble interpretation in that it takes probabilities to be the essence of the quantum state, but unlike the ensemble interpretation, it takes these probabilities to be perfectly applicable to single experimental outcomes, as it interprets them in terms of subjective probability. There are some who say that there are objective variants of the Copenhagen Interpretation that allow for a "real" wave function, but it is questionable whether that view is really consistent with some of Bohr's statements. Bohr emphasized that science is concerned with predictions of the outcomes of experiments, and that any additional propositions offered are not scientific but meta-physical. Bohr was heavily influenced by positivism. On the other hand, Bohr and Heisenberg were not in complete agreement, and they held different views at different times. Heisenberg in particular was prompted to move towards realism.[9] Even if the wave function is not regarded as real, there is still a divide between those who treat it as definitely and entirely subjective, and those who are non-committal or agnostic about the subject. An example of the agnostic view is given by Carl Friedrich von Weizsäcker, who, while participating in a colloquium at Cambridge, denied that the Copenhagen interpretation asserted: "What cannot be observed does not exist." He suggested instead that the Copenhagen interpretation follows the principle: "What is observed certainly exists; about what is not observed we are still free to make suitable assumptions. We use that freedom to avoid paradoxes."[10]

Nature of collapse All versions of the Copenhagen interpretation include at least a formal or methodological version of wave function collapse,[11] in which unobserved eigenvalues are removed from further consideration. (In other words, Copenhagenists have always made the assumption of collapse, even in the early days of quantum physics, in the way that adherents of the Many-worlds interpretation have not.) In more prosaic terms, those who hold to the Copenhagen understanding are willing to say that a wave function involves the various probabilities that a given event will proceed to certain different outcomes. But when one or another of those more- or less-likely outcomes becomes manifest the other probabilities cease to have any function in the real world. So if an electron passes through a double slit apparatus there are various probabilities for where on the detection screen that individual electron will hit. But once it has hit, there is no longer any probability whatsoever that it will hit somewhere else. Many-worlds interpretations say that an electron hits wherever there is a possibility that it might hit, and that each of these hits occurs in a separate universe. An adherent of the subjective view, that the wave function represents nothing but knowledge, would take an equally subjective view of "collapse". Some argue that the concept of the collapse of a "real" wave function was introduced by Heisenberg and later developed by John Von Neumann in 1932.[12] Heisenberg never used the term collapse, preferring to speak of the wavefunction representing our knowledge of a system, and collapse as the "jumping" of the wavefunction to a new state, representing a "jump" in our knowledge which occurs once a particular phenomenon is registered by the experimenter (i.e. when an observation takes place).

87

The Copenhagen Interpretation

Acceptance among physicists According to a poll at a Quantum Mechanics workshop in 1997,[13] the Copenhagen interpretation is the most widely-accepted specific interpretation of quantum mechanics, followed by the many-worlds interpretation.[14] Although current trends show substantial competition from alternative interpretations, throughout much of the twentieth century the Copenhagen interpretation had strong acceptance among physicists. Astrophysicist and science writer John Gribbin describes it as having fallen from primacy after the 1980s.[15]

Consequences The nature of the Copenhagen Interpretation is exposed by considering a number of experiments and paradoxes. 1. Schrödinger's Cat This thought experiment highlights the implications that accepting uncertainty at the microscopic level has on macroscopic objects. A cat is put in a sealed box, with its life or death made dependent on the state of a subatomic particle. Thus a description of the cat during the course of the experiment—having been entangled with the state of a subatomic particle—becomes a "blur" of "living and dead cat." But this can't be accurate because it implies the cat is actually both dead and alive until the box is opened to check on it. But the cat, if he survives, will only remember being alive. Schrödinger resists "so naively accepting as valid a 'blurred model' for representing reality."[16] How can the cat be both alive and dead? The Copenhagen Interpretation: The wave function reflects our knowledge of the system. The wave function means that, once the cat is observed, there is a 50% chance it will be dead, and 50% chance it will be alive. 2. Wigner's Friend Wigner puts his friend in with the cat. The external observer believes the system is in the state . His friend however is convinced that cat is alive, i.e. for him, the cat is in the state

. How can Wigner and his friend see different wave functions?

The Copenhagen Interpretation: Wigner's friend highlights the subjective nature of probability. Each observer (Wigner and his friend) has different information and therefore different wave functions. The distinction between the "objective" nature of reality and the subjective nature of probability has led to a great deal of controversy. Cf. Bayesian versus Frequentist interpretations of probability. 3. Double-Slit Diffraction Light passes through double slits and onto a screen resulting in a diffraction pattern. Is light a particle or a wave? The Copenhagen Interpretation: Light is neither. A particular experiment can demonstrate particle (photon) or wave properties, but not both at the same time (Bohr's Complementarity Principle). The same experiment can in theory be performed with any physical system: electrons, protons, atoms, molecules, viruses, bacteria, cats, humans, elephants, planets, etc. In practice it has been performed for light, electrons, buckminsterfullerene,[17][18] and some atoms. Due to the smallness of Planck's constant it is practically impossible to realize experiments that directly reveal the wave nature of any system bigger than a few atoms but, in general, quantum mechanics considers all matter as possessing both particle and wave behaviors. The greater systems (like viruses, bacteria, cats, etc.) are considered as "classical" ones but only as an approximation, not exact. 4. EPR (Einstein–Podolsky–Rosen) paradox Entangled "particles" are emitted in a single event. Conservation laws ensure that the measured spin of one particle must be the opposite of the measured spin of the other, so that if the spin of one particle is measured, the spin of the other particle is now instantaneously known. The most discomforting aspect of this paradox is

88

The Copenhagen Interpretation that the effect is instantaneous so that something that happens in one galaxy could cause an instantaneous change in another galaxy. But, according to Einstein's theory of special relativity, no information-bearing signal or entity can travel at or faster than the speed of light, which is finite. Thus, it seems as if the Copenhagen interpretation is inconsistent with special relativity. The Copenhagen Interpretation: Assuming wave functions are not real, wave-function collapse is interpreted subjectively. The moment one observer measures the spin of one particle, he knows the spin of the other. However, another observer cannot benefit until the results of that measurement have been relayed to him, at less than or equal to the speed of light. Copenhagenists claim that interpretations of quantum mechanics where the wave function is regarded as real have problems with EPR-type effects, since they imply that the laws of physics allow for influences to propagate at speeds greater than the speed of light. However, proponents of Many worlds[19] and the Transactional interpretation[20][21] (TI) maintain that Copenhagen interpretation is fatally non-local. The claim that EPR effects violate the principle that information cannot travel faster than the speed of light have been countered by noting that they cannot be used for signaling because neither observer can control, or predetermine, what he observes, and therefore cannot manipulate what the other observer measures. However, it should be noted that is a somewhat spurious argument, in that speed of light limitations applies to all information, not to what can or can not be subsequently done with the information. A further argument is that relativistic difficulties about establishing which measurement occurred first also undermine the idea that one observer is causing what the other is measuring. This is totally spurious, since no matter who measured first the other will measure the opposite spin despite the fact that (in theory) the other has a 50% 'probability' (50:50 chance) of measuring the same spin, unless data about the first spin measurement has somehow passed faster than light (of course TI gets around the light speed limit by having information travel backwards in time instead).

Criticism The completeness of quantum mechanics (thesis 1) was attacked by the Einstein-Podolsky-Rosen thought experiment which was intended to show that quantum physics could not be a complete theory. Experimental tests of Bell's inequality using particles have supported the quantum mechanical prediction of entanglement. The Copenhagen Interpretation gives special status to measurement processes without clearly defining them or explaining their peculiar effects. In his article entitled "Criticism and Counterproposals to the Copenhagen Interpretation of Quantum Theory," countering the view of Alexandrov that (in Heisenberg's paraphrase) "the wave function in configuration space characterizes the objective state of the electron." Heisenberg says, Of course the introduction of the observer must not be misunderstood to imply that some kind of subjective features are to be brought into the description of nature. The observer has, rather, only the function of registering decisions, i.e., processes in space and time, and it does not matter whether the observer is an apparatus or a human being; but the registration, i.e., the transition from the "possible" to the "actual," is absolutely necessary here and cannot be omitted from the interpretation of quantum theory.[22] Many physicists and philosophers have objected to the Copenhagen interpretation, both on the grounds that it is non-deterministic and that it includes an undefined measurement process that converts probability functions into non-probabilistic measurements. Einstein's comments "I, at any rate, am convinced that He (God) does not throw dice."[23] and "Do you really think the moon isn't there if you aren't looking at it?"[24] exemplify this. Bohr, in response, said "Einstein, don't tell God what to do". Steven Weinberg in "Einstein's Mistakes", Physics Today, November 2005, page 31, said:

89

The Copenhagen Interpretation All this familiar story is true, but it leaves out an irony. Bohr's version of quantum mechanics was deeply flawed, but not for the reason Einstein thought. The Copenhagen interpretation describes what happens when an observer makes a measurement, but the observer and the act of measurement are themselves treated classically. This is surely wrong: Physicists and their apparatus must be governed by the same quantum mechanical rules that govern everything else in the universe. But these rules are expressed in terms of a wave function (or, more precisely, a state vector) that evolves in a perfectly deterministic way. So where do the probabilistic rules of the Copenhagen interpretation come from? Considerable progress has been made in recent years toward the resolution of the problem, which I cannot go into here. It is enough to say that neither Bohr nor Einstein had focused on the real problem with quantum mechanics. The Copenhagen rules clearly work, so they have to be accepted. But this leaves the task of explaining them by applying the deterministic equation for the evolution of the wave function, the Schrödinger equation, to observers and their apparatus. The problem of thinking in terms of classical measurements of a quantum system becomes particularly acute in the field of quantum cosmology, where the quantum system is the universe.[25] E. T. Jaynes,[26] from a Bayesian point of view, pointed out probability is a measure of a human's information about the physical world. Quantum mechanics under the Copenhagen Interpretation interpreted probability as a physical phenomenon, which is what Jaynes called a Mind Projection Fallacy. A similar view is adopted in Quantum Information Theories.

Alternatives The Ensemble interpretation is similar; it offers an interpretation of the wave function, but not for single particles. The consistent histories interpretation advertises itself as "Copenhagen done right". Although the Copenhagen interpretation is often confused with the idea that consciousness causes collapse, it defines an "observer" merely as that which collapses the wave function.[22] If the wave function is regarded as ontologically real, and collapse is entirely rejected, a many worlds theory results. If wave function collapse is regarded as ontologically real as well, an objective collapse theory is obtained. For an atemporal interpretation that “makes no attempt to give a ‘local’ account on the level of determinate particles”,[27] the conjugate wavefunction, ("advanced" or time-reversed) of the relativistic version of the wavefunction, and the so-called "retarded" or time-forward version[28] are both regarded as real and the transactional interpretation results.[27] Dropping the principle that the wave function is a complete description results in a hidden variable theory. Many physicists have subscribed to the instrumentalist interpretation of quantum mechanics, a position often equated with eschewing all interpretation. It is summarized by the sentence "Shut up and calculate!". While this slogan is sometimes attributed to Paul Dirac[29] or Richard Feynman, it is in fact due to David Mermin.[30]

Notes and references [1] Hermann Wimmel (1992). Quantum physics & observed reality: a critical interpretation of quantum mechanics (http:/ / books. google. com/ books?id=-4sJ_fgyZJEC& pg=PA2). World Scientific. p. 2. ISBN 978-981-02-1010-6. . Retrieved 9 May 2011. [2] J. Mehra and H. Rechenberg, The historical development of quantum theory, Springer-Verlag, 2001, p. 271. [3] Howard, Don (2004). "Who invented the Copenhagen Interpretation? A study in mythology". Philosophy of Science: 669–682. JSTOR 10.1086/425941. [4] Bohm, David (1952). "A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. I & II". Physical Review 85 (2): 166–193. Bibcode 1952PhRv...85..166B. doi:10.1103/PhysRev.85.166. [5] H. Kragh, Quantum generations: A History of Physics in the Twentieth Century, Princeton University Press, 1999, p. 210. ("the term 'Copenhagen interpretation' was not used in the 1930s but first entered the physicist’s vocabulary in 1955 when Heisenberg used it in criticizing certain unorthodox interpretations of quantum mechanics.") [6] Werner Heisenberg, Physics and Philosophy, Harper, 1958

90

The Copenhagen Interpretation [7] In fact Bohr and Heisenberg never totally agreed on how to understand the mathematical formalism of quantum mechanics. Bohr once distanced himself from what he considered to be Heisenberg's more subjective interpretation Stanford Encyclopedia of Philosophy (http:/ / plato. stanford. edu/ entries/ qm-copenhagen/ ) [8] "There seems to be at least as many different Copenhagen interpretations as people who use that term, probably there are more. For example, in two classic articles on the foundations of quantum mechanics, Ballentine (1970) and Stapp(1972) give diametrically opposite definitions of 'Copenhagen.'", Asher Peres (2002). "Popper's experiment and the Copenhagen interpretation". Stud. History Philos. Modern Physics 33 (23): 10078. arXiv:quant-ph/9910078. Bibcode 1999quant.ph.10078P. [9] "Historically, Heisenberg wanted to base quantum theory solely on observable quantities such as the intensity of spectral lines, getting rid of all intuitive (anschauliche) concepts such as particle trajectories in space-time. This attitude changed drastically with his paper in which he introduced the uncertainty relations – there he put forward the point of view that it is the theory which decides what can be observed. His move from positivism to operationalism can be clearly understood as a reaction on the advent of Schrödinger’s wave mechanics which, in particular due to its intuitiveness, became soon very popular among physicists. In fact, the word anschaulich (intuitive) is contained in the title of Heisenberg’s paper.", from Claus Kiefer (2002). "On the interpretation of quantum theory - from Copenhagen to the present day". arXiv:quant-ph/0210152 [quant-ph]. [10] John Cramer on the Copenhagen Interpretation (http:/ / www. npl. washington. edu/ npl/ int_rep/ tiqm/ TI_20. html#2. 0) [11] "To summarize, one can identify the following ingredients as being characteristic for the Copenhagen interpretation(s)[...]Reduction of the wave packet as a formal rule without dynamical significance", Claus Kiefer (2002). "On the interpretation of quantum theory - from Copenhagen to the present day". arXiv:quant-ph/0210152 [quant-ph]. [12] "the “collapse” or “reduction” of the wave function. This was introduced by Heisenberg in his uncertainty paper [3] and later postulated by von Neumann as a dynamical process independent of the Schrodinger equation", Claus Kiefer (2002). "On the interpretation of quantum theory - from Copenhagen to the present day". arXiv:quant-ph/0210152 [quant-ph]. [13] Max Tegmark (1998). "The Interpretation of Quantum Mechanics: Many Worlds or Many Words?". Fortsch.Phys. 46 (6–8): 855–862. arXiv:quant-ph/9709032. Bibcode 1998ForPh..46..855T. doi:10.1002/(SICI)1521-3978(199811)46:6/83.0.CO;2-Q. [14] The Many Worlds Interpretation of Quantum Mechanics (http:/ / www. hep. upenn. edu/ ~max/ everett. ps) [15] Gribbin, J. Q for Quantum [16] Erwin Schrödinger, in an article in the Proceedings of the American Philosophical Society, 124, 323-38. [17] Nairz, Olaf; Brezger, Björn; Arndt, Markus; Zeilinger, Anton (2001). "Diffraction of Complex Molecules by Structures Made of Light". Physical Review Letters 87 (16). arXiv:quant-ph/0110012. Bibcode 2001PhRvL..87p0401N. doi:10.1103/PhysRevLett.87.160401. [18] Brezger, Björn; Hackermüller, Lucia; Uttenthaler, Stefan; Petschinka, Julia; Arndt, Markus; Zeilinger, Anton (2002). "Matter-Wave Interferometer for Large Molecules". Physical Review Letters 88 (10): 100404. arXiv:quant-ph/0202158. Bibcode 2002PhRvL..88j0404B. doi:10.1103/PhysRevLett.88.100404. PMID 11909334. [19] Michael price on nonlocality in Many Worlds (http:/ / www. hedweb. com/ manworld. htm#local) [20] Relativity and Causality in the Transactional Interpretation (http:/ / www. npl. washington. edu/ npl/ int_rep/ tiqm/ TI_38. html#3. 9) [21] Collapse and Nonlocality in the Transactional Interpretation (http:/ / www. npl. washington. edu/ npl/ int_rep/ tiqm/ TI_33. html#3. 7) [22] Werner Heisenberg, Physics and Philosophy, Harper, 1958, p. 137. [23] "God does not throw dice" quote [24] A. Pais, Einstein and the quantum theory, Reviews of Modern Physics 51, 863-914 (1979), p. 907. [25] 'Since the Universe naturally contains all of its observers, the problem arises to come up with an interpretation of quantum theory that contains no classical realms on the fundamental level.', Claus Kiefer (2002). "On the interpretation of quantum theory - from Copenhagen to the present day". arXiv:quant-ph/0210152 [quant-ph]. [26] Jaynes, E. T. (1989). "Clearing up Mysteries--The Original Goal" (http:/ / bayes. wustl. edu/ etj/ articles/ cmystery. pdf). Maximum Entropy and Bayesian Methods: 7. . [27] The Quantum Liar Experiment, RE Kastner, Studies in History and Philosophy of Modern Physics, Vol41, Iss.2,May2010 [28] The non-relativistic Schrödinger equation does not admit advanced solutions. [29] http:/ / home. fnal. gov/ ~skands/ slides/ A-Quantum-Journey. ppt [30] N. David Mermin. "Could Feynman Have Said This?" (http:/ / scitation. aip. org/ journals/ doc/ PHTOAD-ft/ vol_57/ iss_5/ 10_1. shtml). Physics Today 57 (5). .

91

The Copenhagen Interpretation

Further reading • • • • • • •

G. Weihs et al., Phys. Rev. Lett. 81 (1998) 5039 M. Rowe et al., Nature 409 (2001) 791. J.A. Wheeler & W.H. Zurek (eds), Quantum Theory and Measurement, Princeton University Press 1983 A. Petersen, Quantum Physics and the Philosophical Tradition, MIT Press 1968 H. Margeneau, The Nature of Physical Reality, McGraw-Hill 1950 M. Chown, Forever Quantum, New Scientist No. 2595 (2007) 37. T. Schürmann, A Single Particle Uncertainty Relation, Acta Physica Polonica B39 (2008) 587. (http://th-www. if.uj.edu.pl/acta/vol39/pdf/v39p0587.pdf)

External links • Copenhagen Interpretation (Stanford Encyclopedia of Philosophy) (http://plato.stanford.edu/entries/ qm-copenhagen) • Physics FAQ section about Bell's inequality (http://math.ucr.edu/home/baez/physics/Quantum/ bells_inequality.html) • The Copenhagen Interpretation of Quantum Mechanics (http://www.benbest.com/science/quantum.html) • Preprint of Afshar Experiment (http://www.irims.org/quant-ph/030503/) • The Quantum Illusion (http://knol.google.com/k/andy-biddulph/the-quantum-illusion/2na7zaaxgtohe/2/)

92

93

4. Einstein's Objections Principle of Locality In physics, the principle of locality states that an object is influenced directly only by its immediate surroundings. Experiments have shown that quantum mechanically entangled particles must either violate the principle of locality or allow superluminal communication.[1]

Pre-quantum mechanics In the 17th Century Newton's law of universal gravitation was formulated in terms of "action at a distance", thereby violating the principle of locality. It is inconceivable that inanimate Matter should, without the Mediation of something else, which is not material, operate upon, and affect other matter without mutual Contact…That Gravity should be innate, inherent and essential to Matter, so that one body may act upon another at a distance thro' a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it. Gravity must be caused by an Agent acting constantly according to certain laws; but whether this Agent be material or immaterial, I have left to the Consideration of my readers.[2] —Isaac Newton, Letters to Bentley, 1692/3 Coulomb's law of electric forces was initially also formulated as instantaneous action at a distance, but was later superseded by Maxwell's Equations of electromagnetism which obey locality. In 1905 Albert Einstein's Special Theory of Relativity postulated that no material or energy can travel faster than the speed of light, and Einstein thereby sought to reformulate physical laws in a way which obeyed the principle of locality. He later succeeded in producing an alternative theory of gravitation, General Relativity, which obeys the principle of locality. However, a different challenge to the principle of locality subsequently emerged from the theory of Quantum Mechanics, which Einstein himself had helped to create.

Quantum mechanics Einstein's view EPR Paradox Albert Einstein argued that quantum mechanics was an incomplete physical theory. Using the principle of locality, in a famous paper he and his co-authors articulated the Einstein-Podolsky-Rosen Paradox which showed that position and momentum were simultaneous "real" physical properties of a subatomic particle. However, quantum mechanics has nothing to say about these "elements of reality". Thirty years later John Stewart Bell responded with a paper that posited (paraphrased) that no physical theory of local hidden variables, no local realism, can ever reproduce all of the predictions of quantum mechanics (known as Bell's theorem).

Principle of Locality Philosophical view Einstein assumed that the principle of locality was necessary, and that there could be no violations of it. He said: "(...) The following idea characterises the relative independence of objects far apart in space, A and B: external influence on A has no direct influence on B; this is known as the Principle of Local Action, which is used consistently only in field theory. If this axiom were to be completely abolished, the idea of the existence of quasienclosed systems, and thereby the postulation of laws which can be checked empirically in the accepted sense, would become impossible. (...)""Quantum Mechanics and Reality" ("Quanten-Mechanik und Wirklichkeit", Dialectica 2:320-324, 1948)

Local realism Local realism is the combination of the principle of locality with the "realistic" assumption that all objects must objectively have a pre-existing value for any possible measurement before the measurement is made. And so be time independent: “I like to think that the moon is there even if I am not looking at it” ~Albert Einstein

Realism Realism in the sense used by physicists does not equate to realism in metaphysics.[4] The latter is the claim that the world is in some sense mind-independent: that even if the results of a possible measurement do not pre-exist the act of measurement, that does not require that they are the creation of the observer (contrary to the "consciousness causes collapse" interpretation of quantum mechanics). Furthermore, a mind-independent property does not have to be the value of some physical variable such as position or momentum. A property can be dispositional (or potential), i.e. it can be a tendency: in the way that glass objects tend to break, or are disposed to break, even if they do not actually break. Likewise, the mind-independent properties of quantum systems could consist of a tendency to respond to particular measurements with particular values with ascertainable probability.[5] Such an ontology would be metaphysically realistic, without being realistic in the physicist's sense of "local realism" (which would require that a single value be produced with certainty). A closely related term is counterfactual definiteness (CFD), used to refer to the claim that one can meaningfully speak of the definiteness of results of measurements that have not been performed (i.e. the ability to assume the existence of objects, and properties of objects, even when they have not been measured). Local realism is a significant feature of classical mechanics, of general relativity, and of electrodynamics; but quantum mechanics largely rejects this principle due to the theory of distant quantum entanglements, an interpretation rejected by Einstein in the EPR paradox but subsequently apparently quantified by Bell's inequalities.[6] Any theory, such as quantum mechanics, that violates Bell's inequalities must abandon either local realism or counterfactual definiteness; but some physicists dispute that experiments have demonstrated Bell's violations, on the grounds that the sub-class of inhomogeneous Bell inequalities has not been tested or due to experimental limitations in the tests. Different interpretations of quantum mechanics violate different parts of local realism and/or counterfactual definiteness.

Copenhagen interpretation In most of the conventional interpretations, such as the Copenhagen interpretation and the interpretation based on Consistent Histories, where the wavefunction is not assumed to physically exist in real space-time, it is local realism that is rejected. These interpretations propose that actual definite properties of a physical system "do not exist" prior to the measurement; and the wavefunction has a restricted interpretation, as nothing more than a mathematical tool used to calculate the probabilities of experimental outcomes, hence in agreement with positivism in philosophy as the only topic that science should discuss. If the wavefunction is assumed to physically exist in real space-time, the principle of locality is violated during the measurement process via wavefunction collapse. This is a non-local process because Born's Rule, when applied to

94

Principle of Locality the system's wavefunction, yields a probability density for all regions of space and time. Upon actual measurement of the physical system, the probability density vanishes everywhere instantaneously, except where (and when) the measured entity is found to exist. This "vanishing" is postulated to be a real physical process, and clearly non-local (i.e. faster than light) if the wavefunction is considered physically real and the probability density has converged to zero at arbitrarily far distances during the finite time required for the measurement process.

Bohm interpretation The Bohm interpretation preserves realism, hence it needs to violate the principle of locality in order to achieve the required correlations.

Many-worlds interpretation In the many-worlds interpretation both realism and locality are retained, but counterfactual definiteness is rejected by the extension of the notion of reality to allow the existence of parallel universes. Because the differences between the different interpretations are mostly philosophical ones (except for the Bohm and many-worlds interpretations), physicists usually employ language in which the important statements are neutral with regard to all of the interpretations. In this framework, only the measurable action at a distance - a superluminal propagation of real, physical information - would usually be considered in violation of the principle of locality by physicists. Such phenomena have never been seen, and they are not predicted by the current theories.

Relativity Locality is one of the axioms of relativistic quantum field theory, as required for causality. The formalization of locality in this case is as follows: if we have two observables, each localized within two distinct space-time regions which happen to be at a spacelike separation from each other, the observables must commute. Alternatively, a solution to the field equations is local if the underlying equations are either Lorentz invariant or, more generally, generally covariant or locally Lorentz invariant.

References [1] J-D. Bancal, S. Pironio, A. Acín, Y-C. Liang, V. Scarani & N. Gisin (Nature Physics, 2012). Quantum non-locality based on finite-speed causal influences leads to superluminal signalling (http:/ / www. nature. com/ nphys/ journal/ vaop/ ncurrent/ full/ nphys2460. html) [2] Berkovitz, Joseph (2008). "Action at a Distance in Quantum Mechanics" (http:/ / plato. stanford. edu/ archives/ win2008/ entries/ qm-action-distance/ #ActDisCoExiNonSepHol). In Edward N. Zalta. The Stanford Encyclopedia of Philosophy (Winter 2008 ed.). . [3] "Quantum Mechanics and Reality" ("Quanten-Mechanik und Wirklichkeit", Dialectica 2:320-324, 1948) [4] Norsen, T. - Against "Realism" (http:/ / arxiv. org/ abs/ quant-ph/ 0607057v2) [5] Ian Thomson's dispositional quantum mechanics (http:/ / www. generativescience. org/ ) [6] Ben Dov, Y. Local Realism and the Crucial experiment. (http:/ / bendov. info/ eng/ crucial. htm)

External links • Quantum nonlocality vs. Einstein locality (http://www.rzuser.uni-heidelberg.de/~as3/nonlocality.html) by H. D. Zeh

95

EPR Paradox

EPR Paradox The EPR paradox is an early and influential critique leveled against quantum mechanics. Albert Einstein and his colleagues Boris Podolsky and Nathan Rosen (known collectively as EPR) designed a thought experiment intended to reveal what they believed to be inadequacies of quantum mechanics. To that end they pointed to a consequence of quantum mechanics that its supporters had not noticed. According to quantum mechanics, under some conditions a pair of quantum systems may be described by a single wave function, which encodes the probabilities of the outcomes of experiments that may be performed on the two systems, whether jointly or individually. At the time the EPR article was written, it was known from experiments that the outcome of an experiment sometimes cannot be uniquely predicted. An example of such indeterminacy can be seen when a beam of light is incident on a half-silvered mirror. One half of the beam will reflect, the other will pass. But what happens when we keep decreasing the intensity of the beam, so that only one photon is in transit at any time? Half of the photons will pass and another half will be reflected. The routine explanation of this effect was, at that time, provided by Heisenberg's uncertainty principle. Physical quantities come in pairs which are called conjugate quantities. Example of such a conjugate pair are position and momentum of a particle, or components of spin measured around different axes. When one quantity was measured, and became determined, the conjugated quantity became indeterminate. Heisenberg explained this as a disturbance caused by measurement. The EPR paper, written in 1935, has shown that this explanation is inadequate. It considered two entangled particles, referred to as A and B, and pointed out that measuring a quantity of a particle A will cause the conjugated quantity of particle B to become undetermined, even if there was no contact, no classical disturbance. Heisenberg's principle was an attempt to provide a classical explanation of a quantum effect sometimes called non-locality. According to EPR there were two possible explanations. Either there was some interaction between the particles, even though they were separated, or the information about the outcome of all possible measurements was already present in both particles. The EPR authors preferred the second explanation according to which that information was encoded in some 'hidden parameters'. The first explanation, that an effect propagated instantly, across a distance, is in conflict with the theory of relativity. They then concluded that quantum mechanics was incomplete since, in its formalism, there was no space for such hidden parameters. Bell's theorem is generally understood to have demonstrated that their preferred explanation was not viable. Most physicists who have examined the matter concur that experiments, such as those of Alain Aspect and his group, have confirmed that physical probabilities, as predicted by quantum theory, do show the phenomena of Bell-inequality violations that are considered to invalidate EPR's preferred "local hidden-variables" type of explanation for the correlations that EPR first drew attention to.

History of EPR developments The article that first brought forth these matters, "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" was published in 1935.[1] Einstein struggled to the end of his life for a theory that could better comply with his idea of causality, protesting against the view that there exists no objective physical reality other than that which is revealed through measurement interpreted in terms of quantum mechanical formalism. However, since Einstein's death, experiments analogous to the one described in the EPR paper have been carried out, starting in 1976 by French scientists Lamehi-Rachti and Mittig[2] at the Saclay Nuclear Research Centre. These experiments appear to show that the local realism idea is false. [3]

96

EPR Paradox

Quantum mechanics and its interpretation Since the early twentieth century, quantum theory has proved to be successful in describing accurately the physical reality of the mesoscopic and microscopic world, in multiple reproducible physics experiments. Quantum mechanics was developed with the aim of describing atoms and explaining the observed spectral lines in a measurement apparatus. Although disputed, it has yet to be seriously challenged. Philosophical interpretations of quantum phenomena, however, are another matter: the question of how to interpret the mathematical formulation of quantum mechanics has given rise to a variety of different answers from people of different philosophical persuasions (see Interpretations of quantum mechanics). Quantum theory and quantum mechanics do not provide single measurement outcomes in a deterministic way. According to the understanding of quantum mechanics known as the Copenhagen interpretation, measurement causes an instantaneous collapse of the wave function describing the quantum system into an eigenstate of the observable state that was measured. Einstein characterized this imagined collapse in the 1927 Solvay Conference. He presented a thought experiment in which electrons are introduced through a small hole in a sphere whose inner surface serves as a detection screen. The electrons will contact the spherical detection screen in a widely dispersed manner. Those electrons, however, are all individually described by wave fronts that expand in all directions from the point of entry. A wave as it is understood in everyday life would paint a large area of the detection screen, but the electrons would be found to impact the screen at single points and would eventually form a pattern in keeping with the probabilities described by their identical wave functions. Einstein asks what makes each electron's wave front "collapse" at its respective location. Why do the electrons appear as single bright scintillations rather than as dim washes of energy across the surface? Why does any single electron appear at one point rather than some alternative point? The behavior of the electrons gives the impression of some signal having been sent to all possible points of contact that would have nullified all but one of them, or, in other words, would have preferentially selected a single point to the exclusion of all others.[4]

Einstein's opposition Einstein was the most prominent opponent of the Copenhagen interpretation. In his view, quantum mechanics is incomplete. Commenting on this, other writers (such as John von Neumann[5] and David Bohm[6]) have suggested that consequently there would have to be 'hidden' variables responsible for random measurement results, something which was not expressly claimed in the original paper. The 1935 EPR paper [7] condensed the philosophical discussion into a physical argument. The authors claim that given a specific experiment, in which the outcome of a measurement is known before the measurement takes place, there must exist something in the real world, an "element of reality", that determines the measurement outcome. They postulate that these elements of reality are local, in the sense that each belongs to a certain point in spacetime. Each element may only be influenced by events which are located in the backward light cone of its point in spacetime (i.e. the past). These claims are founded on assumptions about nature that constitute what is now known as local realism. Though the EPR paper has often been taken as an exact expression of Einstein's views, it was primarily authored by Podolsky, based on discussions at the Institute for Advanced Study with Einstein and Rosen. Einstein later expressed to Erwin Schrödinger that, "it did not come out as well as I had originally wanted; rather, the essential thing was, so to speak, smothered by the formalism."[8] In 1936 Einstein presented an individual account of his local realist ideas.[9]

97

EPR Paradox

Description of the paradox The original EPR paradox challenges the prediction of quantum mechanics that it is impossible to know both the position and the momentum of a quantum particle. This challenge can be extended to other pairs of physical properties.

EPR paper The original paper purports to describe what must happen to "two systems I and II, which we permit to interact ...", and, after some time, "we suppose that there is no longer any interaction between the two parts." In the words of Kumar (2009), the EPR description involves "two particles, A and B, [which] interact briefly and then move off in opposite directions."[10] According to Heisenberg's uncertainty principle, it is impossible to measure both the momentum and the position of particle B exactly. However, according to Kumar, it is possible to measure the exact position of particle A. By calculation, therefore, with the exact position of particle A known, the exact position of particle B can be known. Also, the exact momentum of particle B can be measured, so the exact momentum of particle A can be worked out. Kumar writes: "EPR argued that they had proved that ... [particle] B can have simultaneously exact values of position and momentum. ... Particle B has a position that is real and a momentum that is real." EPR appeared to have contrived a means to establish the exact values of either the momentum or the position of B due to measurements made on particle A, without the slightest possibility of particle B being physically disturbed.[11] EPR tried to set up a paradox to question the range of true application of Quantum Mechanics: Quantum theory predicts that both values cannot be known for a particle, and yet the EPR thought experiment purports to show that they must all have determinate values. The EPR paper says: "We are thus forced to conclude that the quantum-mechanical description of physical reality given by wave functions is not complete."[12] The EPR paper ends by saying: While we have thus shown that the wave function does not provide a complete description of the physical reality, we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible.

Measurements on an entangled state We have a source that emits electron–positron pairs, with the electron sent to destination A, where there is an observer named Alice, and the positron sent to destination B, where there is an observer named Bob. According to quantum mechanics, we can arrange our source so that each emitted pair occupies a quantum state called a spin singlet. The particles are thus said to be entangled. This can be viewed as a quantum superposition of two states, which we call state I and state II. In state I, the electron has spin pointing upward along the z-axis (+z) and the positron has spin pointing downward along the z-axis (−z). In state II, the electron has spin −z and the positron has spin +z. Therefore, it is impossible (without measuring) to know the definite state of spin of either particle in the spin singlet.

98

EPR Paradox

The EPR thought experiment, performed with electron–positron pairs. A source (center) sends particles toward two observers, electrons to Alice (left) and positrons to Bob (right), who can perform spin measurements.

Alice now measures the spin along the z-axis. She can obtain one of two possible outcomes: +z or −z. Suppose she gets +z. According to the Copenhagen interpretation of quantum mechanics, the quantum state of the system collapses into state I. The quantum state determines the probable outcomes of any measurement performed on the system. In this case, if Bob subsequently measures spin along the z-axis, there is 100% probability that he will obtain −z. Similarly, if Alice gets −z, Bob will get +z. There is, of course, nothing special about choosing the z-axis: according to quantum mechanics the spin singlet state may equally well be expressed as a superposition of spin states pointing in the x direction. Suppose that Alice and Bob had decided to measure spin along the x-axis. We'll call these states Ia and IIa. In state Ia, Alice's electron has spin +x and Bob's positron has spin −x. In state IIa, Alice's electron has spin −x and Bob's positron has spin +x. Therefore, if Alice measures +x, the system 'collapses' into state Ia, and Bob will get −x. If Alice measures −x, the system collapses into state IIa, and Bob will get +x. Whatever axis their spins are measured along, they are always found to be opposite. This can only be explained if the particles are linked in some way. Either they were created with a definite (opposite) spin about every axis—a "hidden variable" argument—or they are linked so that one electron "feels" which axis the other is having its spin measured along, and becomes its opposite about that one axis—an "entanglement" argument. Moreover, if the two particles have their spins measured about different axes, once the electron's spin has been measured about the x-axis (and the positron's spin about the x-axis deduced), the positron's spin about the y-axis will no longer be certain, as if (a) it knows that the measurement has taken place, or (b) it has a definite spin already, about a second axis—a hidden variable. However, it turns out that the predictions of Quantum Mechanics, which have been confirmed by experiment, cannot be explained by any hidden variable theory. This is demonstrated in Bell's theorem.[13] In quantum mechanics, the x-spin and z-spin are "incompatible observables", meaning there is a Heisenberg uncertainty principle operating between them: a quantum state cannot possess a definite value for both of these variables. Suppose Alice measures the z-spin and obtains +z, so that the quantum state collapses into state I. Now, instead of measuring the z-spin as well, Bob measures the x-spin. According to quantum mechanics, when the system is in state I, Bob's x-spin measurement will have a 50% probability of producing +x and a 50% probability of -x. It is impossible to predict which outcome will appear until Bob actually performs the measurement. Here is the crux of the matter. You might imagine that, when Bob measures the x-spin of his positron, he would get an answer with absolute certainty, since prior to this he hasn't disturbed his particle at all. But Bob's positron has a 50% probability of producing +x and a 50% probability of −x—so the outcome is not certain. Bob's positron "knows" that Alice's electron has been measured, and its z-spin detected, and hence B's z-spin calculated, so its x-spin is uncertain.

99

EPR Paradox Put another way, how does Bob's positron know which way to point if Alice decides (based on information unavailable to Bob) to measure x (i.e. to be the opposite of Alice's electron's spin about the x-axis) and also how to point if Alice measures z, since it is only supposed to know one thing at a time? The Copenhagen interpretation rules that say the wave function "collapses" at the time of measurement, so there must be action at a distance (entanglement) or the positron must know more than it's supposed to (hidden variables). Here is the paradox summed up: It is one thing to say that physical measurement of the first particle's momentum affects uncertainty in its own position, but to say that measuring the first particle's momentum affects the uncertainty in the position of the other is another thing altogether. Einstein, Podolsky and Rosen asked how can the second particle "know" to have precisely defined momentum but uncertain position? Since this implies that one particle is communicating with the other instantaneously across space, i.e. faster than light, this is the "paradox". Incidentally, Bell used spin as his example, but many types of physical quantities—referred to as "observables" in quantum mechanics—can be used. The EPR paper used momentum for the observable. Experimental realisations of the EPR scenario often use photon polarization, because polarized photons are easy to prepare and measure.

Locality in the EPR experiment The principle of locality states that physical processes occurring at one place should have no immediate effect on the elements of reality at another location. At first sight, this appears to be a reasonable assumption to make, as it seems to be a consequence of special relativity, which states that information can never be transmitted faster than the speed of light without violating causality. It is generally believed that any theory which violates causality would also be internally inconsistent, and thus useless. It turns out that the usual rules for combining quantum mechanical and classical descriptions violate the principle of locality without violating causality. Causality is preserved because there is no way for Alice to transmit messages (i.e. information) to Bob by manipulating her measurement axis. Whichever axis she uses, she has a 50% probability of obtaining "+" and 50% probability of obtaining "−", completely at random; according to quantum mechanics, it is fundamentally impossible for her to influence what result she gets. Furthermore, Bob is only able to perform his measurement once: there is a fundamental property of quantum mechanics, known as the "no cloning theorem", which makes it impossible for him to make a million copies of the electron he receives, perform a spin measurement on each, and look at the statistical distribution of the results. Therefore, in the one measurement he is allowed to make, there is a 50% probability of getting "+" and 50% of getting "−", regardless of whether or not his axis is aligned with Alice's. However, the principle of locality appeals powerfully to physical intuition, and Einstein, Podolsky and Rosen were unwilling to abandon it. Einstein derided the quantum mechanical predictions as "spooky action at a distance". The conclusion they drew was that quantum mechanics is not a complete theory. In recent years, however, doubt has been cast on EPR's conclusion due to developments in understanding locality and especially quantum decoherence. The word locality has several different meanings in physics. For example, in quantum field theory "locality" means that quantum fields at different points of space do not interact with one another. However, quantum field theories that are "local" in this sense appear to violate the principle of locality as defined by EPR, but they nevertheless do not violate locality in a more general sense. Wavefunction collapse can be viewed as an epiphenomenon of quantum decoherence, which in turn is nothing more than an effect of the underlying local time evolution of the wavefunction of a system and all of its environment. Since the underlying behaviour doesn't violate local causality, it follows that neither does the additional effect of wavefunction collapse, whether real or apparent. Therefore, as outlined in the example above, neither the EPR experiment nor any quantum experiment demonstrates that faster-than-light signaling is possible.

100

EPR Paradox

Resolving the paradox Hidden variables There are several ways to resolve the EPR paradox. The one suggested by EPR is that quantum mechanics, despite its success in a wide variety of experimental scenarios, is actually an incomplete theory. In other words, there is some yet undiscovered theory of nature to which quantum mechanics acts as a kind of statistical approximation (albeit an exceedingly successful one). Unlike quantum mechanics, the more complete theory contains variables corresponding to all the "elements of reality". There must be some unknown mechanism acting on these variables to give rise to the observed effects of "non-commuting quantum observables", i.e. the Heisenberg uncertainty principle. Such a theory is called a hidden variable theory. To illustrate this idea, we can formulate a very simple hidden variable theory for the above thought experiment. One supposes that the quantum spin-singlet states emitted by the source are actually approximate descriptions for "true" physical states possessing definite values for the z-spin and x-spin. In these "true" states, the electron going to Bob always has spin values opposite to the electron going to Alice, but the values are otherwise completely random. For example, the first pair emitted by the source might be "(+z, −x) to Alice and (−z, +x) to Bob", the next pair "(−z, −x) to Alice and (+z, +x) to Bob", and so forth. Therefore, if Bob's measurement axis is aligned with Alice's, he will necessarily get the opposite of whatever Alice gets; otherwise, he will get "+" and "−" with equal probability. Assuming we restrict our measurements to the z- and x-axes, such a hidden variable theory is experimentally indistinguishable from quantum mechanics. In reality, there may be an infinite number of axes along which Alice and Bob can perform their measurements, so there would have to be an infinite number of independent hidden variables. However, this is not a serious problem; we have formulated a very simplistic hidden variable theory, and a more sophisticated theory might be able to patch it up. It turns out that there is a much more serious challenge to the idea of hidden variables. Bell's inequality In 1964, John Bell showed that the predictions of quantum mechanics in the EPR thought experiment are significantly different from the predictions of a particular class of hidden variable theories (the local hidden variable theories). Roughly speaking, quantum mechanics has a much stronger statistical correlation with measurement results performed on different axes than do these hidden variable theories. These differences, expressed using inequality relations known as "Bell's inequalities", are in principle experimentally detectable. Later work by Eberhard showed that the key properties of local hidden variable theories which lead to Bell's inequalities are locality and counter-factual definiteness. Any theory in which these principles apply produces the inequalities. Arthur Fine subsequently showed that any theory satisfying the inequalities can be modeled by a local hidden variable theory. After the publication of Bell's paper, a variety of experiments were devised to test Bell's inequalities (experiments which generally rely on photon polarization measurement). All the experiments conducted to date have found behavior in line with the predictions of standard quantum mechanics theory. However, Bell's theorem does not apply to all possible philosophically realist theories. It is a common misconception that quantum mechanics is inconsistent with all notions of philosophical realism, but realist interpretations of quantum mechanics are possible, although, as discussed above, such interpretations must reject either locality or counter-factual definiteness. Mainstream physics prefers to keep locality, while striving also to maintain a notion of realism that nevertheless rejects counter-factual definiteness. Examples of such mainstream realist interpretations are the consistent histories interpretation and the transactional interpretation. Fine's work showed that, taking locality as a given, there exist scenarios in which two statistical variables are correlated in a manner inconsistent with counter-factual definiteness, and that such scenarios are no more mysterious than any other, despite the inconsistency with counter-factual definiteness seeming 'counter-intuitive'.

101

EPR Paradox Violation of locality is difficult to reconcile with special relativity, and is thought to be incompatible with the principle of causality. On the other hand the Bohm interpretation of quantum mechanics keeps counter-factual definiteness while introducing a conjectured non-local mechanism in form of the 'quantum potential', defined as one of the terms of the Schrödinger equation. Some workers in the field have also attempted to formulate hidden variable theories that exploit loopholes in actual experiments, such as the assumptions made in interpreting experimental data, although no theory has been proposed that can reproduce all the results of quantum mechanics. There are also individual EPR-like experiments that have no local hidden variables explanation. Examples have been suggested by David Bohm and by Lucien Hardy.

Einstein's hope for a purely algebraic theory The Bohm interpretation of quantum mechanics hypothesizes that the state of the universe evolves smoothly through time with no collapsing of quantum wavefunctions. One problem for the Copenhagen interpretation is to precisely define wavefunction collapse. Einstein maintained that quantum mechanics is physically incomplete and logically unsatisfactory. In "The Meaning of Relativity," Einstein wrote, "One can give good reasons why reality cannot at all be represented by a continuous field. From the quantum phenomena it appears to follow with certainty that a finite system of finite energy can be completely described by a finite set of numbers (quantum numbers). This does not seem to be in accordance with a continuum theory and must lead to an attempt to find a purely algebraic theory for the representation of reality. But nobody knows how to find the basis for such a theory." If time, space, and energy are secondary features derived from a substrate below the Planck scale, then Einstein's hypothetical algebraic system might resolve the EPR paradox (although Bell's theorem would still be valid). Edward Fredkin in the Fredkin Finite Nature Hypothesis has suggested an informational basis for Einstein's hypothetical algebraic system. If physical reality is totally finite, then the Copenhagen interpretation might be an approximation to an information processing system below the Planck scale.

"Acceptable theories" and the experiment According to the present view of the situation, quantum mechanics flatly contradicts Einstein's philosophical postulate that any acceptable physical theory must fulfill "local realism". In the EPR paper (1935) the authors realised that quantum mechanics was inconsistent with their assumptions, but Einstein nevertheless thought that quantum mechanics might simply be augmented by hidden variables (i.e. variables which were, at that point, still obscure to him), without any other change, to achieve an acceptable theory. He pursued these ideas for over twenty years until the end of his life, in 1955. In contrast, John Bell, in his 1964 paper, showed that quantum mechanics and the class of hidden variable theories Einstein favored[14] would lead to different experimental results: different by a factor of 3⁄2 for certain correlations. So the issue of "acceptability", up to that time mainly concerning theory, finally became experimentally decidable. There are many Bell test experiments, e.g. those of Alain Aspect and others. They support the predictions of quantum mechanics rather than the class of hidden variable theories supported by Einstein.[15] According to Karl Popper these experiments showed that the class of "hidden variables" Einstein believed in is erroneous.

Implications for quantum mechanics Most physicists today believe that quantum mechanics is correct, and that the EPR paradox is a "paradox" only because classical intuitions do not correspond to physical reality. How EPR is interpreted regarding locality depends on the interpretation of quantum mechanics one uses. In the Copenhagen interpretation, it is usually understood that instantaneous wave function collapse does occur. However, the view that there is no causal instantaneous effect has also been proposed within the Copenhagen interpretation: in this alternate view, measurement affects our ability to define (and measure) quantities in the physical system, not the system itself. In the many-worlds interpretation locality is strictly preserved, since the effects of operations such as measurement affect only the state of the particle

102

EPR Paradox

103

that is measured. However, the results of the measurement are not unique—every possible result is obtained. The EPR paradox has deepened our understanding of quantum mechanics by exposing the fundamentally non-classical characteristics of the measurement process. Prior to the publication of the EPR paper, a measurement was often visualized as a physical disturbance inflicted directly upon the measured system. For instance, when measuring the position of an electron, one imagines shining a light on it, thus disturbing the electron and producing the quantum mechanical uncertainties in its position. Such explanations, which are still encountered in popular expositions of quantum mechanics, are debunked by the EPR paradox, which shows that a "measurement" can be performed on a particle without disturbing it directly, by performing a measurement on a distant entangled particle. In fact, Yakir Aharonov and his collaborators have developed a whole theory of so-called Weak measurement. Technologies relying on quantum entanglement are now being developed. In quantum cryptography, entangled particles are used to transmit signals that cannot be eavesdropped upon without leaving a trace. In quantum computation, entangled quantum states are used to perform computations in parallel, which may allow certain calculations to be performed much more quickly than they ever could be with classical computers.

Mathematical formulation The above discussion can be expressed mathematically using the quantum mechanical formulation of spin. The spin degree of freedom for an electron is associated with a two-dimensional complex Hilbert space H, with each quantum state corresponding to a vector in that space. The operators corresponding to the spin along the x, y, and z direction, denoted Sx, Sy, and Sz respectively, can be represented using the Pauli matrices:

where

stands for Planck's constant divided by 2π.

The eigenstates of Sz are represented as

and the eigenstates of Sx are represented as

The Hilbert space of the electron pair is

, the tensor product of the two electrons' Hilbert spaces. The spin

singlet state is

where the two terms on the right hand side are what we have referred to as state I and state II above. From the above equations, it can be shown that the spin singlet can also be written as

where the terms on the right hand side are what we have referred to as state Ia and state IIa. To illustrate how this leads to the violation of local realism, we need to show that after Alice's measurement of Sz (or Sx), Bob's value of Sz (or Sx) is uniquely determined, and therefore corresponds to an "element of physical reality". This follows from the principles of measurement in quantum mechanics. When Sz is measured, the system state ψ collapses into an eigenvector of Sz. If the measurement result is +z, this means that immediately after measurement the system state undergoes an orthogonal projection of ψ onto the space of states of the form

EPR Paradox For the spin singlet, the new state is

Similarly, if Alice's measurement result is −z, the system undergoes an orthogonal projection onto

which means that the new state is

This implies that the measurement for Sz for Bob's electron is now determined. It will be −z in the first case or +z in the second case. It remains only to show that Sx and Sz cannot simultaneously possess definite values in quantum mechanics. One may show in a straightforward manner that no possible vector can be an eigenvector of both matrices. More generally, one may use the fact that the operators do not commute,

along with the Heisenberg uncertainty relation

References Selected papers • • • • •

A. Aspect, Bell's inequality test: more ideal than ever, Nature 398 189 (1999). [16] J.S. Bell, On the Einstein–Poldolsky–Rosen paradox [17], Physics 1 195-200 (1964). P.H. Eberhard, Bell's theorem without hidden variables. Nuovo Cimento 38B1 75 (1977). P.H. Eberhard, Bell's theorem and the different concepts of locality. Nuovo Cimento 46B 392 (1978). A. Einstein, B. Podolsky, and N. Rosen, Can quantum-mechanical description of physical reality be considered complete? [18] Phys. Rev. 47 777 (1935). [7]

• A. Fine, Hidden Variables, Joint Probability, and the Bell Inequalities. Phys. Rev. Lett. 48, 291 (1982).[19] • A. Fine, Do Correlations need to be explained?, in Philosophical Consequences of Quantum Theory: Reflections on Bell's Theorem, edited by Cushing & McMullin (University of Notre Dame Press, 1986). • L. Hardy, Nonlocality for two particles without inequalities for almost all entangled states. Phys. Rev. Lett. 71 1665 (1993).[20] • M. Mizuki, A classical interpretation of Bell's inequality. Annales de la Fondation Louis de Broglie 26 683 (2001). • P. Pluch, "Theory for Quantum Probability", PhD Thesis University of Klagenfurt (2006) • M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. Itano, C. Monroe and D. J. Wineland, Experimental violation of a Bell's inequality with efficient detection, Nature 409, 791–794 (15 February 2001). [21] • M. Smerlak, C. Rovelli, Relational EPR [22]

104

EPR Paradox Notes [1] Einstein, A; B Podolsky, N Rosen (1935-05-15). "Can Quantum-Mechanical Description of Physical Reality be Considered Complete?". Physical Review 47 (10): 777–780. Bibcode 1935PhRv...47..777E. doi:10.1103/PhysRev.47.777. [2] Advances in atomic and molecular physics, Volume 14 By David Robert Bates (http:/ / books. google. com. au/ books?id=dkaCKHKLo3gC& pg=PA330& lpg=PA330& dq="Saclay"+ "Bell's+ inequality"& source=bl& ots=u-b4s3klA0& sig=1P7sX78b-I9TKtT15KvRSADgLlo& hl=en& ei=VJ7aTpn-FMW8iAeJs-jsDQ& sa=X& oi=book_result& ct=result& resnum=2& ved=0CCEQ6AEwAQ#v=onepage& q="Saclay" "Bell's inequality"& f=false) [3] Gribbin, J (1984). In Search of Schrödinger's cat. Black Swan. ISBN 0-7045-3071-6.. [4] http:/ / plato. stanford. edu/ entries/ qt-epr/ [5] von Neumann, J. (1932/1955). In Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, translated into English by Beyer, R.T., Princeton University Press, Princeton, cited by Baggott, J. (2004) Beyond Measure: Modern physics, philosophy, and the meaning of quantum theory, Oxford University Press, Oxford, ISBN 0-19-852927-9, pages 144–145. [6] Bohm, D. (1951). Quantum Theory (http:/ / books. google. com. au/ books?id=9DWim3RhymsC& printsec=frontcover& dq=david+ bohm+ quantum+ theory& source=bl& ots=6G-2u1wtav& sig=Q1GcoVDLFRmKOmDYFAJte6LzrZU& hl=en& ei=Pv45TNSnLYffcfnS6foO& sa=X& oi=book_result& ct=result& resnum=7& ved=0CEEQ6AEwBg#v=onepage& q& f=false), Prentice-Hall, Englewood Cliffs, page 29, and Chapter 5 section 3, and Chapter 22 Section 19. [7] http:/ / prola. aps. org/ abstract/ PR/ v47/ i10/ p777_1 [8] Quoted in Kaiser, David. "Bringing the human actors back on stage: the personal context of the Einstein–Bohr debate," British Journal for the History of Science 27 (1994): 129–152, on page 147. [9] See "Physics and Reality," originally published in vol. 221, No. 1323—27 of Journal of the Franklin Institute, pp. 313–347, with the Jean Piccard translation starting p.380. The English translation can be downloaded, with different pagination, from: www.kostic.niu.edu/Physics and Reality-Albert Einstein.pdf. The relevant section appears on pp. 371–379. [10] Kumar, M., Quantum, Icon Books, 2009, p. 305. [11] Kumar, M., Quantum, Icon Books, 2009, p. 305–6. [12] Kumar, M., Quantum, Icon Books, 2009, p. 306. [13] George Greenstein and Arthur G. Zajonc, The Quantum Challenge, p. "[Experiments in the early 1980s] have conclusively shown that quantum mechanics is indeed orrect, and that the EPR argument had relied upon incorrect assumptions." [14] Clearing up mysteries: the original goal (http:/ / bayes. wustl. edu/ etj/ articles/ cmystery. pdf). . [15] Aspect A (1999-03-18). "Bell’s inequality test: more ideal than ever" (http:/ / www-ece. rice. edu/ ~kono/ ELEC565/ Aspect_Nature. pdf). Nature 398 (6724): 189–90. Bibcode 1999Natur.398..189A. doi:10.1038/18296. . Retrieved 2010-09-08. [16] http:/ / www-ece. rice. edu/ ~kono/ ELEC565/ Aspect_Nature. pdf [17] http:/ / www. drchinese. com/ David/ Bell_Compact. pdf [18] http:/ / www. drchinese. com/ David/ EPR. pdf [19] http:/ / prola. aps. org/ abstract/ PRL/ v48/ i5/ p291_1 [20] http:/ / prola. aps. org/ abstract/ PRL/ v71/ i11/ p1665_1 [21] http:/ / www. nature. com/ nature/ journal/ v409/ n6822/ full/ 409791a0. html [22] http:/ / arxiv. org/ abs/ quant-ph/ 0604064

Books • John S. Bell (1987) Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press. ISBN 0-521-36869-3. • Arthur Fine (1996) The Shaky Game: Einstein, Realism and the Quantum Theory, 2nd ed. Univ. of Chicago Press. • J.J. Sakurai, J. J. (1994) Modern Quantum Mechanics. Addison-Wesley: 174–187, 223–232. ISBN 0-201-53929-2. • Selleri, F. (1988) Quantum Mechanics Versus Local Realism: The Einstein–Podolsky–Rosen Paradox. New York: Plenum Press. ISBN 0-306-42739-7 • Leon Lederman, L., Teresi, D. (1993). The God Particle: If the Universe is the Answer, What is the Question? Houghton Mifflin Company, pages 21, 187 to 189. • John Gribbin (1984) In Search of Schrödinger's Cat. Black Swan. ISBN 978-0-552-12555-0

105

EPR Paradox

External links • The Einstein–Podolsky–Rosen Argument in Quantum Theory; 1.2 The argument in the text; http://plato.stanford.edu/entries/qt-epr/#1.2 • The original EPR paper. (http://prola.aps.org/abstract/PR/v47/i10/p777_1) • Stanford Encyclopedia of Philosophy: " The Einstein–Podolsky–Rosen Argument in Quantum Theory (http:// plato.stanford.edu/entries/qt-epr/)" by Arthur Fine. • Abner Shimony (2004) " Bell’s Theorem. (http://plato.stanford.edu/entries/bell-theorem/)" • EPR, Bell & Aspect: The Original References. (http://www.drchinese.com/David/EPR_Bell_Aspect.htm) • Does Bell's Inequality Principle rule out local theories of quantum mechanics? (http://math.ucr.edu/home/ baez/physics/Quantum/bells_inequality.html) From the Usenet Physics FAQ. • Theoretical use of EPR in teleportation. (http://www.research.ibm.com/journal/rd/481/brassard.html) • Effective use of EPR in cryptography. (http://www.dhushara.com/book/quantcos/aq/qcrypt.htm) • EPR experiment with single photons interactive. (http://www.QuantumLab.de) • Spooky Actions At A Distance?: Oppenheimer Lecture by Prof. Mermin. (http://www.youtube.com/ watch?v=ta09WXiUqcQ)

Bell's Theorem Bell's theorem is a no-go theorem famous for drawing an important line in the sand between quantum mechanics (QM) and the world as we know it classically. In its simplest form, Bell's theorem states:[1] No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics. When introduced in 1927, the philosophical implications of the new quantum theory were troubling to many prominent physicists of the day, including Albert Einstein. In a well known 1935 paper, he and co-authors Boris Podolsky and Nathan Rosen (collectively EPR) demonstrated by a paradox that QM was incomplete. This provided hope that a more complete (and less troubling) theory might one day be discovered. But that conclusion rested on the seemingly reasonable assumptions of locality and realism (together called "local realism" or "local hidden variables", often interchangeably). In the vernacular of Einstein: locality meant no instantaneous ("spooky") action at a distance; realism meant the moon is there even when not being observed. These assumptions were hotly debated within the physics community, notably with Nobel laureates Einstein on one side and Niels Bohr on the other. In his groundbreaking 1964 paper, "On the Einstein Podolsky Rosen paradox", physicist John Stewart Bell presented an analogy (based on spin measurements on pairs of entangled electrons) to EPR's hypothetical paradox. Using their reasoning, he said, a choice of measurement setting here should not affect the outcome of a measurement there (and vice versa). After providing a mathematical formulation of locality and realism based on this, he showed specific cases where this would be inconsistent with the predictions of QM. In experimental tests following Bell's example, now using quantum entanglement of photons instead of electrons, Alain Aspect et al. (1981) convincingly demonstrated that the predictions of QM are correct in this regard. While this does not demonstrate QM is complete, one is forced to reject either locality or realism (or both). That a relatively simple and elegant theorem could lead to this result has led Henry Stapp to call this theorem "the most profound in science".

106

Bell's Theorem

107

Overview Bell’s theorem states that the concept of local realism, favoured by Einstein[2], yields predictions that disagree with those of quantum mechanical theory. Because numerous experiments agree with the predictions of quantum mechanical theory, and show correlations that are, according to Bell, greater than could be explained by local hidden variables, the experimental results have been taken by many as refuting the concept of local realism as an explanation of the physical phenomena under test. For a hidden variable theory, if Bell's conditions are correct, then the results which are in agreement with quantum mechanical theory appear to evidence superluminal effects, in contradiction to the principle of locality. The theorem applies to any quantum system of two entangled qubits. The most common examples concern systems of particles that are entangled in spin or polarization. Following the argument in the Einstein–Podolsky–Rosen (EPR) paradox paper (but using the example of spin, as in David Bohm's version of the EPR Illustration of Bell test for particles such as photons. A source produces a singlet argument[3][4]), Bell considered an pair, one particle is sent to one location, and the other is sent to another location. A experiment in which there are "a pair of spin measurement of the entangled property is performed at various angles at each location. one-half particles formed somehow in the singlet spin state and moving freely in opposite directions."[3] The two particles travel away from each other to two distant locations, at which measurements of spin are performed, along axes that are independently chosen. Each measurement yields a result of either spin-up (+) or spin-down (−). The probability of the same result being obtained at the two locations varies, depending on the relative angles at which the two spin measurements are made, and is subject to some uncertainty for all relative angles other than perfectly parallel alignments (0° or 180°). Bell's theorem thus applies only to the statistical results from many trials of the experiment. Symbolically, the correlation between results for a single pair can be represented as either "+1" for a match (opposite spins), or "−1" for a non-match. While measuring the spin of these entangled particles along parallel axes will always result in opposite (i.e., perfectly anticorrelated) results, measurement at perpendicular directions will have a 50% chance of matching (i.e., will have a 50% probability of an uncorrelated result). These basic cases are illustrated in the table below. Same axis

Pair 1 Pair 2 Pair 3 Pair 4 … Pair n

Alice, 0°

+

−

−

+

… +

Bob, 0°

-

+

+

-

… -

Correlation: (

+1

+1

+1

+1

… +1

) / n = +1 (100% identical)

Orthogonal axes Pair 1 Pair 2 Pair 3 Pair 4 … Pair n Alice, 0°

+

−

+

−

… −

Bob, 90°

−

−

+

+

… −

Correlation (

+1

-1

-1

+1

… −1

)/n=0 (50% identical)

Bell's Theorem

With the measurements oriented at intermediate angles between these basic cases, the existence of local hidden variables would imply a linear variation in the correlation. However, according to quantum mechanical theory, the correlation varies as the cosine of the angle. Experimental results match the curve predicted by quantum mechanics.[5] Bell achieved his breakthrough by first deriving the results that he posits local realism would necessarily yield. Bell The local realist prediction (solid lines) for quantum correlation for spin (assuming claimed that, without making any 100% detector efficiency). The quantum mechanical prediction is the dotted assumptions about the specific form of the (cosine) curve. theory beyond requirements of basic consistency, the mathematical inequality he discovered was clearly at odds with the results (described above) predicted by quantum mechanics and, later, observed experimentally. If correct, Bell's theorem appears to rule out local hidden variables as a viable explanation of quantum mechanics (though it still leaves the door open for non-local hidden variables). Bell concluded: In a theory in which parameters are added to quantum mechanics to determine the results of individual measurements, without changing the statistical predictions, there must be a mechanism whereby the setting of one measuring device can influence the reading of another instrument, however remote. Moreover, the signal involved must propagate instantaneously, so that a theory could not be Lorentz invariant. —[3] Over the years, Bell's theorem has undergone a wide variety of experimental tests. However, various common deficiencies in the testing of the theorem have been identified, including the detection loophole[6] and the communication loophole.[6] Over the years experiments have been gradually improved to better address these loopholes, but no experiment to date has simultaneously fully addressed all of them.[6] However, it is generally considered unreasonable that such an experiment, if conducted, would give results that are inconsistent with the prior experiments. For example, Anthony Leggett has commented: [While] no single existing experiment has simultaneously blocked all of the so-called ‘‘loopholes’’, each one of those loopholes has been blocked in at least one experiment. Thus, to maintain a local hidden variable theory in the face of the existing experiments would appear to require belief in a very peculiar conspiracy of nature.[7] To date, Bell's theorem is generally regarded as supported by a substantial body of evidence and is treated as a fundamental principle of physics in mainstream quantum mechanics textbooks.[8][9]

Importance of the theorem Bell's theorem, derived in his seminal 1964 paper titled On the Einstein Podolsky Rosen paradox,[3] has been called, on the assumption that the theory is correct, "the most profound in science".[10] Perhaps of equal importance is Bell's deliberate effort to encourage and bring legitimacy to work on the completeness issues, which had fallen into disrepute.[11] Later in his life, Bell expressed his hope that such work would "continue to inspire those who suspect that what is proved by the impossibility proofs is lack of imagination."[12] The title of Bell's seminal article refers to the famous paper by Einstein, Podolsky and Rosen[13] that challenged the completeness of quantum mechanics. In his paper, Bell started from the same two assumptions as did EPR, namely

108

Bell's Theorem (i) reality (that microscopic objects have real properties determining the outcomes of quantum mechanical measurements), and (ii) locality (that reality in one location is not influenced by measurements performed simultaneously at a distant location). Bell was able to derive from those two assumptions an important result, namely Bell's inequality, implying that at least one of the assumptions must be false. In two respects Bell's 1964 paper was a step forward compared to the EPR paper: firstly, it considered more hidden variables than merely the element of physical reality in the EPR paper; and Bell's inequality was, in part, liable to be experimentally tested, thus raising the possibility of testing the local realism hypothesis. Limitations on such tests to date are noted below. Whereas Bell's paper deals only with deterministic hidden variable theories, Bell's theorem was later generalized to stochastic theories[14] as well, and it was also realised[15] that the theorem is not so much about hidden variables as about the outcomes of measurements which could have been done instead of the one actually performed. Existence of these variables is called the assumption of realism, or the assumption of counterfactual definiteness. After the EPR paper, quantum mechanics was in an unsatisfactory position: either it was incomplete, in the sense that it failed to account for some elements of physical reality, or it violated the principle of a finite propagation speed of physical effects. In a modified version of the EPR thought experiment, two hypothetical observers, now commonly referred to as Alice and Bob, perform independent measurements of spin on a pair of electrons, prepared at a source in a special state called a spin singlet state. It is the conclusion of EPR that once Alice measures spin in one direction (e.g. on the x axis), Bob's measurement in that direction is determined with certainty, as being the opposite outcome to that of Alice, whereas immediately before Alice's measurement Bob's outcome was only statistically determined (i.e., was only a probability, not a certainty); thus, either the spin in each direction is an element of physical reality, or the effects travel from Alice to Bob instantly. In QM, predictions are formulated in terms of probabilities — for example, the probability that an electron will be detected in a particular place, or the probability that its spin is up or down. The idea persisted, however, that the electron in fact has a definite position and spin, and that QM's weakness is its inability to predict those values precisely. The possibility existed that some unknown theory, such as a hidden variables theory, might be able to predict those quantities exactly, while at the same time also being in complete agreement with the probabilities predicted by QM. If such a hidden variables theory exists, then because the hidden variables are not described by QM the latter would be an incomplete theory. Two assumptions drove the desire to find a local realist theory: 1. Objects have a definite state that determines the values of all other measurable properties, such as position and momentum. 2. Effects of local actions, such as measurements, cannot travel faster than the speed of light (in consequence of special relativity). Thus if observers are sufficiently far apart, a measurement made by one can have no effect on a measurement made by the other. In the form of local realism used by Bell, the predictions of the theory result from the application of classical probability theory to an underlying parameter space. By a simple argument based on classical probability, he showed that correlations between measurements are bounded in a way that is violated by QM. Bell's theorem seemed to put an end to local realism. This is because, if the theorem is correct, then either quantum mechanics or local realism is wrong, as they are mutually exclusive. The paper noted that "it requires little imagination to envisage the experiments involved actually being made",[3] to determine which of them is correct. It took many years and many improvements in technology to perform tests along the lines Bell envisaged. The tests are, in theory, capable of showing whether local hidden variable theories as envisaged by Bell accurately predict experimental results. The tests are not capable of determining whether Bell has accurately described all local hidden variable theories. The Bell test experiments have been interpreted as showing that the Bell inequalities are violated in favour of QM. The no-communication theorem shows that the observers cannot use the effect to communicate (classical)

109

Bell's Theorem

110

information to each other faster than the speed of light, but the ‘fair sampling’ and ‘no enhancement’ assumptions require more careful consideration (below). That interpretation follows not from any clear demonstration of super-luminal communication in the tests themselves, but solely from Bell's theory that the correctness of the quantum predictions necessarily precludes any local hidden-variable theory. If that theoretical contention is not correct, then the "tests" of Bell's theory to date do not show anything either way about the local or non-local nature of the phenomena.

Bell inequalities Bell inequalities concern measurements made by observers on pairs of particles that have interacted and then separated. According to quantum mechanics they are entangled, while local realism would limit the correlation of subsequent measurements of the particles. Different authors subsequently derived inequalities similar to Bell´s original inequality, and these are here collectively termed Bell inequalities. All Bell inequalities describe experiments in which the predicted result from quantum entanglement differs from that flowing from local realism. The inequalities assume that each quantum-level object has a well-defined state that accounts for all its measurable properties and that distant objects do not exchange information faster than the speed of light. These well-defined states are typically called hidden variables, the properties that Einstein posited when he stated his famous objection to quantum mechanics: "God does not play dice." Bell showed that under quantum mechanics, the mathematics of which contains no local hidden variables, the Bell inequalities can nevertheless be violated: the properties of a particle are not clear, but may be correlated with those of another particle due to quantum entanglement, allowing their state to be well defined only after a measurement is made on either particle. That restriction agrees with the Heisenberg uncertainty principle, a fundamental concept in quantum mechanics. In Bell's words: Theoretical physicists live in a classical world, looking out into a quantum-mechanical world. The latter we describe only subjectively, in terms of procedures and results in our classical domain. (…) Now nobody knows just where the boundary between the classical and the quantum domain is situated. (…) More plausible to me is that we will find that there is no boundary. The wave functions would prove to be a provisional or incomplete description of the quantum-mechanical part. It is this possibility, of a homogeneous account of the world, which is for me the chief motivation of the study of the so-called "hidden variable" possibility. (…) A second motivation is connected with the statistical character of quantum-mechanical predictions. Once the incompleteness of the wave function description is suspected, it can be conjectured that random statistical fluctuations are determined by the extra "hidden" variables — "hidden" because at this stage we can only conjecture their existence and certainly cannot control them. (…) A third motivation is in the peculiar character of some quantum-mechanical predictions, which seem almost to cry out for a hidden variable interpretation. This is the famous argument of Einstein, Podolsky and Rosen. (…) We will find, in fact, that no local deterministic hidden-variable theory can reproduce all the experimental predictions of quantum mechanics. This opens the possibility of bringing the question into the experimental domain, by trying to approximate as well as possible the idealized situations in which local hidden variables and quantum mechanics cannot agree.[16] In probability theory, repeated measurements of system properties can be regarded as repeated sampling of random variables. In Bell's experiment, Alice can choose a detector setting to measure either or and Bob can choose a detector setting to measure either

or

. Measurements of Alice and Bob may be somehow

correlated with each other, but the Bell inequalities say that if the correlation stems from local random variables, there is a limit to the amount of correlation one might expect to see.

Bell's Theorem

111

Original Bell's inequality The original inequality that Bell derived was:[3]

where C is the "correlation" of the particle pairs and a, b and c settings of the apparatus. This inequality is not used in practice. For one thing, it is true only for genuinely "two-outcome" systems, not for the "three-outcome" ones (with possible outcomes of zero as well as +1 and −1) encountered in real experiments. For another, it applies only to a very restricted set of hidden variable theories, namely those for which the outcomes on both sides of the experiment are always exactly anticorrelated when the analysers are parallel, in agreement with the quantum mechanical prediction. A simple limit of Bell's inequality has the virtue of being completely intuitive. If the result of three different statistical coin-flips A, B, and C have the property that: 1. A and B are the same (both heads or both tails) 99% of the time 2. B and C are the same 99% of the time then A and C are the same at least 98% of the time. The number of mismatches between A and B (1/100) plus the number of mismatches between B and C (1/100) are together the maximum possible number of mismatches between A and C (a simple Boole–Fréchet inequality). In quantum mechanics, by letting A, B, and C be the values of the spin of two entangled particles measured relative to some axis at 0 degrees, θ degrees, and 2θ degrees respectively, the overlap of the wavefunction between the different angles is proportional to . The probability that A and B give the same answer is , where is proportional to θ. This is also the probability that B and C give the same answer. But A and C are the same 1 − (2ε)2 of the time. Choosing the angle so that , A and B are 99% correlated, B and C are 99% correlated and A and C are only 96% correlated. Imagine that two entangled particles in a spin singlet are shot out to two distant locations, and the spins of both are measured in the direction A. The spins are 100% correlated (actually, anti-correlated but for this argument that is equivalent). The same is true if both spins are measured in directions B or C. It is safe to conclude that any hidden variables that determine the A,B, and C measurements in the two particles are 100% correlated and can be used interchangeably. If A is measured on one particle and B on the other, the correlation between them is 99%. If B is measured on one and C on the other, the correlation is 99%. This allows us to conclude that the hidden variables determining A and B are 99% correlated and B and C are 99% correlated. But if A is measured in one particle and C in the other, the results are only 96% correlated, which is a contradiction. The intuitive formulation is due to David Mermin, while the small-angle limit is emphasized in Bell's original article.

CHSH inequality In addition to Bell's original inequality,[3] the form given by John Clauser, Michael Horne, Abner Shimony and R. A. Holt,[17] (the CHSH form) is especially important,[17] as it gives classical limits to the expected correlation for the above experiment conducted by Alice and Bob:

where C denotes correlation. Correlation of observables X, Y is defined as

Where

represents the expected or average value of

This is a non-normalized form of the correlation coefficient considered in statistics (see Quantum correlation). To formulate Bell's theorem, we formalize local realism as follows:

Bell's Theorem

112

1. There is a probability space

and the observed outcomes by both Alice and Bob result by random sampling of

the parameter . 2. The values observed by Alice or Bob are functions of the local detector settings and the hidden parameter only. Thus • Value observed by Alice with detector setting is • Value observed by Bob with detector setting is Implicit in assumption 1) above, the hidden parameter space random variable X on with respect to is written

has a probability measure and the expectation of a

where for accessibility of notation we assume that the probability measure has a density. Bell's inequality. The CHSH inequality (1) holds under the hidden variables assumptions above. For simplicity, let us first assume the observed values are +1 or −1; we remove this assumption in Remark 1 below. Let

. Then at least one of

is 0. Thus

and therefore

Remark 1 The correlation inequality (1) still holds if the variables

,

are allowed to take on any real values

between −1 and +1. Indeed, the relevant idea is that each summand in the above average is bounded above by 2. This is easily seen as true in the more general case:

To justify the upper bound 2 asserted in the last inequality, without loss of generality, we can assume that

In that case

Bell's Theorem

113

Remark 2 Though the important component of the hidden parameter in Bell's original proof is associated with the source and is shared by Alice and Bob, there may be others that are associated with the separate detectors, these others being conditionally independent given the first, and with conditional probability distributions only depending on the corresponding local setting (if dependent on the settings at all). This argument was used by Bell in 1971, and again by Clauser and Horne in 1974,[14] to justify a generalisation of the theorem forced on them by the real experiments, in which detectors were never 100% efficient. The derivations were given in terms of the averages of the outcomes over the local detector variables. The formalisation of local realism was thus effectively changed, replacing A and B by averages and retaining the symbol but with a slightly different meaning. It was henceforth restricted (in most theoretical work) to mean only those components that were associated with the source. However, with the extension proved in Remark 1, CHSH inequality still holds even if the instruments themselves contain hidden variables. In that case, averaging over the instrument hidden variables gives new variables:

on

, which still have values in the range [−1, +1] to which we can apply the previous result.

Bell inequalities are violated by quantum mechanical predictions In the usual quantum mechanical formalism, the observables X and Y are represented as self-adjoint operators on a Hilbert space. To compute the correlation, assume that X and Y are represented by matrices in a finite dimensional space and that X and Y commute; this special case suffices for our purposes below. The von Neumann measurement postulate states: a series of measurements of an observable X on a series of identical systems in state produces a distribution of real values. By the assumption that observables are finite matrices, this distribution is discrete. The probability of observing λ is non-zero if and only if λ is an eigenvalue of the matrix X and moreover the probability is

where EX (λ) is the projector corresponding to the eigenvalue λ. The system state immediately after the measurement is

From this, we can show that the correlation of commuting observables X and Y in a pure state

is

We apply this fact in the context of the EPR paradox. The measurements performed by Alice and Bob are spin measurements on electrons. Alice can choose between two detector settings labelled a and a′; these settings correspond to measurement of spin along the z or the x axis. Bob can choose between two detector settings labelled b and b′; these correspond to measurement of spin along the z′ or x′ axis, where the x′ – z′ coordinate system is rotated 135° relative to the x – z coordinate system. The spin observables are represented by the 2 × 2 self-adjoint matrices:

These are the Pauli spin matrices normalized so that the corresponding eigenvalues are +1, −1. As is customary, we denote the eigenvectors of Sx by Let be the spin singlet state for a pair of electrons discussed in the EPR paradox. This is a specially constructed state described by the following vector in the tensor product

Now let us apply the CHSH formalism to the measurements that can be performed by Alice and Bob.

Bell's Theorem

114

Illustration of Bell test for spin 1/2 particles. Source produces spin singlet pairs, one particle of each pair is sent to Alice and the other to Bob. Each performs one of the two spin measurements.

The operators

,

correspond to Bob's spin measurements along x′ and z′. Note that the A operators

commute with the B operators, so we can apply our calculation for the correlation. In this case, we can show that the CHSH inequality fails. In fact, a straightforward calculation shows that

and

so that

Bell's Theorem: If the quantum mechanical formalism is correct, then the system consisting of a pair of entangled electrons cannot satisfy the principle of local realism. Note that is indeed the upper bound for quantum mechanics called Tsirelson's bound. The operators giving this maximal value are always isomorphic to the Pauli matrices.

Bell's Theorem

115

Practical experiments testing Bell's theorem Experimental tests can determine whether the Bell inequalities required by local realism hold up to the empirical evidence.

Scheme of a "two-channel" Bell test The source S produces pairs of "photons", sent in opposite directions. Each photon encounters a two-channel polariser whose orientation (a or b) can be set by the experimenter. Emerging signals from each channel are detected and coincidences of four types (++, −−, +− and −+) counted by the coincidence monitor.

Bell's inequalities are tested by "coincidence counts" from a Bell test experiment such as the optical one shown in the diagram. Pairs of particles are emitted as a result of a quantum process, analysed with respect to some key property such as polarisation direction, then detected. The setting (orientations) of the analysers are selected by the

experimenter. Bell test experiments to date overwhelmingly violate Bell's inequality. Indeed, a table of Bell test experiments performed prior to 1986 is given in 4.5 of Redhead, 1987.[18] Of the thirteen experiments listed, only two reached results contradictory to quantum mechanics; moreover, according to the same source, when the experiments were repeated, "the discrepancies with QM could not be reproduced". Nevertheless, the issue is not conclusively settled. According to Shimony's 2004 Stanford Encyclopedia overview article:[6] Most of the dozens of experiments performed so far have favored Quantum Mechanics, but not decisively because of the 'detection loopholes' or the 'communication loophole.' The latter has been nearly decisively blocked by a recent experiment and there is a good prospect for blocking the former. To explore the 'detection loophole', one must distinguish the classes of homogeneous and inhomogeneous Bell inequality. The standard assumption in Quantum Optics is that "all photons of given frequency, direction and polarization are identical" so that photodetectors treat all incident photons on an equal basis. Such a fair sampling assumption generally goes unacknowledged, yet it effectively limits the range of local theories to those that conceive of the light field as corpuscular. The assumption excludes a large family of local realist theories, in particular, Max Planck's description. We must remember the cautionary words of Albert Einstein[19] shortly before he died: "Nowadays every Tom, Dick and Harry ('jeder Kerl' in German original) thinks he knows what a photon is, but he is mistaken". Those who maintain the concept of duality, or simply of light being a wave, recognize the possibility or actuality that the emitted atomic light signals have a range of amplitudes and, furthermore, that the amplitudes are modified when the signal passes through analyzing devices such as polarizers and beam splitters. It follows that not all signals have the same detection probability.[20]

Two classes of Bell inequalities The fair sampling problem was faced openly in the 1970s. In early designs of their 1973 experiment, Freedman and Clauser[21] used fair sampling in the form of the Clauser-Horne-Shimony-Holt (CHSH[17]) hypothesis. However, shortly afterwards Clauser and Horne[14] made the important distinction between inhomogeneous (IBI) and homogeneous (HBI) Bell inequalities. Testing an IBI requires that we compare certain coincidence rates in two separated detectors with the singles rates of the two detectors. Nobody needed to perform the experiment, because singles rates with all detectors in the 1970s were at least ten times all the coincidence rates. So, taking into account

Bell's Theorem this low detector efficiency, the QM prediction actually satisfied the IBI. To arrive at an experimental design in which the QM prediction violates IBI we require detectors whose efficiency exceeds 82% for singlet states, but have very low dark rate and short dead and resolving times. This is well above the 30% achievable[22] so Shimony’s optimism in the Stanford Encyclopedia, quoted in the preceding section, appears over-stated.

Practical challenges Because detectors don't detect a large fraction of all photons, Clauser and Horne[14] recognized that testing Bell's inequality requires some extra assumptions. They introduced the No Enhancement Hypothesis (NEH): A light signal, originating in an atomic cascade for example, has a certain probability of activating a detector. Then, if a polarizer is interposed between the cascade and the detector, the detection probability cannot increase. Given this assumption, there is a Bell inequality between the coincidence rates with polarizers and coincidence rates without polarizers. The experiment was performed by Freedman and Clauser,[21] who found that the Bell's inequality was violated. So the no-enhancement hypothesis cannot be true in a local hidden variables model. The Freedman-Clauser experiment reveals that local hidden variables imply the new phenomenon of signal enhancement: In the total set of signals from an atomic cascade there is a subset whose detection probability increases as a result of passing through a linear polarizer. This is perhaps not surprising, as it is known that adding noise to data can, in the presence of a threshold, help reveal hidden signals (this property is known[23] as stochastic resonance). One cannot conclude that this is the only local-realist alternative to Quantum Optics, but it does show that the word loophole is biased. Moreover, the analysis leads us to recognize that the Bell-inequality experiments, rather than showing a breakdown of realism or locality, are capable of revealing important new phenomena.

Theoretical challenges Most advocates of the hidden variables idea believe that experiments have ruled out local hidden variables. They are ready to give up locality, explaining the violation of Bell's inequality by means of a non-local hidden variable theory, in which the particles exchange information about their states. This is the basis of the Bohm interpretation of quantum mechanics, which requires that all particles in the universe be able to instantaneously exchange information with all others. A 2007 experiment ruled out a large class of non-Bohmian non-local hidden variable theories.[24] If the hidden variables can communicate with each other faster than light, Bell's inequality can easily be violated. Once one particle is measured, it can communicate the necessary correlations to the other particle. Since in relativity the notion of simultaneity is not absolute, this is unattractive. One idea is to replace instantaneous communication with a process that travels backwards in time along the past Light cone. This is the idea behind a transactional interpretation of quantum mechanics, which interprets the statistical emergence of a quantum history as a gradual coming to agreement between histories that go both forward and backward in time.[25] A few advocates of deterministic models have not given up on local hidden variables. For example, Gerard 't Hooft has argued that the superdeterminism loophole cannot be dismissed.[26][27] The quantum mechanical wavefunction can also provide a local realistic description, if the wavefunction values are interpreted as the fundamental quantities that describe reality. Such an approach is called a many-worlds interpretation of quantum mechanics. In this view, two distant observers both split into superpositions when measuring a spin. The Bell inequality violations are no longer counterintuitive, because it is not clear which copy of the observer B observer A will see when going to compare notes. If reality includes all the different outcomes, locality in physical space (not outcome space) places no restrictions on how the split observers can meet up.

116

Bell's Theorem This implies that there is a subtle assumption in the argument that realism is incompatible with quantum mechanics and locality. The assumption, in its weakest form, is called counterfactual definiteness. This states that if the results of an experiment are always observed to be definite, there is a quantity that determines what the outcome would have been even if you don't do the experiment. Many worlds interpretations are not only counterfactually indefinite, they are factually indefinite. The results of all experiments, even ones that have been performed, are not uniquely determined. E. T. Jaynes[28] pointed out two hidden assumptions in Bell Inequality that could limit its generality. According to him: 1. Bell interpreted conditional probability P(X|Y) as a causal inference, i.e. Y exerted a causal inference on X in reality. However, P(X|Y) actually only means logical inference (deduction). Causes cannot travel faster than light or backward in time, but deduction can. 2. Bell's inequality does not apply to some possible hidden variable theories. It only applies to a certain class of local hidden variable theories. In fact, it might have just missed the kind of hidden variable theories that Einstein is most interested in. However Jaynes later admitted that he had misunderstood Bell's argument.

Final remarks The violations of Bell's inequalities, due to quantum entanglement, just provide the definite demonstration of something that was already strongly suspected, that quantum physics cannot be represented by any version of the classical picture of physics.[29] Some earlier elements that had seemed incompatible with classical pictures included apparent complementarity and (hypothesized) wavefunction collapse. Complementarity is now seen not as an independent ingredient of the quantum picture but rather as a direct consequence of the Quantum decoherence expected from the quantum formalism itself. The possibility of wavefunction collapse is now seen as one possible problematic ingredient of some interpretations, rather than as an essential part of quantum mechanics. The Bell violations show that no resolution of such issues can avoid the ultimate strangeness of quantum behavior.[30] The EPR paper "pinpointed" the unusual properties of the entangled states, e.g. the above-mentioned singlet state, which is the foundation for present-day applications of quantum physics, such as quantum cryptography; one application involves the measurement of quantum entanglement as a physical source of bits for Rabin's oblivious transfer protocol. This strange non-locality was originally supposed to be a Reductio ad absurdum, because the standard interpretation could easily do away with action-at-a-distance by simply assigning to each particle definite spin-states. Bell's theorem showed that the "entangledness" prediction of quantum mechanics has a degree of non-locality that cannot be explained away by any local theory. In well-defined Bell experiments (see the paragraph on "test experiments") one can now falsify either quantum mechanics or Einstein's quasi-classical assumptions: currently many experiments of this kind have been performed, and the experimental results support quantum mechanics, though some believe that detectors give a biased sample of photons, so that until nearly every photon pair generated is observed there will be loopholes. What is powerful about Bell's theorem is that it doesn't refer to any particular physical theory. What makes Bell's theorem unique and powerful is that it shows that nature violates the most general assumptions behind classical pictures, not just details of some particular models. No combination of local deterministic and local random variables can reproduce the phenomena predicted by quantum mechanics and repeatedly observed in experiments. [31]

117

Bell's Theorem

Notes [1] C.B. Parker (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). McGraw Hill. p. 542. ISBN 0-07-051400-3. [2] C.B. Parker (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). McGraw Hill. p. 541. ISBN 0-07-051400-3. [3] Bell, John (1964). "On the Einstein Podolsky Rosen Paradox" (http:/ / www. drchinese. com/ David/ Bell_Compact. pdf). Physics 1 (3): 195–200. . [4] Bohm, David Quantum Theory. Prentice-Hall, 1951. [5] C.B. Parker (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). McGraw Hill. p. 542. ISBN 0-07-051400-3. [6] Article on Bell's Theorem (http:/ / plato. stanford. edu/ entries/ bell-theorem) by Abner Shimony in the Stanford Encyclopedia of Philosophy, (2004). [7] Leggett, Anthony (2003). "Nonlocal Hidden-Variable Theories and Quantum Mechanics: An Incompatibility Theorem". Foundations of Physics 33 (10): 1469-1493. [8] Griffiths, David J. (1998). Introduction to Quantum Mechanics (2nd ed.). Pearson/Prentice Hall. pp. 423. [9] Merzbacher, Eugene (2005). Quantum Mechanics (3rd ed.). John Wiley & Sons. pp. 18, 362. [10] Stapp, 1975 [11] Bell, JS, "On the impossible pilot wave." Foundations of Physics (1982) 12:989-99. Reprinted in Speakable and unspeakable in quantum mechanics: collected papers on quantum philosophy. CUP, 2004, p. 160. [12] Bell, JS, "On the impossible pilot wave." Foundations of Physics (1982) 12:989-99. Reprinted in Speakable and unspeakable in quantum mechanics: collected papers on quantum philosophy. CUP, 2004, p. 161. [13] Einstein, A.; Podolsky, B.; Rosen, N. (1935). "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?". Physical Review 47 (10): 777. Bibcode 1935PhRv...47..777E. doi:10.1103/PhysRev.47.777. [14] Clauser, John F. (1974). "Experimental consequences of objective local theories". Physical Review D 10 (2): 526. Bibcode 1974PhRvD..10..526C. doi:10.1103/PhysRevD.10.526. [15] Eberhard, P. H. (1977). "Bell's theorem without hidden variables". Nuovo Cimento B 38: 75–80. Bibcode 1977NCimB..38...75E. doi:10.1007/BF02726212. [16] Bell, JS, Speakable and unspeakable in quantum mechanics: Introduction remarks at Naples-Amalfi meeting., 1984. Reprinted in Speakable and unspeakable in quantum mechanics: collected papers on quantum philosophy. CUP, 2004, p. 29. [17] Clauser, John; Horne, Michael; Shimony, Abner; Holt, Richard (1969). "Proposed Experiment to Test Local Hidden-Variable Theories". Physical Review Letters 23 (15): 880. Bibcode 1969PhRvL..23..880C. doi:10.1103/PhysRevLett.23.880. [18] M. Redhead, Incompleteness, Nonlocality and Realism, Clarendon Press (1987) [19] A. Einstein in Correspondance Einstein–Besso, p.265 (Herman, Paris, 1979) [20] Marshall and Santos, Semiclassical optics as an alternative to nonlocality (http:/ / www. crisisinphysics. co. uk/ optrev. pdf) Recent Research Developments in Optics 2:683-717 (2002) ISBN 81-7736-140-6 [21] Freedman, Stuart J.; Clauser, John F. (1972). "Experimental Test of Local Hidden-Variable Theories". Physical Review Letters 28 (14): 938. Bibcode 1972PhRvL..28..938F. doi:10.1103/PhysRevLett.28.938. [22] Giorgio Brida; Marco Genovese; Marco Gramegna; Fabrizio Piacentini; Enrico Predazzi; Ivano Ruo-Berchera (2007). "Experimental tests of hidden variable theories from dBB to Stochastic Electrodynamics". Journal of Physics: Conference Series 67 (12047): 012047. arXiv:quant-ph/0612075. Bibcode 2007JPhCS..67a2047G. doi:10.1088/1742-6596/67/1/012047. [23] Gammaitoni, Luca; Hänggi, Peter; Jung, Peter; Marchesoni, Fabio (1998). "Stochastic resonance". Reviews of Modern Physics 70: 223. Bibcode 1998RvMP...70..223G. doi:10.1103/RevModPhys.70.223. [24] Gröblacher, Simon; Paterek, Tomasz; Kaltenbaek, Rainer; Brukner, Časlav; Żukowski, Marek; Aspelmeyer, Markus; Zeilinger, Anton (2007). "An experimental test of non-local realism". Nature 446 (7138): 871–5. arXiv:0704.2529. Bibcode 2007Natur.446..871G. doi:10.1038/nature05677. PMID 17443179. [25] Cramer, John (1986). "The transactional interpretation of quantum mechanics". Reviews of Modern Physics 58 (3): 647. Bibcode 1986RvMP...58..647C. doi:10.1103/RevModPhys.58.647. [26] Gerard 't Hooft (2009). "Entangled quantum states in a local deterministic theory". arXiv:0908.3408 [quant-ph]. [27] Gerard 't Hooft (2007). "The Free-Will Postulate in Quantum Mechanics". arXiv:quant-ph/0701097 [quant-ph]. [28] Jaynes, E. T. (1989). "Clearing up Mysteries--The Original Goal" (http:/ / bayes. wustl. edu/ etj/ articles/ cmystery. pdf). Maximum Entropy and Bayesian Methods: 12. . [29] Roger Penrose (2007). The Road to Reality. Vintage books. p. 583. ISBN 0-679-77631-1. [30] E. Abers (2004). Quantum Mechanics. Addison Wesley. p. 193-195. ISBN 9780131461000. [31] R.G. Lerner, G.L. Trigg (1991). Encyclopaedia of Physics (2nd ed.). VHC publishers. p. 495. ISBN 3-527-26954-1 (Verlagsgesellschaft) 0-89573-752-3 (VHC Inc.).

118

Bell's Theorem

References • A. Aspect et al., Experimental Tests of Realistic Local Theories via Bell's Theorem, Phys. Rev. Lett. 47, 460 (1981) • A. Aspect et al., Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A New Violation of Bell's Inequalities, Phys. Rev. Lett. 49, 91 (1982). • A. Aspect et al., Experimental Test of Bell's Inequalities Using Time-Varying Analyzers, Phys. Rev. Lett. 49, 1804 (1982). • A. Aspect and P. Grangier, About resonant scattering and other hypothetical effects in the Orsay atomic-cascade experiment tests of Bell inequalities: a discussion and some new experimental data, Lettere al Nuovo Cimento 43, 345 (1985) • B. D'Espagnat, The Quantum Theory and Reality (http://www.sciam.com/media/pdf/197911_0158.pdf), Scientific American, 241, 158 (1979) • J. S. Bell, On the problem of hidden variables in quantum mechanics, Rev. Mod. Phys. 38, 447 (1966) • J. S. Bell, On the Einstein Podolsky Rosen Paradox, Physics 1, 3, 195-200 (1964) • J. S. Bell, Introduction to the hidden variable question, Proceedings of the International School of Physics 'Enrico Fermi', Course IL, Foundations of Quantum Mechanics (1971) 171–81 • J. S. Bell, Bertlmann’s socks and the nature of reality, Journal de Physique, Colloque C2, suppl. au numero 3, Tome 42 (1981) pp C2 41–61 • J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press 1987) [A collection of Bell's papers, including all of the above.] • J. F. Clauser and A. Shimony, Bell's theorem: experimental tests and implications, Reports on Progress in Physics 41, 1881 (1978) • J. F. Clauser and M. A. Horne, Phys. Rev D 10, 526–535 (1974) • E. S. Fry, T. Walther and S. Li, Proposal for a loophole-free test of the Bell inequalities, Phys. Rev. A 52, 4381 (1995) • E. S. Fry, and T. Walther, Atom based tests of the Bell Inequalities — the legacy of John Bell continues, pp 103–117 of Quantum [Un]speakables, R.A. Bertlmann and A. Zeilinger (eds.) (Springer, Berlin-Heidelberg-New York, 2002) • R. B. Griffiths, Consistent Quantum Theory', Cambridge University Press (2002). • L. Hardy, Nonlocality for 2 particles without inequalities for almost all entangled states. Physical Review Letters 71 (11) 1665–1668 (1993) • M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2000) • P. Pearle, Hidden-Variable Example Based upon Data Rejection, Physical Review D 2, 1418–25 (1970) • A. Peres, Quantum Theory: Concepts and Methods, Kluwer, Dordrecht, 1993. • P. Pluch, Theory of Quantum Probability, PhD Thesis, University of Klagenfurt, 2006. • B. C. van Frassen, Quantum Mechanics, Clarendon Press, 1991. • M.A. Rowe, D. Kielpinski, V. Meyer, C.A. Sackett, W.M. Itano, C. Monroe, and D.J. Wineland, Experimental violation of Bell's inequalities with efficient detection,(Nature, 409, 791–794, 2001). • S. Sulcs, The Nature of Light and Twentieth Century Experimental Physics, Foundations of Science 8, 365–391 (2003) • S. Gröblacher et al., An experimental test of non-local realism,(Nature, 446, 871–875, 2007). • D. N. Matsukevich, P. Maunz, D. L. Moehring, S. Olmschenk, and C. Monroe, Bell Inequality Violation with Two Remote Atomic Qubits, Phys. Rev. Lett. 100, 150404 (2008). • The comic Dilbert, by Scott Adams, refers to Bell's Theorem in the 1992-09-21 (http://www.dilbert.com/strips/ comic/1992-09-21/) and 1992-09-22 (http://www.dilbert.com/strips/comic/1992-09-22/) strips.

119

Bell's Theorem

Further reading The following are intended for general audiences. • • • • • • • • • •

Amir D. Aczel, Entanglement: The greatest mystery in physics (Four Walls Eight Windows, New York, 2001). A. Afriat and F. Selleri, The Einstein, Podolsky and Rosen Paradox (Plenum Press, New York and London, 1999) J. Baggott, The Meaning of Quantum Theory (Oxford University Press, 1992) N. David Mermin, "Is the moon there when nobody looks? Reality and the quantum theory", in Physics Today, April 1985, pp. 38–47. Louisa Gilder, The Age of Entanglement: When Quantum Physics Was Reborn (New York: Alfred A. Knopf, 2008) Brian Greene, The Fabric of the Cosmos (Vintage, 2004, ISBN 0-375-72720-5) Nick Herbert, Quantum Reality: Beyond the New Physics (Anchor, 1987, ISBN 0-385-23569-0) D. Wick, The infamous boundary: seven decades of controversy in quantum physics (Birkhauser, Boston 1995) R. Anton Wilson, Prometheus Rising (New Falcon Publications, 1997, ISBN 1-56184-056-4) Gary Zukav "The Dancing Wu Li Masters" (Perennial Classics, 2001, ISBN 0-06-095968-1)

External links • An explanation of Bell's Theorem (http://www.ncsu.edu/felder-public/kenny/papers/bell.html), based on N. D. Mermin's article, Mermin, N. D. (1981). "Bringing home the atomic world: Quantum mysteries for anybody". American Journal of Physics 49 (10): 940. Bibcode 1981AmJPh..49..940M. doi:10.1119/1.12594. • Mermin: Spooky Actions At A Distance? Oppenheimer Lecture (http://www.youtube.com/ watch?v=ta09WXiUqcQ) • Quantum Entanglement (http://www.ipod.org.uk/reality/reality_entangled.asp) Includes a simple explanation of Bell's Inequality. • Bell's theorem on arXiv.org (http://xstructure.inr.ac.ru/x-bin/theme3.py?level=2&index1=369244) • Interactive experiments with single photons: entanglement and Bell´s theorem (http://www.didaktik.physik. uni-erlangen.de/quantumlab/english/index.html) • Bell's Inequalities: Obscurantist Obfuscation or Condign Confabulation? (http://groups.google.com/groups/ profile?hl=en&show=more&enc_user=8YcXCQ4AAABUc-oUoA1Uy7yFEaUY6YXQ&group=sci.physics)

120

121

5. Schrödinger's Objections Schrödinger's Cat Schrödinger's cat is a thought experiment, sometimes described as a paradox, devised by Austrian physicist Erwin Schrödinger in 1935. It illustrates what he saw as the problem of the Copenhagen interpretation of quantum mechanics applied to everyday objects, resulting in a contradiction with common sense. The scenario presents a cat that might be alive or dead, depending on an earlier random event. Schrödinger's Cat: A cat, a flask of poison and a radioactive source are placed in a sealed Although the original "experiment" was box. If an internal monitor detects radioactivity (i.e. a single atom decaying), the flask is imaginary, similar principles have been shattered, releasing the poison that kills the cat. There is a supposed fifty-percent chance researched and used in practical of this happening. The Copenhagen interpretation of quantum mechanics implies that after a while, the cat is simultaneously alive and dead. Yet, when we look in the box, we applications. The thought experiment is see the cat either alive or dead, not both alive and dead. This poses the question of when also often featured in theoretical exactly superposition ends and reality collapses into one possibility or the other. discussions of the interpretations of quantum mechanics. In the course of developing this experiment, Schrödinger coined the term Verschränkung (entanglement).

Origin and motivation Schrödinger intended his thought experiment as a discussion of the EPR article—named after its authors Einstein, Podolsky, and Rosen—in 1935.[1] The EPR article highlighted the strange nature of quantum entanglement, which is a characteristic of a quantum state that is a combination of the states of two systems (for example, two subatomic particles), that once interacted but were then separated and are not each in a definite state. The Copenhagen interpretation implies that the state of the two systems undergoes collapse into a definite state when one of the systems is measured. Schrödinger and Einstein exchanged letters about Einstein's EPR article, in the course of which Einstein pointed out that the state of an unstable keg of gunpowder will, after a while, contain a superposition of both exploded and unexploded states. To further illustrate, Schrödinger describes how one could, in principle, transpose the superposition of an atom to large-scale systems. He proposed a scenario with a cat in a sealed box, wherein the cat's life or death depended on the state of a subatomic particle. According to Schrödinger, the Copenhagen interpretation implies that the cat remains both alive and dead (to the universe outside the box) until the box is opened. Schrödinger did not wish to promote the idea of dead-and-alive cats as a serious possibility; quite the reverse, the paradox is a classic reductio ad absurdum.[2] The thought experiment illustrates quantum mechanics and the mathematics necessary to describe quantum states. Intended as a critique of just the Copenhagen interpretation (the prevailing orthodoxy in 1935), the Schrödinger cat thought experiment remains a typical touchstone for limited interpretations of quantum mechanics. Physicists often use the way each interpretation deals with Schrödinger's cat as a way of illustrating and comparing

Schrödinger's Cat the particular features, strengths, and weaknesses of each interpretation.

The thought experiment Schrödinger wrote:[3][2] One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter, there is a tiny bit of radioactive substance, so small that perhaps in the course of the hour, one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges, and through a relay releases a hammer that shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts. It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy, which can then be resolved by direct observation. That prevents us from so naively accepting as valid a "blurred model" for representing reality. In itself, it would not embody anything unclear or contradictory. There is a difference between a shaky or out-of-focus photograph and a snapshot of clouds and fog banks. —Erwin Schrödinger, Die gegenwärtige Situation in der Quantenmechanik (The present situation in quantum mechanics), Naturwissenschaften (translated by John D. Trimmer in Proceedings of the American Philosophical Society) Schrödinger's famous thought experiment poses the question, when does a quantum system stop existing as a superposition of states and become one or the other? (More technically, when does the actual quantum state stop being a linear combination of states, each of which resembles different classical states, and instead begins to have a unique classical description?) If the cat survives, it remembers only being alive. But explanations of the EPR experiments that are consistent with standard microscopic quantum mechanics require that macroscopic objects, such as cats and notebooks, do not always have unique classical descriptions. The thought experiment illustrates this apparent paradox. Our intuition says that no observer can be in a mixture of states—yet the cat, it seems from the thought experiment, can be such a mixture. Is the cat required to be an observer, or does its existence in a single well-defined classical state require another external observer? Each alternative seemed absurd to Albert Einstein, who was impressed by the ability of the thought experiment to highlight these issues. In a letter to Schrödinger dated 1950, he wrote: You are the only contemporary physicist, besides Laue, who sees that one cannot get around the assumption of reality, if only one is honest. Most of them simply do not see what sort of risky game they are playing with reality—reality as something independent of what is experimentally established. Their interpretation is, however, refuted most elegantly by your system of radioactive atom + amplifier + charge of gunpowder + cat in a box, in which the psi-function of the system contains both the cat alive and blown to bits. Nobody really doubts that the presence or absence of the cat is something independent of the act of observation.[4] Note that the charge of gunpowder is not mentioned in Schrödinger's setup, which uses a Geiger counter as an amplifier and hydrocyanic poison instead of gunpowder. The gunpowder had been mentioned in Einstein's original suggestion to Schrödinger 15 years before, and apparently Einstein had carried it forward to the present discussion.

122

Schrödinger's Cat

123

Interpretations of the experiment Since Schrödinger's time, other interpretations of quantum mechanics have been proposed that give different answers to the questions posed by Schrödinger's cat of how long superpositions last and when (or whether) they collapse.

Copenhagen interpretation The most commonly held interpretation of quantum mechanics is the Copenhagen interpretation.[5] In the Copenhagen interpretation, a system stops being a superposition of states and becomes either one or the other when an observation takes place. This experiment makes apparent the fact that the nature of measurement, or observation, is not well-defined in this interpretation. The experiment can be interpreted to mean that while the box is closed, the system simultaneously exists in a superposition of the states "decayed nucleus/dead cat" and "undecayed nucleus/living cat," and that only when the box is opened and an observation performed does the wave function collapse into one of the two states. However, one of the main scientists associated with the Copenhagen interpretation, Niels Bohr, never had in mind the observer-induced collapse of the wave function, so that Schrödinger's Cat did not pose any riddle to him. The cat would be either dead or alive long before the box is opened by a conscious observer.[6] Analysis of an actual experiment found that measurement alone (for example by a Geiger counter) is sufficient to collapse a quantum wave function before there is any conscious observation of the measurement.[7] The view that the "observation" is taken when a particle from the nucleus hits the detector can be developed into objective collapse theories. The thought experiment requires an "unconscious observation" by the detector in order for magnification to occur. In contrast, the many worlds approach denies that collapse ever occurs.

Many-worlds interpretation and consistent histories In 1957, Hugh Everett formulated the many-worlds interpretation of quantum mechanics, which does not single out observation as a special process. In the many-worlds interpretation, both alive and dead states of the cat persist after the box is opened, but are decoherent from each other. In other words, when the box is opened, the observer and the already-dead cat split into an observer looking at a box with a dead cat, and an observer looking at a box with a live cat. But since the dead and alive states are decoherent, there is no effective communication or interaction between them.

The quantum-mechanical "Schrödinger's cat" paradox according to the many-worlds interpretation. In this interpretation, every event is a branch point. The cat is both alive and dead—regardless of whether the box is opened—but the "alive" and "dead" cats are in different branches of the universe that are equally real but cannot interact with each other.

When opening the box, the observer becomes entangled with the cat, so "observer states" corresponding to the cat's being alive and dead are formed; each observer state is entangled or linked with the cat so that the "observation of the cat's state" and the "cat's state" correspond with each other. Quantum decoherence ensures that the different outcomes have no interaction with each other. The same mechanism of quantum decoherence is also important for the interpretation in terms of consistent histories. Only the "dead cat" or "alive cat" can be a part of a consistent history in this interpretation.

Schrödinger's Cat Roger Penrose criticises this: "I wish to make it clear that, as it stands, this is far from a resolution of the cat paradox. For there is nothing in the formalism of quantum mechanics that demands that a state of consciousness cannot involve the simultaneous perception of a live and a dead cat",[8] However, the mainstream view (without necessarily endorsing many-worlds) is that decoherence is the mechanism that forbids such simultaneous perception.[9][10] A variant of the Schrödinger's Cat experiment, known as the quantum suicide machine, has been proposed by cosmologist Max Tegmark. It examines the Schrödinger's Cat experiment from the point of view of the cat, and argues that by using this approach, one may be able to distinguish between the Copenhagen interpretation and many-worlds.

Ensemble interpretation The ensemble interpretation states that superpositions are nothing but subensembles of a larger statistical ensemble. The state vector would not apply to individual cat experiments, but only to the statistics of many similarly prepared cat experiments. Proponents of this interpretation state that this makes the Schrödinger's Cat paradox a trivial non-issue. This interpretation serves to discard the idea that a single physical system in quantum mechanics has a mathematical description that corresponds to it in any way.

Relational interpretation The relational interpretation makes no fundamental distinction between the human experimenter, the cat, or the apparatus, or between animate and inanimate systems; all are quantum systems governed by the same rules of wavefunction evolution, and all may be considered "observers." But the relational interpretation allows that different observers can give different accounts of the same series of events, depending on the information they have about the system.[11] The cat can be considered an observer of the apparatus; meanwhile, the experimenter can be considered another observer of the system in the box (the cat plus the apparatus). Before the box is opened, the cat, by nature of it being alive or dead, has information about the state of the apparatus (the atom has either decayed or not decayed); but the experimenter does not have information about the state of the box contents. In this way, the two observers simultaneously have different accounts of the situation: To the cat, the wavefunction of the apparatus has appeared to "collapse"; to the experimenter, the contents of the box appear to be in superposition. Not until the box is opened, and both observers have the same information about what happened, do both system states appear to "collapse" into the same definite result, a cat that is either alive or dead.

Objective collapse theories According to objective collapse theories, superpositions are destroyed spontaneously (irrespective of external observation) when some objective physical threshold (of time, mass, temperature, irreversibility, etc.) is reached. Thus, the cat would be expected to have settled into a definite state long before the box is opened. This could loosely be phrased as "the cat observes itself," or "the environment observes the cat." Objective collapse theories require a modification of standard quantum mechanics to allow superpositions to be destroyed by the process of time evolution.

124

Schrödinger's Cat

Applications and tests The experiment as described is a purely theoretical one, and the machine proposed is not known to have been constructed. However, successful experiments involving similar principles, e.g. superpositions of relatively large (by the standards of quantum physics) objects have been performed.[12] These experiments do not show that a cat-sized object can be superposed, but the known upper limit on "cat states" has been pushed upwards by them. In many cases the state is short-lived, even when cooled to near absolute zero. • A "cat state" has been achieved with photons.[13] • A beryllium ion has been trapped in a superposed state.[14] • An experiment involving a superconducting quantum interference device ("SQUID") has been linked to theme of the thought experiment: " The superposition state does not correspond to a billion electrons flowing one way and a billion others flowing the other way. Superconducting electrons move en masse. All the superconducting electrons in the SQUID flow both ways around the loop at once when they are in the Schrödinger’s cat state.".[15] • A piezoelectric "tuning fork" has been constructed, which can be placed into a superposition of vibrating and non vibrating states. The resonator comprises about 10 trillion atoms.[16] • An experiment involving a flu virus has been proposed.[17] In quantum computing the phrase "cat state" often refers to the special entanglement of qubits wherein the qubits are in an equal superposition of all being 0 and all being 1; e.g.,

Extensions Wigner's friend is a variant on the experiment with two external observers: the first opens and inspects the box and then communicates his observations to a second observer. The issue here is, does the wave function "collapse" when the first observer opens the box, or only when the second observer is informed of the first observer's observations? In another extension, prominent physicists have gone so far as to suggest that astronomers observing dark energy in the universe in 1998 may have "reduced its life expectancy" through a pseudo-Schrödinger's Cat scenario, although this is a controversial viewpoint.[18][19]

References [1] EPR article: Can Quantum-Mechanical Description Reality Be Considered Complete? (http:/ / prola. aps. org/ abstract/ PR/ v47/ i10/ p777_1) [2] Schrödinger, Erwin (November 1935). "Die gegenwärtige Situation in der Quantenmechanik (The present situation in quantum mechanics)". Naturwissenschaften. [3] Schroedinger: "The Present Situation in Quantum Mechanics" (http:/ / www. tu-harburg. de/ rzt/ rzt/ it/ QM/ cat. html#sect5) [4] Pay link to Einstein letter (http:/ / www. jstor. org/ pss/ 687649) [5] Hermann Wimmel (1992). Quantum physics & observed reality: a critical interpretation of quantum mechanics (http:/ / books. google. com/ books?id=-4sJ_fgyZJEC& pg=PA2). World Scientific. p. 2. ISBN 978-981-02-1010-6. . Retrieved 9 May 2011. [6] Faye, J (2008-01-24). "Copenhagen Interpretation of Quantum Mechanics" (http:/ / plato. stanford. edu/ entries/ qm-copenhagen/ ). Stanford Encyclopedia of Philosophy. The Metaphysics Research Lab Center for the Study of Language and Information, Stanford University. . Retrieved 2010-09-19. [7] Carpenter RHS, Anderson AJ (2006). "The death of Schroedinger's Cat and of consciousness-based wave-function collapse" (http:/ / web. archive. org/ web/ 20061130173850/ http:/ / www. ensmp. fr/ aflb/ AFLB-311/ aflb311m387. pdf). Annales de la Fondation Louis de Broglie (http:/ / web. archive. org/ web/ 20080618174026/ http:/ / www. ensmp. fr/ aflb/ AFLB-Web/ en-annales-index. htm) 31 (1): 45–52. Archived from the original (http:/ / www. ensmp. fr/ aflb/ AFLB-311/ aflb311m387. pdf) on 2006-11-30. . Retrieved 2010-09-10. [8] Penrose, R. The Road to Reality, p 807. [9] Wojciech H. Zurek, Decoherence, einselection, and the quantum origins of the classical, Reviews of Modern Physics 2003, 75, 715 or (http:/ / arxiv. org/ abs/ quant-ph/ 0105127) [10] Wojciech H. Zurek, "Decoherence and the transition from quantum to classical", Physics Today, 44, pp 36–44 (1991) [11] Rovelli, Carlo (1996). "Relational Quantum Mechanics". International Journal of Theoretical Physics 35: 1637–1678. arXiv:quant-ph/9609002. Bibcode 1996IJTP...35.1637R. doi:10.1007/BF02302261.

125

Schrödinger's Cat [12] What is the World's Biggest Schrodinger Cat? (http:/ / physics. stackexchange. com/ questions/ 3309/ what-is-the-worlds-biggest-schrodinger-cat) [13] Schr%C%B6dingers Cat Now Made of Light (http:/ / www. science20. com/ news_articles/ schrödingers_cat_now_made_light) [14] C. Monroe, et. al. A “Schrodinger Cat” Superposition State of an Atom (http:/ / www. quantumsciencephilippines. com/ seminar/ seminar-topics/ SchrodingerCatAtom. pdf) [15] Physics World: Schrodinger's cat comes into view (http:/ / physicsworld. com/ cws/ article/ news/ 2815) [16] Scientific American : Macro-Weirdness: "Quantum Microphone" Puts Naked-Eye Object in 2 Places at Once: A new device tests the limits of Schrödinger's cat (http:/ / www. scientificamerican. com/ article. cfm?id=quantum-microphone) [17] How to Create Quantum Superpositions of Living Things (http:/ / www. technologyreview. com/ blog/ arxiv/ 24101/ )> [18] Chown, Marcus (2007-11-22). "Has observing the universe hastened its end?" (http:/ / www. newscientist. com/ channel/ fundamentals/ mg19626313. 800-has-observing-the-universe-hastened-its-end. html). New Scientist. . Retrieved 2007-11-25. [19] Krauss, Lawrence M.; James Dent (April 30, 2008). "Late Time Behavior of False Vacuum Decay: Possible Implications for Cosmology and Metastable Inflating States". Phys. Rev. Lett. (US: APS) 100 (17). arXiv:0711.1821. Bibcode 2008PhRvL.100q1301K. doi:10.1103/PhysRevLett.100.171301.

External links • Schrödinger's cat in audio (http://soundcloud.com/siftpodcast/schr-dingers-cat) produced by Sift (http:// siftpodcast.com/) • Erwin Schrödinger, The Present Situation in Quantum Mechanics (Translation) (http://www.tu-harburg.de/rzt/ rzt/it/QM/cat.html) • The EPR paper (http://prola.aps.org/abstract/PR/v47/i10/p777_1) • Viennese Meow (the cat's perspective - short story) (http://primastoria.com/story/viennese-meow/) • The story of Schroedinger's cat (an epic poem) (http://www.straightdope.com/classics/a1_122.html); The Straight Dope • Tom Leggett (Aug. 1, 2000) New life for Schrödinger's cat, Physics World, UK (http://physicsworld.com/cws/ article/print/525) Experiments at two universities claim to observe superposition in large scale systems • Information Philosopher on Schrödinger's cat (http://www.informationphilosopher.com/solutions/ experiments/schrodingerscat/) More diagrams and an information creation explanation. • A YouTube video explaining Schrödingers cat (http://www.youtube.com/watch?v=CrxqTtiWxs4)

126

127

6. Measurement Problems The Measurement Problem The measurement problem in quantum mechanics is the unresolved problem of how (or if) wavefunction collapse occurs. The inability to observe this process directly has given rise to different interpretations of quantum mechanics, and poses a key set of questions that each interpretation must answer. The wavefunction in quantum mechanics evolves deterministically according to the Schrödinger equation as a linear superposition of different states, but actual measurements always find the physical system in a definite state. Any future evolution is based on the state the system was discovered to be in when the measurement was made, meaning that the measurement "did something" to the process under examination. Whatever that "something" may be does not appear to be explained by the basic theory. To express matters differently (to paraphrase Steven Weinberg [1][2]), the Schrödinger wave equation determines the wavefunction at any later time. If observers and their measuring apparatus are themselves described by a deterministic wave function, why can we not predict precise results for measurements, but only probabilities? As a general question: How can one establish a correspondence between quantum and classical reality?[3]

Example The best known is the "paradox" of the Schrödinger's cat: a cat is apparently evolving into a linear superposition of basis vectors that can be characterized as an "alive cat" and states that can be described as a "dead cat". Each of these possibilities is associated with a specific nonzero probability amplitude; the cat seems to be in some kind of "combination" state (specifically, a "superposition"). However, a single, particular observation of the cat does not measure the probabilities: it always finds either a living cat, or a dead cat. After the measurement the cat is definitively alive or dead. The question is: How are the probabilities converted into an actual, sharply well-defined outcome?

Interpretations Hugh Everett's many-worlds interpretation attempts to solve the problem by suggesting there is only one wavefunction, the superposition of the entire universe, and it never collapses—so there is no measurement problem. Instead, the act of measurement is simply an interaction between quantum entities, e.g. observer, measuring instrument, electron/positron etc, which entangle to form a single larger entity, for instance living cat/happy scientist. Everett also attempted to demonstrate the way that in measurements the probabilistic nature of quantum mechanics would appear; work later extended by Bryce DeWitt. De Broglie–Bohm theory tries to solve the measurement problem very differently: this interpretation contains not only the wavefunction, but also the information about the position of the particle(s). The role of the wavefunction is to generate the velocity field for the particles. These velocities are such that the probability distribution for the particle remains consistent with the predictions of the orthodox quantum mechanics. According to de Broglie–Bohm theory, interaction with the environment during a measurement procedure separates the wave packets in configuration space which is where apparent wavefunction collapse comes from even though there is no actual collapse. Erich Joos and Heinz-Dieter Zeh claim that the latter approach was put on firm ground in the 1980s by the phenomenon of quantum decoherence.[4] Zeh further claims that decoherence makes it possible to identify the fuzzy

The Measurement Problem boundary between the quantum microworld and the world where the classical intuition is applicable.[5] Quantum decoherence was proposed in the context of the many-worlds interpretation, but it has also become an important part of some modern updates of the Copenhagen interpretation based on consistent histories,[6] [7] . Quantum decoherence does not describe the actual process of the wavefunction collapse, but it explains the conversion of the quantum probabilities (that exhibit interference effects) to the ordinary classical probabilities. See, for example, Zurek,[3] Zeh[5] and Schlosshauer.[8] The present situation is slowly clarifying, as described in a recent paper by Schlosshauer as follows:[9] Several decoherence-unrelated proposals have been put forward in the past to elucidate the meaning of probabilities and arrive at the Born rule … It is fair to say that no decisive conclusion appears to have been reached as to the success of these derivations. … As it is well known, [many papers by Bohr insist upon] the fundamental role of classical concepts. The experimental evidence for superpositions of macroscopically distinct states on increasingly large length scales counters such a dictum. Only the physical interactions between systems then determine a particular decomposition into classical states from the view of each particular system. Thus classical concepts are to be understood as locally emergent in a relative-state sense and should no longer claim a fundamental role in the physical theory.

References and notes [1] Steven Weinberg (1998). The Oxford History of the Twentieth Century (http:/ / books. google. com/ ?id=uYTW5ZWrwWAC& pg=PA22& dq=observer+ measurement+ "S+ Weinberg") (Michael Howard & William Roger Louis, editors ed.). Oxford University Press. p. 26. ISBN 0-19-820428-0. . [2] Steven Weinberg: Einstein's Mistakes (http:/ / scitation. aip. org/ journals/ doc/ PHTOAD-ft/ vol_58/ iss_11/ 31_1. shtml) in Physics Today (2005); see subsection "Contra quantum mechanics" [3] Wojciech Hubert Zurek Decoherence, einselection, and the quantum origins of the classical Reviews of Modern Physics, Vol. 75, July 2003 (http:/ / hubcap. clemson. edu/ ~daw/ D_PHYS455/ RevModPhys. v75p715y03. pdf) [4] Joos, E., and H. D. Zeh, "The emergence of classical properties through interaction with the environment" (1985), Z. Phys. B 59, 223. [5] H D Zeh (http:/ / arxiv. org/ abs/ quant-ph/ 9506020v3) in E. Joos .... (2003). Decoherence and the Appearance of a Classical World in Quantum Theory (http:/ / books. google. com/ ?id=6eTHcxeNxdUC& printsec=frontcover& dq=isbn=3540613943#PPT21,M1) (2nd Edition; Erich Joos, H. D. Zeh, C. Kiefer, Domenico Giulini, J. Kupsch, I. O. Stamatescu (editors) ed.). Springer-Verlag. Chapter 2. ISBN 3-540-00390-8. . [6] V. P. Belavkin (1994). "Nondemolition principle of quantum measurement theory". Foundations of Physics 24 (5): 685–714. arXiv:quant-ph/0512188. doi:10.1007/BF02054669. [7] V. P. Belavkin (2001). "Quantum noise, bits and jumps: uncertainties, decoherence, measurements and filtering". Progress in Quantum Electronics 25 (1): 1–53. arXiv:quant-ph/0512208. doi:10.1016/S0079-6727(00)00011-2. [8] Maximilian Schlosshauer (2005). "Decoherence, the measurement problem, and interpretations of quantum mechanics". Rev. Mod. Phys. 76 (4): 1267–1305. arXiv:quant-ph/0312059. Bibcode 2004RvMP...76.1267S. doi:10.1103/RevModPhys.76.1267. [9] M Schlosshauer: Experimental motivation and empirical consistency in minimal no-collapse quantum mechanics, Annals of Physics, Volume 321, Issue 1, January 2006, Pages 112-149 (http:/ / www. citebase. org/ fulltext?format=application/ pdf& identifier=oai:arXiv. org:quant-ph/ 0506199)

Further reading • R. Buniy, S. Hsu and A. Zee On the origin of probability in quantum mechanics (2006) (http://duende.uoregon. edu/~hsu/talks/probability_qm.pdf)

External links • The Quantum Measurement Problem (http://www.shantena.com/en/physicslectures/quantummeasurement) Two presentations: a non-technical and a more technical presentation.

128

Measurement in Quantum Mechanics

Measurement in Quantum Mechanics The framework of quantum mechanics requires a careful definition of measurement. The issue of measurement lies at the heart of the problem of the interpretation of quantum mechanics, for which there is currently no consensus.

Measurement from a practical point of view Measurement is viewed in different ways in the many interpretations of quantum mechanics; however, despite the considerable philosophical differences, they almost universally agree on the practical question of what results from a routine quantum-physics laboratory measurement. To describe this, a simple framework to use is the Copenhagen interpretation, and it will be implicitly used in this section; the utility of this approach has been verified countless times, and all other interpretations are necessarily constructed so as to give the same quantitative predictions as this in almost every case.

Qualitative overview The quantum state of a system is a mathematical object that fully describes the quantum system. One typically imagines some experimental apparatus and procedure which "prepares" this quantum state; the mathematical object then reflects the setup of the apparatus. Once the quantum state has been prepared, some aspect of it is measured (for example, its position or energy). If the experiment is repeated, so as to measure the same aspect of the same quantum state prepared in the same way, the result of the measurement will often be different. The expected result of the measurement is in general described by a probability distribution that specifies the likelihoods that the various possible results will be obtained.[1] (This distribution can be either discrete or continuous, depending on what is being measured.) The measurement process is often said to be random and indeterministic. (However, there is considerable dispute over this issue; in some interpretations of quantum mechanics, the result merely appears random and indeterministic, in other interpretations the indeterminism is core and irreducible.) This is because an important aspect of measurement is wavefunction collapse, the nature of which varies according to the interpretation adopted. What is universally agreed, however, is that if the measurement is repeated, without re-preparing the state, one finds the same result as the first measurement.[2] As a result, after measuring some aspect of the quantum state, we normally update the quantum state to reflect the result of the measurement; it is this updating that ensures that if an immediate re-measurement is repeated without re-preparing the state, one finds the same result as the first measurement. The updating of the quantum state model is called wavefunction collapse.

Quantitative details The mathematical relationship between the quantum state and the probability distribution is, again, widely accepted among physicists, and has been experimentally confirmed countless times. This section summarizes this relationship, which is stated in terms of the mathematical formulation of quantum mechanics. Measurable quantities ("observables") as operators It is a postulate of quantum mechanics that all measurements have an associated operator (called an observable operator, or just an observable), with the following properties: 1. The observable is a Hermitian (self-adjoint) operator mapping a Hilbert space (namely, the state space, which consists of all possible quantum states) into itself. 2. The observable's eigenvalues are real. The possible outcomes of the measurement are precisely the eigenvalues of the given observable.

129

Measurement in Quantum Mechanics

130

3. For each eigenvalue there are one or more corresponding eigenvectors (which in this context are called eigenstates), which will make up the state of the system after the measurement. 4. The observable has a set of eigenvectors which span the state space. It follows that each observable generates an orthonormal basis of eigenvectors (called an eigenbasis). Physically, this is the statement that any quantum state can always be represented as a superposition of the eigenstates of an observable. Important examples of observables are: • The Hamiltonian operator, representing the total energy of the system; with the special case of the nonrelativistic Hamiltonian operator:

.

• The momentum operator: • The position operator:

(in the position basis).

, where

(in the momentum basis).

Operators can be noncommuting. Two Hermitian operators commute if (and only if) there is at least one basis of vectors, each of which is an eigenvector of both operators (this is sometimes called a simultaneous eigenbasis). Noncommuting observables are said to be incompatible and cannot in general be measured simultaneously. In fact, they are related by an uncertainty principle, as a consequence of the Robertson–Schrödinger relation. Measurement probabilities and wavefunction collapse There are a few possible ways to mathematically describe the measurement process (both the probability distribution and the collapsed wavefunction). The most convenient description depends on the spectrum (i.e., set of eigenvalues) of the observable. Discrete, nondegenerate spectrum Let

be an observable, and suppose that it has discrete eigenstates

and corresponding eigenvalues

, no two of which are equal.

Assume the system is prepared in state follows that (where

(in bra-ket notation) for

. Since the eigenstates of an observable form a basis (the eigenbasis), it

can be written in terms of the eigenstates as are complex numbers). Then measuring

can yield any of the results

, with

corresponding probabilities given by

Usually

is assumed to be normalized, in which case this expression reduces to

If the result of the measurement is

, then the system's quantum state after the measurement is

so any repeated measurement of

will yield the same result

collapse.)

. (This phenomenon is called wavefunction

Measurement in Quantum Mechanics

131

Continuous, nondegenerate spectrum Let

be an observable, and suppose that it has a continuous spectrum of eigenvalues filling the interval (a,b).

Assume further that each eigenvalue x in this range is associated with a unique eigenstate Assume the system is prepared in state

(where

, which can be written in terms of the eigenbasis as

is a complex-valued function). Then measuring

with probability density function

Again,

.

can yield a result anywhere in the interval (a,b),

; i.e., a result between y and z will occur with probability

is often assumed to be normalized, in which case this expression reduces to

If the result of the measurement is x, then the new wave function will be

Alternatively, it is often possible and convenient to analyze a continuous-spectrum measurement by taking it to be the limit of a different measurement with a discrete spectrum. For example, an analysis of scattering involves a continuous spectrum of energies, but by adding a "box" potential (which bounds the volume in which the particle can be found), the spectrum becomes discrete. By considering larger and larger boxes, this approach need not involve any approximation, but rather can be regarded as an equally valid formalism in which this problem can be analyzed. Degenerate spectra If there are multiple eigenstates with the same eigenvalue (called degeneracies), the analysis is a bit less simple to state, but not essentially different. In the discrete case, for example, instead of finding a complete eigenbasis, it is a bit more convenient to write the Hilbert space as a direct sum of eigenspaces. The probability of measuring a particular eigenvalue is the squared component of the state vector in the corresponding eigenspace, and the new state after measurement is the projection of the original state vector into the appropriate eigenspace. Density matrix formulation Instead of performing quantum-mechanics computations in terms of wavefunctions (kets), it is sometimes necessary to describe a quantum-mechanical system in terms of a density matrix. The analysis in this case is formally slightly different, but the physical content is the same, and indeed this case can be derived from the wavefunction formulation above. The result for the discrete, degenerate case, for example, is as follows: Let

be an observable, and suppose that it has discrete eigenvalues

eigenspaces

respectively. Let

be the projection operator into the space

, associated with .

Assume the system is prepared in the state described by the density matrix ρ. Then measuring the results where

, with corresponding probabilities given by denotes trace. If the result of the measurement is n, then the new density matrix will be

Alternatively, one can say that the measurement process results in the new density matrix

can yield any of

Measurement in Quantum Mechanics where the difference is that

132 is the density matrix describing the entire ensemble, whereas

matrix describing the sub-ensemble whose measurement result was

is the density

.

Statistics of measurement As detailed above, the result of measuring a quantum-mechanical system is described by a probability distribution. Some properties of this distribution are as follows: Suppose we take a measurement corresponding to observable

, on a state whose quantum state is

.

• The mean (average) value of the measurement is (see Expectation value (quantum mechanics)) . • The variance of the measurement is

• The standard deviation of the measurement is

These are direct consequences of the above formulas for measurement probabilities. Example Suppose that we have a particle in a 1-dimensional box, set up initially in the ground state computed from the time-independent Schrödinger equation, the energy of this state is particle's mass and L is the box length), and the spatial wavefunction is

. As can be (where m is the . If the energy

is now measured, the result will always certainly be , and this measurement will not affect the wavefunction. Next suppose that the particle's position is measured. The position x will be measured with probability density

If the measurement result was x=S, then the wavefunction after measurement will be the position eigenstate . If the particle's position is immediately measured again, the same position will be obtained. The new wavefunction

can, like any wavefunction, be written as a superposition of eigenstates of any

observable. In particular, using energy eigenstates,

, we have

If we now leave this state alone, it will smoothly evolve in time according to the Schrödinger equation. But suppose instead that an energy measurement is immediately taken. Then the possible energy values will be measured with relative probabilities:

and moreover if the measurement result is

, then the new state will be the energy eigenstate

.

So in this example, due to the process of wavefunction collapse, a particle initially in the ground state can end up in any energy level, after just two subsequent non-commuting measurements are made.

Measurement in Quantum Mechanics

133

Wavefunction collapse The process in which a quantum state becomes one of the eigenstates of the operator corresponding to the measured observable is called "collapse", or "wavefunction collapse". The final eigenstate appears randomly with a probability equal to the square of its overlap with the original state.[1] The process of collapse has been studied in many experiments, most famously in the double-slit experiment. The wavefunction collapse raises serious questions regarding "the measurement problem",[3] as well as questions of determinism and locality, as demonstrated in the EPR paradox and later in GHZ entanglement. (See below.) In the last few decades, major advances have been made toward a theoretical understanding of the collapse process. This new theoretical framework, called quantum decoherence, supersedes previous notions of instantaneous collapse and provides an explanation for the absence of quantum coherence after measurement. While this theory correctly predicts the form and probability distribution of the final eigenstates, it does not explain the randomness inherent in the choice of final state.

von Neumann measurement scheme The von Neumann measurement scheme, the ancestor of quantum decoherence theory, describes measurements by taking into account the measuring apparatus which is also treated as a quantum object. Let the quantum state be in the superposition , where are eigenstates of the operator that needs to be measured. In order to make the measurement, the measured system described by described by the quantum state

needs to interact with the measuring apparatus

, so that the total wave function before the interaction is

. During the

interaction of object and measuring instrument the unitary evolution is supposed to realize the following transition from the initial to the final total wave function:

where

are orthonormal states of the measuring apparatus. The unitary evolution above is referred to as

premeasurement. The relation with wave function collapse is established by calculating the final density operator of the object from the final total wave function. This density operator is interpreted by von Neumann as describing an ensemble of objects being after the measurement with probability

in the state

The transition

is often referred to as weak von Neumann projection, the wave function collapse or strong von Neumann projection

being thought to correspond to an additional selection of a subensemble by means of observation. In case the measured observable has a degenerate spectrum, weak von Neumann projection is generalized to Lüders projection

in which the vectors

for fixed n are the degenerate eigenvectors of the measured observable. For an arbitrary

state described by a density operator Lüders projection is given by

Measurement in Quantum Mechanics

134

Measurements of the second kind In a measurement of the second kind the unitary evolution during the interaction of object and measuring instrument is supposed to be given by

in which the states

of the object are determined by specific properties of the interaction between object and

measuring instrument. They are normalized but not necessarily mutually orthogonal. The relation with wave function collapse is analogous to that obtained for measurements of the first kind, the final state of the object now being with probability

Note that many present-day measurement procedures are measurements of the second kind,

some even functioning correctly only as a consequence of being of the second kind. For instance, a photon counter, detecting a photon by absorbing and hence annihilating it, thus ideally leaving the electromagnetic field in the vacuum state rather than in the state corresponding to the number of detected photons; also the Stern–Gerlach [4] experiment would not function at all if it really were a measurement of the first kind.

Decoherence in quantum measurement One can also introduce the interaction with the environment

, so that, in a measurement of the first kind, after the

interaction the total wave function takes a form

which is related to the phenomenon of decoherence. The above is completely described by the Schrödinger equation and there are not any interpretational problems with this. Now the problematic wavefunction collapse does not need to be understood as a process on the level of the measured system, but can also be understood as a process or as a process

on the level of the measuring apparatus,

on the level of the environment. Studying these processes provides considerable insight into

the measurement problem by avoiding the arbitrary boundary between the quantum and classical worlds, though it does not explain the presence of randomness in the choice of final eigenstate. If the set of states ,

, or

represents a set of states that do not overlap in space, the appearance of collapse can be generated by either the Bohm interpretation or the Everett interpretation which both deny the reality of wavefunction collapse. Both of these are stated to predict the same probabilities for collapses to various states as the conventional interpretation by their supporters. The Bohm interpretation is held to be correct only by a small minority of physicists, since there are difficulties with the generalization for use with relativistic quantum field theory. However, there is no proof that the Bohm interpretation is inconsistent with quantum field theory, and work to reconcile the two is ongoing. The Everett interpretation easily accommodates relativistic quantum field theory.

Philosophical problems of quantum measurements What physical interaction constitutes a measurement? Until the advent of quantum decoherence theory in the late 20th century, a major conceptual problem of quantum mechanics and especially the Copenhagen interpretation was the lack of a distinctive criterion for a given physical interaction to qualify as "a measurement" and cause a wavefunction to collapse. This is best illustrated by the Schrödinger's cat paradox. Certain aspects of this question are now well understood in the framework of quantum decoherence theory, such as an understanding of weak measurements, and quantifying what measurements or interactions are sufficient to destroy quantum coherence. Nevertheless, there remains less than universal agreement among physicists on some aspects of the question of what constitutes a measurement.

Measurement in Quantum Mechanics

Does measurement actually determine the state? The question of whether (and in what sense) a measurement actually determines the state is one which differs among the different interpretations of quantum mechanics. (It is also closely related to the understanding of wavefunction collapse.) For example, in most versions of the Copenhagen interpretation, the measurement determines the state, and after measurement the state is definitely what was measured. But according to the many-worlds interpretation, measurement determines the state in a more restricted sense: In other "worlds", other measurement results were obtained, and the other possible states still exist.

Is the measurement process random or deterministic? As described above, there is universal agreement that quantum mechanics appears random, in the sense that all experimental results yet uncovered can be predicted and understood in the framework of quantum mechanics measurements being fundamentally random. Nevertheless, it is not settled[5] whether this is true, fundamental randomness, or merely "emergent" randomness resulting from underlying hidden variables which deterministically cause measurement results to happen a certain way each time. This continues to be an area of active research.[6] If there are hidden variables, they would have to be "nonlocal".

Does the measurement process violate locality? In physics, the Principle of locality is the concept that information cannot travel faster than the speed of light (also see special relativity). It is known experimentally (see Bell's theorem, which is related to the EPR paradox) that if quantum mechanics is deterministic (due to hidden variables, as described above), then it is nonlocal (i.e. violates the principle of locality). Nevertheless, there is not universal agreement among physicists on whether quantum mechanics is nondeterministic, nonlocal, or both.[5]

References [1] J. J. Sakurai (1994). Modern Quantum Mechanics (2nd ed.). p. 24. ISBN 0201539292. [2] J. J. Sakurai (1994). Modern Quantum Mechanics (2nd ed.). p. 25. ISBN 0201539292. [3] George S. Greenstein and Arthur G. Zajonc (2006). The Quantum Challenge: Modern Research On The Foundations Of Quantum Mechanics (http:/ / books. google. com/ books?id=5t0tm0FB1CsC& pg=PA215& lpg=PA215& dq=wave+ function+ collapse& source=bl& ots=a7iUGurRDC& sig=o1ddjY7lQrj4EQdvS49xcceWq2M& hl=en& ei=RfgtSsDNL4WgM8u-rf4J& sa=X& oi=book_result& ct=result& resnum=7#PPA215,M1) (2nd ed.). ISBN 076372470X. . [4] M.O. Scully, W.E. Lamb, A. Barut (1987). "On the theory of the Stern–Gerlach apparatus" (http:/ / www. springerlink. com/ content/ t4266804k832p42p/ fulltext. pdf). Foundations of Physics 17: 575–583. . Retrieved 9 November 2012. [5] Hrvoje Nikolić (2007). "Quantum mechanics: Myths and facts" (http:/ / arxiv. org/ pdf/ quant-ph/ 0609163). Foundation of Physics 37: 1563-1611. . Retrieved 9 November 2012. [6] S. Gröblacher et al. (2007). "An experimental test of non-local realism" (http:/ / dx. doi. org/ 10. 1038/ nature05677). Nature 446 (871). . Retrieved 9 November 2012.

Further reading • John A. Wheeler and Wojciech Hubert Zurek, eds. (1983). Quantum Theory and Measurement. Princeton University Press. ISBN 0-691-08316-9. • Vladimir B. Braginsky and Farid Ya. Khalili (1992). Quantum Measurement. Cambridge University Press. ISBN 0-521-41928-X. • George S. Greenstein and Arthur G. Zajonc (2006). The Quantum Challenge: Modern Research On The Foundations Of Quantum Mechanics (2nd ed.). ISBN 076372470X.

135

Measurement in Quantum Mechanics

External links • " The Double Slit Experiment (http://physicsweb.org/article/world/15/9/1)". (physicsweb.org) • " Measurement in Quantum Mechanics (http://plato.stanford.edu/entries/qt-measurement/)" Henry Krips in the Stanford Encyclopedia of Philosophy • Decoherence, the measurement problem, and interpretations of quantum mechanics (http://arxiv.org/abs/ quant-ph/0312059) • Measurements and Decoherence (http://arxiv.org/abs/quant-ph/0505070) • The conditions for discrimination between quantum states with minimum error (http://arxiv.org/pdf/0810. 1919) • Quantum behavior of measurement apparatus (http://arxiv.org/abs/1001.3032)

136

137

7. Advanced Concepts Quantum Number Quantum numbers describe values of conserved quantities in the dynamics of the quantum system. Perhaps the most peculiar aspect of quantum mechanics is the quantization of observable quantities, since quantum numbers are discrete sets of integers or half-integers. This is distinguished from classical mechanics where the values can range continuously. Quantum numbers often describe specifically the energies of electrons in atoms, but other possibilities include angular momentum, spin, etc. Any quantum system can have one or more quantum numbers; it is thus difficult to list all possible quantum numbers.[1]

How many quantum numbers? The question of how many quantum numbers are needed to describe any given system has no universal answer, hence for each system, one must find the answer for a full analysis of the system. A quantized system requires at least one quantum number. The dynamics of any quantum system are described by a quantum Hamiltonian, H. There is one quantum number of the system corresponding to the energy, i.e., the eigenvalue of the Hamiltonian. There is also one quantum number for each operator O that commutes with the Hamiltonian (i.e. satisfies the relation HO = OH). These are all the quantum numbers that the system can have. Note that the operators O defining the quantum numbers should be independent of each other. Often, there is more than one way to choose a set of independent operators. Consequently, in different situations different sets of quantum numbers may be used for the description of the same system.

Spatial and angular momentum numbers To completely describe an electron in an atom, four quantum numbers are needed: energy, angular momentum, magnetic moment and spin.

Traditional nomenclatures Many different models have been proposed throughout the history of quantum mechanics, but the most prominent system of nomenclature spawned from the Hund-Mulliken molecular orbital theory of Friedrich Hund, Robert S. Mulliken, and contributions from Schrödinger, Slater and John Lennard-Jones. This system of nomenclature incorporated Bohr energy levels, Hund-Mulliken orbital theory, and observations on electron spin based on spectroscopy and Hund's rules.[2] This model describes electrons using four quantum numbers, n, ℓ, mℓ, ms, given below. It is also the common nomenclature in the classical description of nuclear particle states (e.g. protons and neutrons). Molecular orbitals require different quantum numbers, because the Hamiltonian and its symmetries are quite different. 1. The principal quantum number: n The first describes the electron shell, or energy level, of an atom. The value of n ranges from 1 to the shell containing the outermost electron of that atom, i.e.[3]

n = 1, 2, ... . For example, in caesium (Cs), the outermost valence electron is in the shell with energy level 6, so an electron in caesium can have an n value from 1 to 6. For particles in a time-independent potential (see Schrödinger equation), it also labels the nth eigenvalue of Hamiltonian (H), i.e. the energy, E with the contribution due to angular

Quantum Number momentum (the term involving J2) left out. This number therefore has a dependence only on the distance between the electron and the nucleus (i.e., the radial coordinate, r). The average distance increases with n, and hence quantum states with different principal quantum numbers are said to belong to different shells. 2. The azimuthal quantum number: ℓ The second (also known as the angular quantum number or orbital quantum number) describes the subshell, and gives the magnitude of the orbital angular momentum through the relation L2 = ħ2 ℓ (ℓ + 1). In chemistry and spectroscopy, "ℓ = 0" is called an s orbital, "ℓ = 1" a p orbital, "ℓ = 2" a d orbital, and "ℓ = 3" an f orbital. The value of ℓ ranges from 0 to n − 1, because the first p orbital (ℓ = 1) appears in the second electron shell (n = 2), the first d orbital (ℓ = 2) appears in the third shell (n = 3), and so on:[4]

ℓ = 0, 1, 2,..., n − 1. A quantum number beginning in 3, 0, … describes an electron in the s orbital of the third electron shell of an atom. In chemistry, this quantum number is very important, since it specifies the shape of an atomic orbital and strongly influences chemical bonds and bond angles. 3. The magnetic quantum number: mℓ The third describes the specific orbital (or "cloud") within that subshell, and yields the projection of the orbital angular momentum along a specified axis: Lz = mℓ ħ. The values of mℓ range from −ℓ to ℓ, with integer steps between them:[5] The s subshell (ℓ = 0) contains only one orbital, and therefore the mℓ of an electron in an s subshell will always be 0. The p subshell (ℓ = 1) contains three orbitals (in some systems, depicted as three "dumbbell-shaped" clouds), so the mℓ of an electron in a p subshell will be −1, 0, or 1. The d subshell (ℓ = 2) contains five orbitals, with mℓ values of −2, −1, 0, 1, and 2. 4. The spin projection quantum number: ms The fourth describes the spin (intrinsic angular momentum) of the electron within that orbital, and gives the projection of the spin angular momentum S along the specified axis: Sz = ms ħ.

Analogously, the values of ms range from −s to s, where s is the spin quantum number, an intrinsic property of particles:[6]

ms = −s, −s + 1, −s + 2,...,s − 2, s − 1, s.

An electron has spin s = ½, consequently ms will be ±½, corresponding with "spin" and "opposite spin." Each electron in any individual orbital must have different spins because of the Pauli exclusion principle, therefore an orbital never contains more than two electrons. Note that, since atoms and electrons are in a state of constant motion, there is no universal fixed value for mℓ and ms values. Therefore, the mℓ and ms values are defined somewhat arbitrarily. The only requirement is that the naming schematic used within a particular set of calculations or descriptions must be consistent (e.g. the orbital occupied by the first electron in a p subshell could be described as mℓ = −1 or mℓ = 0, or mℓ = 1, but the mℓ value of the other electron in that orbital must be the same, and the mℓ assigned to electrons in other orbitals must be different). These rules are summarized as follows:

138

Quantum Number

139

Name

Symbol

Orbital meaning

Range of values

Value examples

principal quantum number

n

shell

1≤n

n = 1, 2, 3, …

azimuthal quantum number (angular momentum)

ℓ

subshell (s orbital is listed as 0, p orbital as 1 etc.)

0≤ℓ≤n−1

for n = 3: ℓ = 0, 1, 2 (s, p, d)

magnetic quantum number, (projection of angular momentum)

mℓ

energy shift (orientation of the subshell's shape)

−ℓ ≤ mℓ ≤ ℓ

for ℓ = 2: mℓ = −2, −1, 0, 1, 2

spin projection quantum number

ms

spin of the electron (−½ = "spin down", ½ = "spin up")

−s ≤ ms ≤ s

for an electron s = ½, so ms = −½, ½

Example: The quantum numbers used to refer to the outermost valence electrons of the Carbon (C) atom, which are located in the 2p atomic orbital, are; n = 2 (2nd electron shell), ℓ = 1 (p orbital subshell), mℓ = 1, 0 or −1, ms = ½ (parallel spins). Results from spectroscopy indicated that up to two electrons can occupy a single orbital. However two electrons can never have the same exact quantum state nor the same set of quantum numbers according to Hund's Rules, which addresses the Pauli exclusion principle. A fourth quantum number with two possible values was added as an ad hoc assumption to resolve the conflict; this supposition could later be explained in detail by relativistic quantum mechanics and from the results of the renowned Stern-Gerlach experiment.

Total angular momenta numbers Total momentum of a particle When one takes the spin-orbit interaction into consideration, the L and S operators no longer commute with the Hamiltonian, and their eigenvalues therefore change over time. Thus another set of quantum numbers should be used. This set includes[7][8] 1. The total angular momentum quantum number:

j = |ℓ ± s| which gives the total angular momentum through the relation J2 = ħ2 j (j + 1). 2. The projection of the total angular momentum along a specified axis:

mj = −j, −j + 1, −j + 2,...,j − 2, j − 1, j

analogous to the above, and satisfies

mj = mℓ + ms and |mℓ + ms| ≤ j. 3. Parity This is the eigenvalue under reflection, and is positive (+1) for states which came from even ℓ and negative (−1) for states which came from odd ℓ. The former is also known as even parity and the latter as odd parity, and is given by P = (−1)ℓ. For example, consider the following eight states, defined by their quantum numbers:

Quantum Number

140

n ℓ mℓ

ms

ℓ + s ℓ - s ml + ms

#1. 2 1

1 +1/2

3/2

1/2

3/2

#2. 2 1

1 -1/2

3/2

1/2

1/2

#3. 2 1

0 +1/2

3/2

1/2

1/2

#4. 2 1

0 -1/2

3/2

1/2

-1/2

#5. 2 1 -1 +1/2

3/2

1/2

-1/2

#6. 2 1 -1 -1/2

3/2

1/2

-3/2

#7. 2 0

0 +1/2

1/2 -1/2

1/2

#8. 2 0

0 -1/2

1/2 -1/2

-1/2

The quantum states in the system can be described as linear combination of these eight states. However, in the presence of spin-orbit interaction, if one wants to describe the same system by eight states which are eigenvectors of the Hamiltonian (i.e. each represents a state which does not mix with others over time), we should consider the following eight states: j = 3/2, mj =

3/2,

odd parity (coming from state (1) above)

j = 3/2, mj =

1/2,

odd parity (coming from states (2) and (3) above)

j = 3/2, mj = -1/2,

odd parity (coming from states (4) and (5) above)

j = 3/2, mj = -3/2,

odd parity (coming from state (6) above)

j = 1/2, mj =

1/2,

odd parity (coming from states (2) and (3) above)

j = 1/2, mj = -1/2,

odd parity (coming from states (4) and (5) above)

j = 1/2, mj =

1/2, even parity (coming from state (7) above)

j = 1/2, mj = -1/2, even parity (coming from state (8) above)

Nuclear angular momentum quantum numbers In nuclei, the entire assembly of protons and neutrons (nucleons) has a resultant angular momentum due to the angular momenta of each nucleon, usually denoted I. If the total angular momentum of a neutron is jn = ℓ + s and for a proton is jp = ℓ + s (where s for protons and neutrons happens to be ½ again) then the nuclear angular momentum quantum numbers I are given by:

I = |jn − jp|, |jn − jp| + 1, |jn − jp| + 2,..., |jn − jp| − 2, |jn − jp| − 1, |jn − jp| Parity with the number I is used to label nuclear angular momentum states, examples for some isotopes of Hydrogen (H), Carbon (C), and Sodium (Na) are;[9]

Quantum Number

141

H11 I = (1/2)+ C69 H12 I = 1+

I = (3/2)− Na1120 I = 2+

C610 I = 0+

Na1121 I = (3/2)+

H13 I = (1/2)+ C611 I = (3/2)− Na1122 I = 3+ C612 I = 0+

Na1123 I = (3/2)+

C613 I = (1/2)− Na1124 I = 4+ C614 I = 0+

Na1125 I = (5/2)+

C615 I = (1/2)+ Na1126 I = 3+

The reason for the unusual fluctuations in I, even by differences of just one nucleon, are due to the odd/even numbers of protons and neutrons - pairs of nucleons have a total angular momentum of zero (just like electrons in orbitals), leaving an odd/even numbers of unpaired nucleons. The property of nuclear spin is an important factor for the operation of NMR spectroscopy in organic chemistry[10], and MRI in nuclear medicine[11], due to the nuclear magnetic moment interacting with an external magnetic field.

Elementary particles Elementary particles contain many quantum numbers which are usually said to be intrinsic to them. However, it should be understood that the elementary particles are quantum states of the standard model of particle physics, and hence the quantum numbers of these particles bear the same relation to the Hamiltonian of this model as the quantum numbers of the Bohr atom does to its Hamiltonian. In other words, each quantum number denotes a symmetry of the problem. It is more useful in quantum field theory to distinguish between spacetime and internal symmetries. Typical quantum numbers related to spacetime symmetries are spin (related to rotational symmetry), the parity, C-parity and T-parity (related to the Poincaré symmetry of spacetime). Typical internal symmetries are lepton number and baryon number or the electric charge. (For a full list of quantum numbers of this kind see the article on flavour.) A minor but often confusing point is as follows: most conserved quantum numbers are additive, so in an elementary particle reaction, the sum of the quantum numbers should be the same before and after the reaction. However, some, usually called a parity, are multiplicative; i.e., their product is conserved. All multiplicative quantum numbers belong to a symmetry (like parity) in which applying the symmetry transformation twice is equivalent to doing nothing. These are all examples of an abstract group called Z2.

References and external links [1] [2] [3] [4] [5] [6] [7] [8] [9]

McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN 0-07-051400-3 Chemistry, Matter, and the Universe, R.E. Dickerson, I. Geis, W.A. Benjamin Inc. (USA), 1976, ISBN 0-19-855148-7 Concepts of Modern Physics (4th Edition), A. Beiser, Physics, McGraw-Hill (International), 1987, ISBN 0-07-100144-1 Molecular Quantum Mechanics Parts I and II: An Introduction to QUANTUM CHEMISRTY (Volume 1), P.W. Atkins, Oxford University Press, 1977, ISBN 0-19-855129-0 Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd Edition), R. Eisberg, R. Resnick, John Wiley & Sons, 1985, ISBN 978-0-471-87373-0 Quantum Mechanics (2nd edition), Y. Peleg, R. Pnini, E. Zaarur, E. Hecht, Schuam's Outlines, McGraw Hill (USA), 2010, ISBN 978-0-07-162358-2 Molecular Quantum Mechanics Parts I and II: An Introduction to QUANTUM CHEMISTRY (Volume 1), P.W. Atkins, Oxford University Press, 1977, ISBN 0-19-855129-0 Molecular Quantum Mechanics Part III: An Introduction to QUANTUM CHEMISTRY (Volume 2), P.W. Atkins, Oxford University Press, 1977 Introductory Nuclear Physics, K.S. Krane, 1988, John Wiley & Sons Inc, ISBN 978-0-471-80553-3

Quantum Number [10] Molecular Quantum Mechanics Part III: An Introduction to QUANTUM CHEMISTRY (Volume 2), P.W. Atkins, Oxford University Press, 1977 [11] Introductory Nuclear Physics, K.S. Krane, 1988, John Wiley & Sons Inc, ISBN 978-0-471-80553-3

General principles • Dirac, Paul A.M. (1982). Principles of quantum mechanics. Oxford University Press. ISBN 0-19-852011-5.

Atomic physics • Quantum numbers for the hydrogen atom (http://hyperphysics.phy-astr.gsu.edu/hbase/qunoh.html)

Particle physics • Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-805326-X. • Halzen, Francis and Martin, Alan D. (1984). QUARKS AND LEPTONS: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 0-471-88741-2. • The particle data group (http://pdg.lbl.gov/) • Lecture notes on quantum numbers (http://www.physics.byu.edu/faculty/durfee/courses/Summer2009/ physics222/AtomicQuantumNumbers.pdf)

Quantum Information In quantum mechanics, quantum information is physical information that is held in the "state" of a quantum system. The most popular unit of quantum information is the qubit, a two-level quantum system. However, unlike classical digital states (which are discrete), a two-state quantum system can actually be in a superposition of the two states at any given time. Quantum information differs from classical information in several respects, among which we note the following: • It cannot be read without the state becoming the measured value, • An arbitrary state cannot be cloned, • The state may be in a superposition of basis values. However, despite this, the amount of information that can be retrieved in a single qubit is equal to one bit. It is in the processing of information (quantum computation) that the differentiation occurs. The ability to manipulate quantum information enables us to perform tasks that would be unachievable in a classical context, such as unconditionally secure transmission of information. Quantum information processing is the most general field that is concerned with quantum information. There are certain tasks which classical computers cannot perform "efficiently" (that is, in polynomial time) according to any known algorithm. However, a quantum computer can compute the answer to some of these problems in polynomial time; one well-known example of this is Shor's factoring algorithm. Other algorithms can speed up a task less dramatically—for example, Grover's search algorithm which gives a quadratic speed-up over the best possible classical algorithm. Quantum information, and changes in quantum information, can be quantitatively measured by using an analogue of Shannon entropy, called the von Neumann entropy. Given a statistical ensemble of quantum mechanical systems with the density matrix , it is given by

Many of the same entropy measures in classical information theory can also be generalized to the quantum case, such as Holevo entropy [1] and the conditional quantum entropy.

142

Quantum Information

143

Quantum information theory The theory of quantum information is a result of the effort to generalize classical information theory to the quantum world. Quantum information theory aims to investigate the following question: What happens if information is stored in a state of a quantum system? One of the strengths of classical information theory is that physical representation of information can be disregarded: There is no need for an 'ink-on-paper' information theory or a 'DVD information' theory. This is because it is always possible to efficiently transform information from one representation to another. However, this is not the case for quantum information: it is not possible, for example, to write down on paper the previously unknown information contained in the polarisation of a photon. In general, quantum mechanics does not allow us to read out the state of a quantum system with arbitrary precision. The existence of Bell correlations between quantum systems cannot be converted into classical information. It is only possible to transform quantum information between quantum systems of sufficient information capacity. The information content of a message can, for this reason, be measured in terms of the minimum number n of two-level systems which are needed to store the message:

consists of n qubits. In its original theoretical sense,

the term qubit is thus a measure for the amount of information. A two-level quantum system can carry at most one qubit, in the same sense a classical binary digit can carry at most one classical bit. As a consequence of the noisy-channel coding theorem, noise limits the information content of an analog information carrier to be finite. It is very difficult to protect the remaining finite information content of analog information carriers against noise. The example of classical analog information shows that quantum information processing schemes must necessarily be tolerant against noise, otherwise there would not be a chance for them to be useful. It was a big breakthrough for the theory of quantum information, when quantum error correction codes and fault-tolerant quantum computation schemes were discovered.

Journals Among the journals in this field are • International Journal of Quantum Information • Journal of Quantum Chemistry • Applied Mathematics & Information Sciences

External links and references • • • • • • • •

Lectures at the Institut Henri Poincaré (slides and videos) [2] Quantum Information Theory at ETH Zurich [3] Quantum Information [4] Perimeter Institute for Theoretical Physics Center for Quantum Computation [5] - The CQC, part of Cambridge University, is a group of researchers studying quantum information, and is a useful portal for those interested in this field. Quantum Information Group [6] The quantum information research group at the University of Nottingham. Qwiki [7] - A quantum physics wiki devoted to providing technical resources for practicing quantum information scientists. Quantiki [8] - A wiki portal for quantum information with introductory tutorials. Charles H. Bennett and Peter W. Shor, "Quantum Information Theory," IEEE Transactions on Information Theory, Vol 44, pp 2724–2742, Oct 1998

• Institute for Quantum Computing [9] - The Institute for Quantum Computing, based in Waterloo, ON Canada, is a research institute working in conjunction with the University of Waterloo [10] and Perimeter Institute [11] on the subject of Quantum Information. • Quantum information can be negative [12]

Quantum Information Gregg Jaeger's book on Quantum Information [13](published by Springer, New York, 2007, ISBN 0-387-35725-4) The International Conference on Quantum Information (ICQI) [14] New Trends in Quantum Computation, Stony Brook, 2010 [15] Institute of Quantum Information [16] Caltech Quantum Information Theory [17] Imperial College Quantum Information [18] University College London Quantum Information Technology [19] Toshiba Research International Journal of Quantum Information [20] World Scientific Quantum Information Processing [21] Springer USC Center for Quantum Information Science & Technology [22] Center for Quantum Information and Control [23] Theoretical and experimental groups from University of New Mexico and University of Arizona. • Mark M. Wilde, "From Classical to Quantum Shannon Theory", arXiv:1106.1445 [24]. • Group of Quantum Information Theory [25] Kyungnam University in Korea • • • • • • • • • • •

References [1] http:/ / www. mi. ras. ru/ ~holevo/ eindex. html [2] http:/ / www. quantware. ups-tlse. fr/ IHP2006/ [3] http:/ / www. qit. ethz. ch/ [4] http:/ / www. perimeterinstitute. ca/ research/ research-areas/ quantum-information/ more-quantum-information [5] http:/ / cam. qubit. org/ [6] http:/ / www. maths. nottingham. ac. uk/ research/ appliedmathematics/ quantuminformation/ [7] http:/ / qwiki. caltech. edu/ [8] http:/ / www. quantiki. org [9] http:/ / www. iqc. ca/ [10] http:/ / www. uwaterloo. ca [11] http:/ / www. perimeterinstitute. ca/ [12] http:/ / www. damtp. cam. ac. uk/ user/ jono/ negative-information. html [13] http:/ / www. springer. com/ east/ home?SGWID=5-102-22-173664707-0& changeHeader=true [14] http:/ / osa. org/ meetings/ topicalmeetings/ icqi/ default. aspx [15] http:/ / insti. physics. sunysb. edu/ itp/ conf/ simons-qcomputation2/ program. html [16] http:/ / www. iqi. caltech. edu/ [17] http:/ / www3. imperial. ac. uk/ quantuminformation [18] http:/ / www. theory. phys. ucl. ac. uk/ quinfo [19] http:/ / www. toshiba-europe. com/ research/ crl/ qig/ index. html [20] http:/ / www. worldscinet. com/ ijqi/ ijqi. shtml [21] http:/ / www. springer. com/ new+ %26+ forthcoming+ titles+ %28default%29/ journal/ 11128 [22] http:/ / cqist. usc. edu/ [23] http:/ / www. cquic. org/ [24] http:/ / arxiv. org/ abs/ 1106. 1445 [25] http:/ / qubit. kyungnam. ac. kr/

144

Quantum Statistical Mechanics

145

Quantum Statistical Mechanics Quantum statistical mechanics is the study of statistical ensembles of quantum mechanical systems. A statistical ensemble is described by a density operator S, which is a non-negative, self-adjoint, trace-class operator of trace 1 on the Hilbert space H describing the quantum system. This can be shown under various mathematical formalisms for quantum mechanics. One such formalism is provided by quantum logic.

Expectation From classical probability theory, we know that the expectation of a random variable X is completely determined by its distribution DX by

assuming, of course, that the random variable is integrable or that the random variable is non-negative. Similarly, let A be an observable of a quantum mechanical system. A is given by a densely defined self-adjoint operator on H. The spectral measure of A defined by

uniquely determines A and conversely, is uniquely determined by A. EA is a boolean homomorphism from the Borel subsets of R into the lattice Q of self-adjoint projections of H. In analogy with probability theory, given a state S, we introduce the distribution of A under S which is the probability measure defined on the Borel subsets of R by

Similarly, the expected value of A is defined in terms of the probability distribution DA by

Note that this expectation is relative to the mixed state S which is used in the definition of DA. Remark. For technical reasons, one needs to consider separately the positive and negative parts of A defined by the Borel functional calculus for unbounded operators. One can easily show:

Note that if S is a pure state corresponding to the vector ψ,

Von Neumann entropy Of particular significance for describing randomness of a state is the von Neumann entropy of S formally defined by . Actually, the operator S log2 S is not necessarily trace-class. However, if S is a non-negative self-adjoint operator not of trace class we define Tr(S) = +∞. Also note that any density operator S can be diagonalized, that it can be represented in some orthonormal basis by a (possibly infinite) matrix of the form

Quantum Statistical Mechanics

146

and we define

The convention is that

, since an event with probability zero should not contribute to the entropy. This

value is an extended real number (that is in [0, ∞]) and this is clearly a unitary invariant of S. Remark. It is indeed possible that H(S) = +∞ for some density operator S. In fact T be the diagonal matrix

T is non-negative trace class and one can show T log2 T is not trace-class. Theorem. Entropy is a unitary invariant. In analogy with classical entropy (notice the similarity in the definitions), H(S) measures the amount of randomness in the state S. The more dispersed the eigenvalues are, the larger the system entropy. For a system in which the space H is finite-dimensional, entropy is maximized for the states S which in diagonal form have the representation

For such an S, H(S) = log2 n. The state S is called the maximally mixed state. Recall that a pure state is one of the form

for ψ a vector of norm 1. Theorem. H(S) = 0 if and only if S is a pure state. For S is a pure state if and only if its diagonal form has exactly one non-zero entry which is a 1. Entropy can be used as a measure of quantum entanglement.

Gibbs canonical ensemble Consider an ensemble of systems described by a Hamiltonian H with average energy E. If H has pure-point spectrum and the eigenvalues of H go to + ∞ sufficiently fast, e-r H will be a non-negative trace-class operator for every positive r. The Gibbs canonical ensemble is described by the state

Where β is such that the ensemble average of energy satisfies

and

This is called the partition function; it is the quantum mechanical version of the canonical partition function of classical statistical mechanics. The probability that a system chosen at random from the ensemble will be in a state

Quantum Statistical Mechanics corresponding to energy eigenvalue

147 is

Under certain conditions, the Gibbs canonical ensemble maximizes the von Neumann entropy of the state subject to the energy conservation requirement.

References • J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955. • F. Reif, Statistical and Thermal Physics, McGraw-Hill, 1965.

148

8. Advanced Topics Quantum Field Theory Quantum field theory (QFT) provides a theoretical framework for constructing quantum mechanical models of systems classically represented by an infinite number of degrees of freedom, that is, fields and (in a condensed matter context) many-body systems. It is the natural and quantitative language of particle physics and condensed matter physics. Most theories in modern particle physics, including the Standard Model of elementary particles and their interactions, are formulated as relativistic quantum field theories. Quantum field theories are used in many contexts, and are especially vital in elementary particle physics, where the particle count/number may change over the course of a reaction. They are also used in the description of critical phenomena and quantum phase transitions, such as in the BCS theory of superconductivity. In perturbative quantum field theory, the forces between particles are mediated by other particles. The electromagnetic force between two electrons is caused by an exchange of photons. Intermediate vector bosons mediate the weak force and gluons mediate the strong force. There is currently no complete quantum theory of the remaining fundamental force, gravity, but many of the proposed theories postulate the existence of a graviton particle that mediates it. These force-carrying particles are virtual particles and, by definition, cannot be detected while carrying the force, because such detection will imply that the force is not being carried. In addition, the notion of "force mediating particle" comes from perturbation theory, and thus does not make sense in a context of bound states. In QFT, photons are not thought of as "little billiard balls" but are rather viewed as field quanta – necessarily chunked ripples in a field, or "excitations", that "look like" particles. Fermions, like the electron, can also be described as ripples/excitations in a field, where each kind of fermion has its own field. In summary, the classical visualisation of "everything is particles and fields", in quantum field theory, resolves into "everything is particles", which then resolves into "everything is fields". In the end, particles are regarded as excited states of a field (field quanta). The gravitational field and the electromagnetic field are the only two fundamental fields in Nature that have infinite range and a corresponding classical low-energy limit, which greatly diminishes and hides their "particle-like" excitations. Albert Einstein, in 1905, attributed "particle-like" and discrete exchanges of momenta and energy, characteristic of "field quanta", to the electromagnetic field. Originally, his principal motivation was to explain the thermodynamics of radiation. Although it is often claimed that the photoelectric and Compton effects require a quantum description of the EM field, this is now understood to be untrue, and proper proof of the quantum nature of radiation is now taken up into modern quantum optics as in the antibunching effect.[1] The word "photon" was coined in 1926 by physical chemist Gilbert Newton Lewis (see also the articles photon antibunching and laser). There are several theories using the QFT framework, such as quantum electrodynamics and quantum chromodynamics. Within a theory, there is one field for each type of particle in that theory, and interaction terms between the fields. For example, QED has one electron field and one photon field; QCD has one field for each type of quark, etc. The interaction terms are similar in spirit to those in Maxwell's equations, being interactions between fields. However unlike Maxwell's theory, QFT fields generally exist in superpositions of states. In the "low-energy limit", the quantum field-theoretic description of the electromagnetic field, quantum electrodynamics, does not exactly reduce to James Clerk Maxwell's 1864 theory of classical electrodynamics. Small quantum corrections due to virtual electron-positron pairs give rise to small non-linear corrections to the Maxwell equations, although the "classical limit" of quantum electrodynamics has not been as widely explored as that of quantum mechanics.

Quantum Field Theory Presumably, the as yet unknown correct quantum field-theoretic treatment of the gravitational field will become and "look exactly like" Einstein's general theory of relativity in the "low-energy limit", or, more generally, like the Einstein-Yang-Mills-Dirac System. Indeed, quantum field theory itself is possibly the low-energy-effective-field-theory limit of a more fundamental theory such as the highly speculative superstring theory. Compare in this context the article effective field theory.

History Foundations The early development of the field involved Dirac, Fock, Pauli, Heisenberg, Bogolyubov. This phase of development culminated with the construction of the theory of quantum electrodynamics in the 1950s.

Gauge theory Gauge theory was formulated and quantized, leading to the unification of forces embodied in the standard model of particle physics. This effort started in the 1950s with the work of Yang and Mills, was carried on by Martinus Veltman and a host of others during the 1960s and completed by the 1970s through the work of Gerard 't Hooft, Frank Wilczek, David Gross and David Politzer.

Grand synthesis Parallel developments in the understanding of phase transitions in condensed matter physics led to the study of the renormalization group. This in turn led to the grand synthesis of theoretical physics which unified theories of particle and condensed matter physics through quantum field theory. This involved the work of Michael Fisher and Leo Kadanoff in the 1970s which led to the seminal reformulation of quantum field theory by Kenneth G. Wilson.

Principles Classical and quantum fields A classical field is a function defined over some region of space and time.[2] Two physical phenomena which are described by classical fields are Newtonian gravitation, described by Newtonian gravitational field g(x, t), and classical electromagnetism, described by the electric and magnetic fields E(x, t) and B(x, t). Because such fields can in principle take on distinct values at each point in space, they are said to have infinite degrees of freedom.[2] Classical field theory does not, however, account for the quantum-mechanical aspects of such physical phenomena. For instance, it is known from quantum mechanics that certain aspects of electromagnetism involve discrete particles—photons—rather than continuous fields. The business of quantum field theory is to write down a field that is, like a classical field, a function defined over space and time, but which also accommodates the observations of quantum mechanics. This is a quantum field. It is not immediately clear how to write down such a quantum field, since quantum mechanics has a structure very unlike a field theory. In its most general formulation, quantum mechanics is a theory of abstract operators (observables) acting on an abstract state space (Hilbert space), where the observables represent physically observable quantities and the state space represents the possible states of the system under study.[3] For instance, the fundamental observables associated with the motion of a single quantum mechanical particle are the position and momentum operators and . Field theory, in contrast, treats x as a way to index the field rather than as an operator.[4] There are two common ways of developing a quantum field: the path integral formalism and canonical quantization.[5] The latter of these is pursued in this article.

149

Quantum Field Theory Lagrangian formalism Quantum field theory frequently makes use of the Lagrangian formalism from classical field theory. This formalism is analogous to the Lagrangian formalism used in classical mechanics to solve for the motion of a particle under the influence of a field. In classical field theory, one writes down a Lagrangian density, , involving a field, φ(x,t), and possibly its first derivatives (∂φ/∂t and ∇φ), and then applies a field-theoretic form of the Euler–Lagrange equation. Writing coordinates (t, x) = (x0, x1, x2, x3) = xμ, this form of the Euler–Lagrange equation is[2]

where a sum over μ is performed according to the rules of Einstein notation. By solving this equation, one arrives at the "equations of motion" of the field.[2] For example, if one begins with the Lagrangian density

and then applies the Euler–Lagrange equation, one obtains the equation of motion

This equation is Newton's law of universal gravitation, expressed in differential form in terms of the gravitational potential φ(t, x) and the mass density ρ(t, x). Despite the nomenclature, the "field" under study is the gravitational potential, φ, rather than the gravitational field, g. Similarly, when classical field theory is used to study electromagnetism, the "field" of interest is the electromagnetic four-potential (V/c, A), rather than the electric and magnetic fields E and B. Quantum field theory uses this same Lagrangian procedure to determine the equations of motion for quantum fields. These equations of motion are then supplemented by commutation relations derived from the canonical quantization procedure described below, thereby incorporating quantum mechanical effects into the behavior of the field.

Single- and many-particle quantum mechanics In quantum mechanics, a particle (such as an electron or proton) is described by a complex wavefunction, ψ(x, t), whose time-evolution is governed by the Schrödinger equation:

Here m is the particle's mass and V(x) is the applied potential. Physical information about the behavior of the particle is extracted from the wavefunction by constructing probability density functions for various quantities; for example, the p.d.f. for the particle's position is ψ*(x)ψ(x), and the p.d.f. for the particle's momentum is −iħψ*(x)∂ψ/∂t. This treatment of quantum mechanics, where a particle's wavefunction evolves against a classical background potential V(x), is sometimes called first quantization. This description of quantum mechanics can be extended to describe the behavior of multiple particles, so long as the number and the type of particles remain fixed. The particles are described by a wavefunction ψ(x1, x2, ..., xN, t) which is governed by an extended version of the Schrödinger equation. Often one is interested in the case where then N particles are all of the same type (for example, the 18 electrons orbiting a neutral argon nucleus). As described in the article on identical particles, this implies that the state of the entire system must be either symmetric (bosons) or antisymmetric (fermions) when the coordinates of its constituent particles are exchanged. These multi-particle states are rather complicated to write. For example, the general quantum state of a system of N bosons is written as

150

Quantum Field Theory where

151

are the single-particle states, Nj is the number of particles occupying state j, and the sum is taken over

all possible permutations p acting on N elements. In general, this is a sum of N! (N factorial) distinct terms. There are several shortcomings to the above description of quantum mechanics which are addressed by quantum field theory. First, it is unclear how to extend quantum mechanics to include the effects of special relativity.[6] Attempted replacements for the Schrödinger equation, such as the Klein-Gordon equation or the Dirac equation, have many unsatisfactory qualities; for instance, they possess energy eigenvalues that extend to –∞, so that there seems to be no easy definition of a ground state. It turns out that such inconsistencies arise from relativistic wavefunctions having a probabilistic interpretation in position space, as probability conservation is not a relativistically covariant concept. The second shortcoming, related to the first, is that in quantum mechanics there is no mechanism to describe particle creation and annihilation;[7] this is crucial for describing phenomena such as pair production which result from the conversion between mass and energy according to the relativistic relation E = mc2.

Second quantization In this section, we will describe a method for constructing a quantum field theory called second quantization. This basically involves choosing a way to index the quantum mechanical degrees of freedom in the space of multiple identical-particle states. It is based on the Hamiltonian formulation of quantum mechanics; several other approaches exist, such as the Feynman path integral,[8] which uses a Lagrangian formulation. For an overview, see the article on quantization. Bosons For simplicity, we will first discuss second quantization for bosons, which form perfectly symmetric quantum states. Let us denote the mutually orthogonal single-particle states by and so on. For example, the 3-particle state with one particle in state

and two in state

is

The first step in second quantization is to express such quantum states in terms of occupation numbers, by listing the number of particles occupying each of the single-particle states etc. This is simply another way of labelling the states. For instance, the above 3-particle state is denoted as The next step is to expand the N-particle state space to include the state spaces for all possible values of N. This extended state space, known as a Fock space, is composed of the state space of a system with no particles (the so-called vacuum state), plus the state space of a 1-particle system, plus the state space of a 2-particle system, and so forth. There is a one-to-one correspondence between the occupation number representation and valid boson states in the Fock space. At this point, the quantum mechanical system has become a quantum field in the sense we described above. The field's elementary degrees of freedom are the occupation numbers, and each occupation number is indexed by a number indicating which of the single-particle states it refers to:

The properties of this quantum field can be explored by defining creation and annihilation operators, which add and subtract particles. They are analogous to ladder operators in the quantum harmonic oscillator problem, which added and subtracted energy quanta. However, these operators literally create and annihilate particles of a given quantum state. The bosonic annihilation operator and creation operator have the following effects:

Quantum Field Theory

152

It can be shown that these are operators in the usual quantum mechanical sense, i.e. linear operators acting on the Fock space. Furthermore, they are indeed Hermitian conjugates, which justifies the way we have written them. They can be shown to obey the commutation relation

where

stands for the Kronecker delta. These are precisely the relations obeyed by the ladder operators for an

infinite set of independent quantum harmonic oscillators, one for each single-particle state. Adding or removing bosons from each state is therefore analogous to exciting or de-exciting a quantum of energy in a harmonic oscillator. Applying an annihilation operator

followed by its corresponding creation operator

returns the number

of particles in the kth single-particle eigenstate: The combination of operators

is known as the number operator for the kth eigenstate.

The Hamiltonian operator of the quantum field (which, through the Schrödinger equation, determines its dynamics) can be written in terms of creation and annihilation operators. For instance, for a field of free (non-interacting) bosons, the total energy of the field is found by summing the energies of the bosons in each energy eigenstate. If the kth single-particle energy eigenstate has energy and there are bosons in this state, then the total energy of these bosons is

. The energy in the entire field is then a sum over

:

This can be turned into the Hamiltonian operator of the field by replacing operator,

with the corresponding number

. This yields

Fermions It turns out that a different definition of creation and annihilation must be used for describing fermions. According to the Pauli exclusion principle, fermions cannot share quantum states, so their occupation numbers Ni can only take on the value 0 or 1. The fermionic annihilation operators c and creation operators are defined by their actions on a Fock state thus

These obey an anticommutation relation:

One may notice from this that applying a fermionic creation operator twice gives zero, so it is impossible for the particles to share single-particle states, in accordance with the exclusion principle.

Quantum Field Theory

153

Field operators We have previously mentioned that there can be more than one way of indexing the degrees of freedom in a quantum field. Second quantization indexes the field by enumerating the single-particle quantum states. However, as we have discussed, it is more natural to think about a "field", such as the electromagnetic field, as a set of degrees of freedom indexed by position. To this end, we can define field operators that create or destroy a particle at a particular point in space. In particle physics, these operators turn out to be more convenient to work with, because they make it easier to formulate theories that satisfy the demands of relativity. Single-particle states are usually enumerated in terms of their momenta (as in the particle in a box problem.) We can construct field operators by applying the Fourier transform to the creation and annihilation operators for these states. For example, the bosonic field annihilation operator is

The bosonic field operators obey the commutation relation

where

stands for the Dirac delta function. As before, the fermionic relations are the same, with the

commutators replaced by anticommutators. The field operator is not the same thing as a single-particle wavefunction. The former is an operator acting on the Fock space, and the latter is a quantum-mechanical amplitude for finding a particle in some position. However, they are closely related, and are indeed commonly denoted with the same symbol. If we have a Hamiltonian with a space representation, say

where the indices i and j run over all particles, then the field theory Hamiltonian (in the non-relativistic limit and for negligible self-interactions) is

This looks remarkably like an expression for the expectation value of the energy, with

playing the role of the

wavefunction. This relationship between the field operators and wavefunctions makes it very easy to formulate field theories starting from space-projected Hamiltonians.

Implications Unification of fields and particles The "second quantization" procedure that we have outlined in the previous section takes a set of single-particle quantum states as a starting point. Sometimes, it is impossible to define such single-particle states, and one must proceed directly to quantum field theory. For example, a quantum theory of the electromagnetic field must be a quantum field theory, because it is impossible (for various reasons) to define a wavefunction for a single photon.[9] In such situations, the quantum field theory can be constructed by examining the mechanical properties of the classical field and guessing the corresponding quantum theory. For free (non-interacting) quantum fields, the quantum field theories obtained in this way have the same properties as those obtained using second quantization, such as well-defined creation and annihilation operators obeying commutation or anticommutation relations. Quantum field theory thus provides a unified framework for describing "field-like" objects (such as the electromagnetic field, whose excitations are photons) and "particle-like" objects (such as electrons, which are treated

Quantum Field Theory

154

as excitations of an underlying electron field), so long as one can treat interactions as "perturbations" of free fields. There are still unsolved problems relating to the more general case of interacting fields that may or may not be adequately described by perturbation theory. For more on this topic, see Haag's theorem. Physical meaning of particle indistinguishability The second quantization procedure relies crucially on the particles being identical. We would not have been able to construct a quantum field theory from a distinguishable many-particle system, because there would have been no way of separating and indexing the degrees of freedom. Many physicists prefer to take the converse interpretation, which is that quantum field theory explains what identical particles are. In ordinary quantum mechanics, there is not much theoretical motivation for using symmetric (bosonic) or antisymmetric (fermionic) states, and the need for such states is simply regarded as an empirical fact. From the point of view of quantum field theory, particles are identical if and only if they are excitations of the same underlying quantum field. Thus, the question "why are all electrons identical?" arises from mistakenly regarding individual electrons as fundamental objects, when in fact it is only the electron field that is fundamental. Particle conservation and non-conservation During second quantization, we started with a Hamiltonian and state space describing a fixed number of particles (N), and ended with a Hamiltonian and state space for an arbitrary number of particles. Of course, in many common situations N is an important and perfectly well-defined quantity, e.g. if we are describing a gas of atoms sealed in a box. From the point of view of quantum field theory, such situations are described by quantum states that are eigenstates of the number operator , which measures the total number of particles present. As with any quantum mechanical observable,

is conserved if it commutes with the Hamiltonian. In that case, the quantum state is

trapped in the N-particle subspace of the total Fock space, and the situation could equally well be described by ordinary N-particle quantum mechanics. (Strictly speaking, this is only true in the noninteracting case or in the low energy density limit of renormalized quantum field theories) For example, we can see that the free-boson Hamiltonian described above conserves particle number. Whenever the Hamiltonian operates on a state, each particle destroyed by an annihilation operator ak is immediately put back by the creation operator . On the other hand, it is possible, and indeed common, to encounter quantum states that are not eigenstates of

,

which do not have well-defined particle numbers. Such states are difficult or impossible to handle using ordinary quantum mechanics, but they can be easily described in quantum field theory as quantum superpositions of states having different values of N. For example, suppose we have a bosonic field whose particles can be created or destroyed by interactions with a fermionic field. The Hamiltonian of the combined system would be given by the Hamiltonians of the free boson and free fermion fields, plus a "potential energy" term such as

where

and ak denotes the bosonic creation and annihilation operators,

and ck denotes the fermionic creation

and annihilation operators, and Vq is a parameter that describes the strength of the interaction. This "interaction term" describes processes in which a fermion in state k either absorbs or emits a boson, thereby being kicked into a different eigenstate k+q. (In fact, this type of Hamiltonian is used to describe interaction between conduction electrons and phonons in metals. The interaction between electrons and photons is treated in a similar way, but is a little more complicated because the role of spin must be taken into account.) One thing to notice here is that even if we start out with a fixed number of bosons, we will typically end up with a superposition of states with different numbers of bosons at later times. The number of fermions, however, is conserved in this case. In condensed matter physics, states with ill-defined particle numbers are particularly important for describing the various superfluids. Many of the defining characteristics of a superfluid arise from the notion that its quantum state is

Quantum Field Theory a superposition of states with different particle numbers. In addition, the concept of a coherent state (used to model the laser and the BCS ground state) refers to a state with an ill-defined particle number but a well-defined phase.

Axiomatic approaches The preceding description of quantum field theory follows the spirit in which most physicists approach the subject. However, it is not mathematically rigorous. Over the past several decades, there have been many attempts to put quantum field theory on a firm mathematical footing by formulating a set of axioms for it. These attempts fall into two broad classes. The first class of axioms, first proposed during the 1950s, include the Wightman, Osterwalder-Schrader, and Haag-Kastler systems. They attempted to formalize the physicists' notion of an "operator-valued field" within the context of functional analysis, and enjoyed limited success. It was possible to prove that any quantum field theory satisfying these axioms satisfied certain general theorems, such as the spin-statistics theorem and the CPT theorem. Unfortunately, it proved extraordinarily difficult to show that any realistic field theory, including the Standard Model, satisfied these axioms. Most of the theories that could be treated with these analytic axioms were physically trivial, being restricted to low-dimensions and lacking interesting dynamics. The construction of theories satisfying one of these sets of axioms falls in the field of constructive quantum field theory. Important work was done in this area in the 1970s by Segal, Glimm, Jaffe and others. During the 1980s, a second set of axioms based on geometric ideas was proposed. This line of investigation, which restricts its attention to a particular class of quantum field theories known as topological quantum field theories, is associated most closely with Michael Atiyah and Graeme Segal, and was notably expanded upon by Edward Witten, Richard Borcherds, and Maxim Kontsevich. However, most of the physically relevant quantum field theories, such as the Standard Model, are not topological quantum field theories; the quantum field theory of the fractional quantum Hall effect is a notable exception. The main impact of axiomatic topological quantum field theory has been on mathematics, with important applications in representation theory, algebraic topology, and differential geometry. Finding the proper axioms for quantum field theory is still an open and difficult problem in mathematics. One of the Millennium Prize Problems—proving the existence of a mass gap in Yang-Mills theory—is linked to this issue.

Associated phenomena In the previous part of the article, we described the most general properties of quantum field theories. Some of the quantum field theories studied in various fields of theoretical physics possess additional special properties, such as renormalizability, gauge symmetry, and supersymmetry. These are described in the following sections.

Renormalization Early in the history of quantum field theory, it was found that many seemingly innocuous calculations, such as the perturbative shift in the energy of an electron due to the presence of the electromagnetic field, give infinite results. The reason is that the perturbation theory for the shift in an energy involves a sum over all other energy levels, and there are infinitely many levels at short distances that each give a finite contribution. Many of these problems are related to failures in classical electrodynamics that were identified but unsolved in the 19th century, and they basically stem from the fact that many of the supposedly "intrinsic" properties of an electron are tied to the electromagnetic field that it carries around with it. The energy carried by a single electron—its self energy—is not simply the bare value, but also includes the energy contained in its electromagnetic field, its attendant cloud of photons. The energy in a field of a spherical source diverges in both classical and quantum mechanics, but as discovered by Weisskopf with help from Furry, in quantum mechanics the divergence is much milder, going only as the logarithm of the radius of the sphere.

155

Quantum Field Theory The solution to the problem, presciently suggested by Stueckelberg, independently by Bethe after the crucial experiment by Lamb, implemented at one loop by Schwinger, and systematically extended to all loops by Feynman and Dyson, with converging work by Tomonaga in isolated postwar Japan, comes from recognizing that all the infinities in the interactions of photons and electrons can be isolated into redefining a finite number of quantities in the equations by replacing them with the observed values: specifically the electron 's mass and charge: this is called renormalization. The technique of renormalization recognizes that the problem is essentially purely mathematical, that extremely short distances are at fault. In order to define a theory on a continuum, first place a cutoff on the fields, by postulating that quanta cannot have energies above some extremely high value. This has the effect of replacing continuous space by a structure where very short wavelengths do not exist, as on a lattice. Lattices break rotational symmetry, and one of the crucial contributions made by Feynman, Pauli and Villars, and modernized by 't Hooft and Veltman, is a symmetry-preserving cutoff for perturbation theory (this process is called regularization). There is no known symmetrical cutoff outside of perturbation theory, so for rigorous or numerical work people often use an actual lattice. On a lattice, every quantity is finite but depends on the spacing. When taking the limit of zero spacing, we make sure that the physically observable quantities like the observed electron mass stay fixed, which means that the constants in the Lagrangian defining the theory depend on the spacing. Hopefully, by allowing the constants to vary with the lattice spacing, all the results at long distances become insensitive to the lattice, defining a continuum limit. The renormalization procedure only works for a certain class of quantum field theories, called renormalizable quantum field theories. A theory is perturbatively renormalizable when the constants in the Lagrangian only diverge at worst as logarithms of the lattice spacing for very short spacings. The continuum limit is then well defined in perturbation theory, and even if it is not fully well defined non-perturbatively, the problems only show up at distance scales that are exponentially small in the inverse coupling for weak couplings. The Standard Model of particle physics is perturbatively renormalizable, and so are its component theories (quantum electrodynamics/electroweak theory and quantum chromodynamics). Of the three components, quantum electrodynamics is believed to not have a continuum limit, while the asymptotically free SU(2) and SU(3) weak hypercharge and strong color interactions are nonperturbatively well defined. The renormalization group describes how renormalizable theories emerge as the long distance low-energy effective field theory for any given high-energy theory. Because of this, renormalizable theories are insensitive to the precise nature of the underlying high-energy short-distance phenomena. This is a blessing because it allows physicists to formulate low energy theories without knowing the details of high energy phenomenon. It is also a curse, because once a renormalizable theory like the standard model is found to work, it gives very few clues to higher energy processes. The only way high energy processes can be seen in the standard model is when they allow otherwise forbidden events, or if they predict quantitative relations between the coupling constants.

Gauge freedom A gauge theory is a theory that admits a symmetry with a local parameter. For example, in every quantum theory the global phase of the wave function is arbitrary and does not represent something physical. Consequently, the theory is invariant under a global change of phases (adding a constant to the phase of all wave functions, everywhere); this is a global symmetry. In quantum electrodynamics, the theory is also invariant under a local change of phase, that is – one may shift the phase of all wave functions so that the shift may be different at every point in space-time. This is a local symmetry. However, in order for a well-defined derivative operator to exist, one must introduce a new field, the gauge field, which also transforms in order for the local change of variables (the phase in our example) not to affect the derivative. In quantum electrodynamics this gauge field is the electromagnetic field. The change of local gauge of variables is termed gauge transformation. In quantum field theory the excitations of fields represent particles. The particle associated with excitations of the gauge field is the gauge boson, which is the photon in the case of quantum electrodynamics.

156

Quantum Field Theory

157

The degrees of freedom in quantum field theory are local fluctuations of the fields. The existence of a gauge symmetry reduces the number of degrees of freedom, simply because some fluctuations of the fields can be transformed to zero by gauge transformations, so they are equivalent to having no fluctuations at all, and they therefore have no physical meaning. Such fluctuations are usually called "non-physical degrees of freedom" or gauge artifacts; usually some of them have a negative norm, making them inadequate for a consistent theory. Therefore, if a classical field theory has a gauge symmetry, then its quantized version (i.e. the corresponding quantum field theory) will have this symmetry as well. In other words, a gauge symmetry cannot have a quantum anomaly. If a gauge symmetry is anomalous (i.e. not kept in the quantum theory) then the theory is non-consistent: for example, in quantum electrodynamics, had there been a gauge anomaly, this would require the appearance of photons with longitudinal polarization and polarization in the time direction, the latter having a negative norm, rendering the theory inconsistent; another possibility would be for these photons to appear only in intermediate processes but not in the final products of any interaction, making the theory non-unitary and again inconsistent (see optical theorem). In general, the gauge transformations of a theory consist of several different transformations, which may not be commutative. These transformations are together described by a mathematical object known as a gauge group. Infinitesimal gauge transformations are the gauge group generators. Therefore the number of gauge bosons is the group dimension (i.e. number of generators forming a basis). All the fundamental interactions in nature are described by gauge theories. These are: • Quantum chromodynamics, whose gauge group is SU(3). The gauge bosons are eight gluons. • The electroweak theory, whose gauge group is U(1) × SU(2), (a direct product of U(1) and SU(2)). • Gravity, whose classical theory is general relativity, admits the equivalence principle, which is a form of gauge symmetry. However, it is explicitly non-renormalizable.

Multivalued gauge transformations The gauge transformations which leave the theory invariant involve by definition only single-valued gauge functions which satisfy the Schwarz integrability criterion

An interesting extension of gauge transformations arises if the gauge functions

are allowed to be multivalued

functions which violate the integrability criterion. These are capable of changing the physical field strengths and are therefore no proper symmetry transformations. Nevertheless, the transformed field equations describe correctly the physical laws in the presence of the newly generated field strengths. See the textbook by H. Kleinert cited below for the applications to phenomena in physics.

Supersymmetry Supersymmetry assumes that every fundamental fermion has a superpartner that is a boson and vice versa. It was introduced in order to solve the so-called Hierarchy Problem, that is, to explain why particles not protected by any symmetry (like the Higgs boson) do not receive radiative corrections to its mass driving it to the larger scales (GUT, Planck...). It was soon realized that supersymmetry has other interesting properties: its gauged version is an extension of general relativity (Supergravity), and it is a key ingredient for the consistency of string theory. The way supersymmetry protects the hierarchies is the following: since for every particle there is a superpartner with the same mass, any loop in a radiative correction is cancelled by the loop corresponding to its superpartner, rendering the theory UV finite. Since no superpartners have yet been observed, if supersymmetry exists it must be broken (through a so-called soft term, which breaks supersymmetry without ruining its helpful features). The simplest models of this breaking require that the energy of the superpartners not be too high; in these cases, supersymmetry is expected to be observed by experiments at the Large Hadron Collider.

Quantum Field Theory

Notes [1] [2] [3] [4] [5] [6] [7] [8]

People.whitman.edu (http:/ / people. whitman. edu/ ~beckmk/ QM/ grangier/ Thorn_ajp. pdf) David Tong, Lectures on Quantum Field Theory (http:/ / www. damtp. cam. ac. uk/ user/ tong/ qft. html), chapter 1. Srednicki, Mark. Quantum Field Theory (1st ed.). p. 19. Srednicki, Mark. Quantum Field Theory (1st ed.). pp. 25–6. Zee, Anthony. Quantum Field Theory in a Nutshell (2nd ed.). p. 61. David Tong, Lectures on Quantum Field Theory (http:/ / www. damtp. cam. ac. uk/ user/ tong/ qft. html), Introduction. Zee, Anthony. Quantum Field Theory in a Nutshell (2nd ed.). p. 3. Abraham Pais, Inward Bound: Of Matter and Forces in the Physical World ISBN 0-19-851997-4. Pais recounts how his astonishment at the rapidity with which Feynman could calculate using his method. Feynman's method is now part of the standard methods for physicists. [9] Newton, T.D.; Wigner, E.P. (1949). "Localized states for elementary particles". Reviews of Modern Physics 21 (3): 400–406. Bibcode 1949RvMP...21..400N. doi:10.1103/RevModPhys.21.400.

References Further reading General readers: • Weinberg, S. Quantum Field Theory, Vols. I to III, 2000, Cambridge University Press: Cambridge, UK. • Feynman, R.P. (2001) [1964]. The Character of Physical Law. MIT Press. ISBN 0-262-56003-8. • Feynman, R.P. (2006) [1985]. QED: The Strange Theory of Light and Matter. Princeton University Press. ISBN 0-691-12575-9. • Gribbin, J. (1998). Q is for Quantum: Particle Physics from A to Z. Weidenfeld & Nicolson. ISBN 0-297-81752-3. • Schumm, Bruce A. (2004) Deep Down Things. Johns Hopkins Univ. Press. Chpt. 4. Introductory texts: • • • • • • •

• • • • • • •

Bogoliubov, N.; Shirkov, D. (1982). Quantum Fields. Benjamin-Cummings. ISBN 0-8053-0983-7. Frampton, P.H. (2000). Gauge Field Theories. Frontiers in Physics (2nd ed.). Wiley. Greiner, W; Müller, B. (2000). Gauge Theory of Weak Interactions. Springer. ISBN 3-540-67672-4. Itzykson, C.; Zuber, J.-B. (1980). Quantum Field Theory. McGraw-Hill. ISBN 0-07-032071-3. Kane, G.L. (1987). Modern Elementary Particle Physics. Perseus Books. ISBN 0-201-11749-5. Kleinert, H.; Schulte-Frohlinde, Verena (2001). Critical Properties of φ4-Theories (http://users.physik. fu-berlin.de/~kleinert/re.html#B6). World Scientific. ISBN 981-02-4658-7. Kleinert, H. (2008). Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation (http://users. physik.fu-berlin.de/~kleinert/public_html/kleiner_reb11/psfiles/mvf.pdf). World Scientific. ISBN 978-981-279-170-2. Loudon, R (1983). The Quantum Theory of Light. Oxford University Press. ISBN 0-19-851155-8. Mandl, F.; Shaw, G. (1993). Quantum Field Theory. John Wiley & Sons. ISBN 00471941867 . Peskin, M.; Schroeder, D. (1995). An Introduction to Quantum Field Theory. Westview Press. ISBN 0-201-50397-2. Ryder, L.H. (1985). Quantum Field Theory. Cambridge University Press. ISBN 0-521-33859-X. Srednicki, Mark (2007) Quantum Field Theory. (http://www.cambridge.org/us/catalogue/catalogue. asp?isbn=0521864496) Cambridge Univ. Press. Ynduráin, F.J. (1996). Relativistic Quantum Mechanics and Introduction to Field Theory (1st ed.). Springer. ISBN 978-3-540-60453-2. Zee, A. (2003). Quantum Field Theory in a Nutshell. Princeton University Press. ISBN 0-691-01019-6.

Advanced texts:

158

Quantum Field Theory • Bogoliubov, N.; Logunov, A.A.; Oksak, A.I.; Todorov, I.T. (1990). General Principles of Quantum Field Theory. Kluwer Academic Publishers. ISBN 978-0-7923-0540-8. • Weinberg, S. (1995). The Quantum Theory of Fields. 1–3. Cambridge University Press. Articles: • Gerard 't Hooft (2007) " The Conceptual Basis of Quantum Field Theory (http://www.phys.uu.nl/~thooft/ lectures/basisqft.pdf)" in Butterfield, J., and John Earman, eds., Philosophy of Physics, Part A. Elsevier: 661-730. • Frank Wilczek (1999) " Quantum field theory (http://arxiv.org/abs/hep-th/9803075)", Reviews of Modern Physics 71: S83-S95. Also doi=10.1103/Rev. Mod. Phys. 71.

External links • Hazewinkel, Michiel, ed. (2001), "Quantum field theory" (http://www.encyclopediaofmath.org/index. php?title=p/q076300), Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • Stanford Encyclopedia of Philosophy: " Quantum Field Theory (http://plato.stanford.edu/entries/ quantum-field-theory/)", by Meinard Kuhlmann. • Siegel, Warren, 2005. Fields. (http://insti.physics.sunysb.edu/~siegel/errata.html) A free text, also available from arXiv:hep-th/9912205. • Quantum Field Theory (http://www.nat.vu.nl/~mulders/QFT-0.pdf) by P. J. Mulders

String Theory String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything (TOE), a self-contained mathematical model that describes all fundamental forces and forms of matter. String theory posits that the elementary particles (i.e., electrons and quarks) within an atom are not 0-dimensional objects, but rather 1-dimensional oscillating lines ("strings"). The earliest string model, the bosonic string, incorporated only bosons, although this view developed to the superstring theory, which posits that a connection (a "supersymmetry") exists between bosons and fermions. String theories also require the existence of several extra dimensions to the universe that have been compactified into extremely small scales, in addition to the four known spacetime dimensions. The theory has its origins in an effort to understand the strong force, the dual resonance model (1969). Subsequent to this, five superstring theories were developed that incorporated fermions and possessed other properties necessary for a theory of everything. Since the mid-1990s, in particular due to insights from dualities shown to relate the five theories, an eleven-dimensional theory called M-theory is believed to encompass all of the previously distinct superstring theories. Many theoretical physicists (among them Stephen Hawking, Edward Witten, Juan Maldacena and Leonard Susskind) believe that string theory is a step towards the correct fundamental description of nature. This is because string theory allows for the consistent combination of quantum field theory and general relativity, agrees with general insights in quantum gravity (such as the holographic principle and black hole thermodynamics), and because it has passed many non-trivial checks of its internal consistency.[1][2][3][4] According to Hawking in particular, "M-theory is the only candidate for a complete theory of the universe."[5] Nevertheless, other physicists, such as Feynman and Glashow, have criticized string theory for not providing novel experimental predictions at accessible energy scales.[6]

159

String Theory

Overview String theory posits that the electrons and quarks within an atom are not 0-dimensional objects, but made up of 1-dimensional strings. These strings can oscillate, giving the observed particles their flavor, charge, mass and spin. Among the modes of oscillation of the string is a massless, spin-two state—a graviton. The existence of this graviton state and the fact that the equations describing string theory include Einstein's equations for general relativity mean that string theory is a quantum theory of gravity. Since string theory is widely believed[7] to be mathematically consistent, many hope that it fully describes our universe, making it a theory of everything. String theory is known to contain configurations that describe all the observed fundamental forces and matter but with a zero cosmological constant and some new fields.[8] Other configurations have different values of the cosmological constant, and are metastable but long-lived. This leads many to believe that there is at least one metastable solution that is quantitatively identical with the standard model, with a small cosmological constant, containing dark matter and a plausible mechanism for cosmic inflation. It is not yet known whether string theory has such a solution, nor how much freedom the theory allows to choose the details. String theories also include objects other than strings, called branes. The word brane, derived from "membrane", refers to a variety of interrelated objects, such as D-branes, black p-branes and Neveu–Schwarz 5-branes. These are extended objects that are charged sources for differential form generalizations of the vector potential electromagnetic field. These objects are related to one another by a variety of dualities. Black hole-like black p-branes are identified with D-branes, which are endpoints for strings, and this identification is called Gauge-gravity duality. Research on this equivalence has led to new insights on quantum chromodynamics, the fundamental theory of the strong nuclear force.[9][10][11][12] The strings make closed loops unless they encounter D-branes, where they can open up into 1-dimensional lines. The endpoints of the string cannot break off the D-brane, but they can slide around on it.

160

String Theory

161

The full theory does not yet have a satisfactory definition in all circumstances, since the scattering of strings is most straightforwardly defined by a perturbation theory. The complete quantum mechanics of high dimensional branes is not easily defined, and the behavior of string theory in cosmological settings (time-dependent backgrounds) is not fully worked out. It is also not clear as to whether there is any principle by which string theory selects its vacuum state, the spacetime configuration that determines the properties of our universe (see string theory landscape).

Basic properties String theory can be formulated in terms of an action principle, either the Nambu-Goto action or the Polyakov action, which describe how strings propagate through space and time. In the absence of external interactions, string dynamics are governed by tension and kinetic energy, which combine to produce oscillations. The quantum mechanics of strings implies these oscillations exist in discrete vibrational modes, the spectrum of the theory. On distance scales larger than the string radius, each oscillation mode behaves as a different species of particle, with its mass, spin and charge determined by the string's dynamics. Splitting and recombination of strings correspond to particle emission and absorption, giving rise to the interactions between particles. An analogy for strings' modes of vibration is a guitar string's production of multiple distinct musical notes. In the analogy, different notes correspond to different particles. One difference is the guitar string exists in 3 dimensions, so that there are only two dimensions transverse to the string. Fundamental strings exist in 9 dimensions and the strings can vibrate in any direction, meaning that the spectrum of vibrational modes is much richer.

Levels of magnification: 1. Macroscopic level – Matter 2. Molecular level 3. Atomic level – Protons, neutrons, and electrons 4. Subatomic level – Electron 5. Subatomic level – Quarks 6. String level

String theory includes both open strings, which have two distinct endpoints, and closed strings making a complete loop. The two types of string behave in slightly different ways, yielding two different spectra. For example, in most string theories one of the closed string modes is the graviton, and one of the open string modes is the photon. Because the two ends of an open string can always meet and connect, forming a closed string, there are no string theories without closed strings. The earliest string model, the bosonic string, incorporated only bosonic degrees of freedom. This model describes, in low enough energies, a quantum gravity theory, which also includes (if open strings are incorporated as well) gauge fields such as the photon (or, in more general terms, any gauge theory). However, this model has problems. What is most significant is that the theory has a fundamental instability, believed to result in the decay (at least partially) of spacetime itself. In addition, as the name implies, the spectrum of particles contains only bosons, particles which, like the photon, obey particular rules of behavior. In broad terms, bosons are the constituents of radiation, but not of matter, which is made of fermions. Investigating how a string theory may include fermions in its spectrum led to the invention of supersymmetry, a mathematical relation between bosons and fermions. String theories that include fermionic vibrations are now known as superstring theories; several kinds have been described, but all are now thought to be different limits of M-theory.

String Theory Some qualitative properties of quantum strings can be understood in a fairly simple fashion. For example, quantum strings have tension, much like regular strings made of twine; this tension is considered a fundamental parameter of the theory. The tension of a quantum string is closely related to its size. Consider a closed loop of string, left to move through space without external forces. Its tension will tend to contract it into a smaller and smaller loop. Classical intuition suggests that it might shrink to a single point, but this would violate Heisenberg's uncertainty principle. The characteristic size of the string loop will be a balance between the tension force, acting to make it small, and the uncertainty effect, which keeps it "stretched". As a consequence, the minimum size of a string is related to the string tension.

Worldsheet A point-like particle's motion may be described by drawing a graph of its position (in one or two dimensions of space) against time. The resulting picture depicts the worldline of the particle (its 'history') in spacetime. By analogy, a similar graph depicting the progress of a string as time passes by can be obtained; the string (a one-dimensional object — a small line — by itself) will trace out a surface (a two-dimensional manifold), known as the worldsheet. The different string modes (representing different particles, such as photon or graviton) are surface waves on this manifold. A closed string looks like a small loop, so its worldsheet will look like a pipe or, in more general terms, a Riemann surface (a two-dimensional oriented manifold) with no boundaries (i.e., no edge). An open string looks like a short line, so its worldsheet will look like a strip or, in more general terms, a Riemann surface with a boundary. Strings can split and connect. This is reflected by the form of their worldsheet (in more accurate terms, by its topology). For example, if a closed string splits, its worldsheet will look like a single pipe splitting (or connected) to two pipes (often referred to as a pair of pants — see drawing at right). If a closed string splits and its two parts later reconnect, its worldsheet will look like a single pipe splitting to two and then reconnecting, which also looks like a torus connected to two pipes (one Interaction in the subatomic world: world lines of point-like particles in the representing the ingoing string, and the Standard Model or a world sheet swept up by closed strings in string theory other — the outgoing one). An open string doing the same thing will have its worldsheet looking like a ring connected to two strips. Note that the process of a string splitting (or strings connecting) is a global process of the worldsheet, not a local one: Locally, the worldsheet looks the same everywhere, and it is not possible to determine a single point on the worldsheet where the splitting occurs. Therefore, these processes are an integral part of the theory, and are described by the same dynamics that controls the string modes. In some string theories (namely, closed strings in Type I and some versions of the bosonic string), strings can split and reconnect in an opposite orientation (as in a Möbius strip or a Klein bottle). These theories are called unoriented. In formal terms, the worldsheet in these theories is a non-orientable surface.

162

String Theory

163

Dualities Before the 1990s, string theorists believed there were five distinct superstring theories: open type I, closed type I, closed type IIA, closed type IIB, and the two flavors of heterotic string theory (SO(32) and E8×E8).[13] The thinking was that out of these five candidate theories, only one was the actual correct theory of everything, and that theory was the one whose low energy limit, with ten spacetime dimensions compactified down to four, matched the physics observed in our world today. It is now believed that this picture was incorrect and that the five superstring theories are connected to one another as if they are each a special case of some more fundamental theory (thought to be M-theory). These theories are related by transformations that are called dualities. If two theories are related by a duality transformation, it means that the first theory can be transformed in some way so that it ends up looking just like the second theory. The two theories are then said to be dual to one another under that kind of transformation. Put differently, the two theories are mathematically different descriptions of the same phenomena. These dualities link quantities that were also thought to be separate. Large and small distance scales, as well as strong and weak coupling strengths, are quantities that have always marked very distinct limits of behavior of a physical system in both classical field theory and quantum particle physics. But strings can obscure the difference between large and small, strong and weak, and this is how these five very different theories end up being related. T-duality relates the large and small distance scales between string theories, whereas S-duality relates strong and weak coupling strengths between string theories. U-duality links T-duality and S-duality. String theories Type

Spacetime dimensions

Details

Bosonic

26

Only bosons, no fermions, meaning only forces, no matter, with both open and closed strings; major flaw: a particle with imaginary mass, called the tachyon, representing an instability in the theory.

I

10

Supersymmetry between forces and matter, with both open and closed strings; no tachyon; group symmetry is SO(32)

IIA

10

Supersymmetry between forces and matter, with only closed strings bound to D-branes; no tachyon; massless fermions are non-chiral

IIB

10

Supersymmetry between forces and matter, with only closed strings bound to D-branes; no tachyon; massless fermions are chiral

HO

10

Supersymmetry between forces and matter, with closed strings only; no tachyon; heterotic, meaning right moving and left moving strings differ; group symmetry is SO(32)

HE

10

Supersymmetry between forces and matter, with closed strings only; no tachyon; heterotic, meaning right moving and left moving strings differ; group symmetry is E8×E8

Note that in the type IIA and type IIB string theories closed strings are allowed to move everywhere throughout the ten-dimensional spacetime (called the bulk), while open strings have their ends attached to D-branes, which are membranes of lower dimensionality (their dimension is odd — 1, 3, 5, 7 or 9 — in type IIA and even — 0, 2, 4, 6 or 8 — in type IIB, including the time direction).

Extra dimensions Number of dimensions An intriguing feature of string theory is that it predicts extra dimensions. In classical string theory the number of dimensions is not fixed by any consistency criterion. However, to make a consistent quantum theory, string theory is required to live in a spacetime of the so-called "critical dimension": we must have 26 spacetime dimensions for the bosonic string and 10 for the superstring. This is necessary to ensure the vanishing of the conformal anomaly of the worldsheet conformal field theory. Modern understanding indicates that there exist less-trivial ways of satisfying this criterion. Cosmological solutions exist in a wider variety of dimensionalities, and these different dimensions are

String Theory related by dynamical transitions. The dimensions are more precisely different values of the "effective central charge", a count of degrees of freedom that reduces to dimensionality in weakly curved regimes.[14][15] One such theory is the 11-dimensional M-theory, which requires spacetime to have eleven dimensions,[16] as opposed to the usual three spatial dimensions and the fourth dimension of time. The original string theories from the 1980s describe special cases of M-theory where the eleventh dimension is a very small circle or a line, and if these formulations are considered as fundamental, then string theory requires ten dimensions. But the theory also describes universes like ours, with four observable spacetime dimensions, as well as universes with up to 10 flat space dimensions, and also cases where the position in some of the dimensions is not described by a real number, but by a completely different type of mathematical quantity. So the notion of spacetime dimension is not fixed in string theory: it is best thought of as different in different circumstances.[17] Nothing in Maxwell's theory of electromagnetism or Einstein's theory of relativity makes this kind of prediction; these theories require physicists to insert the number of dimensions "by both hands", and this number is fixed and independent of potential energy. String theory allows one to relate the number of dimensions to scalar potential energy. In technical terms, this happens because a gauge anomaly exists for every separate number of predicted dimensions, and the gauge anomaly can be counteracted by including nontrivial potential energy into equations to solve motion. Furthermore, the absence of potential energy in the "critical dimension" explains why flat spacetime solutions are possible. This can be better understood by noting that a photon included in a consistent theory (technically, a particle carrying a force related to an unbroken gauge symmetry) must be massless. The mass of the photon that is predicted by string theory depends on the energy of the string mode that represents the photon. This energy includes a contribution from the Casimir effect, namely from quantum fluctuations in the string. The size of this contribution depends on the number of dimensions, since for a larger number of dimensions there are more possible fluctuations in the string position. Therefore, the photon in flat spacetime will be massless—and the theory consistent—only for a particular number of dimensions.[18] When the calculation is done, the critical dimensionality is not four as one may expect (three axes of space and one of time). The subset of X is equal to the relation of photon fluctuations in a linear dimension. Flat space string theories are 26-dimensional in the bosonic case, while superstring and M-theories turn out to involve 10 or 11 dimensions for flat solutions. In bosonic string theories, the 26 dimensions come from the Polyakov equation.[19] Starting from any dimension greater than four, it is necessary to consider how these are reduced to four dimensional spacetime. Compact dimensions Two ways have been proposed to resolve this apparent contradiction. The first is to compactify the extra dimensions; i.e., the 6 or 7 extra dimensions are so small as to be undetectable by present-day experiments. To retain a high degree of supersymmetry, these compactification spaces must be very special, as reflected in their holonomy. A 6-dimensional manifold must have SU(3) structure, a particular case (torsionless) of this being SU(3) holonomy, making it a Calabi–Yau space, and a 7-dimensional manifold must have G2 structure, with G2 holonomy again being a specific, simple, case. Such spaces have been studied in attempts to relate string theory to the 4-dimensional Standard Model, in part due to the computational simplicity afforded by the assumption of Calabi–Yau manifold (3D projection) supersymmetry. More recently, progress has been made constructing more realistic compactifications without the degree of symmetry of Calabi–Yau or G2 manifolds.

164

String Theory A standard analogy for this is to consider multidimensional space as a garden hose. If the hose is viewed from a sufficient distance, it appears to have only one dimension, its length. Indeed, think of a ball just small enough to enter the hose. Throwing such a ball inside the hose, the ball would move more or less in one dimension; in any experiment we make by throwing such balls in the hose, the only important movement will be one-dimensional, that is, along the hose. However, as one approaches the hose, one discovers that it contains a second dimension, its circumference. Thus, an ant crawling inside it would move in two dimensions (and a fly flying in it would move in three dimensions). This "extra dimension" is only visible within a relatively close range to the hose, or if one "throws in" small enough objects. Similarly, the extra compact dimensions are only "visible" at extremely small distances, or by experimenting with particles with extremely small wavelengths (of the order of the compact dimension's radius), which in quantum mechanics means very high energies (see wave-particle duality). Brane-world scenario Another possibility is that we are "stuck" in a 3+1 dimensional (three spatial dimensions plus one time dimension) subspace of the full universe. Properly localized matter and Yang-Mills gauge fields will typically exist if the sub-spacetime is an exceptional set of the larger universe.[20] These "exceptional sets" are ubiquitous in Calabi–Yau n-folds and may be described as subspaces without local deformations, akin to a crease in a sheet of paper or a crack in a crystal, the neighborhood of which is markedly different from the exceptional subspace itself. However, until the work of Randall and Sundrum,[21] it was not known that gravity can be properly localized to a sub-spacetime. In addition, spacetime may be stratified, containing strata of various dimensions, allowing us to inhabit the 3+1-dimensional stratum—such geometries occur naturally in Calabi–Yau compactifications.[22] Such sub-spacetimes are D-branes, hence such models are known as brane-world scenarios. Effect of the hidden dimensions In either case, gravity acting in the hidden dimensions affects other non-gravitational forces such as electromagnetism. In fact, Kaluza's early work demonstrated that general relativity in five dimensions actually predicts the existence of electromagnetism. However, because of the nature of Calabi–Yau manifolds, no new forces appear from the small dimensions, but their shape has a profound effect on how the forces between the strings appear in our four-dimensional universe. In principle, therefore, it is possible to deduce the nature of those extra dimensions by requiring consistency with the standard model, but this is not yet a practical possibility. It is also possible to extract information regarding the hidden dimensions by precision tests of gravity, but so far these have only put upper limitations on the size of such hidden dimensions.

D-branes Another key feature of string theory is the existence of D-branes. These are membranes of different dimensionality (anywhere from a zero dimensional membrane—which is in fact a point—and up, including 2-dimensional membranes, 3-dimensional volumes, and so on). D-branes are defined by the fact that worldsheet boundaries are attached to them. D-branes have mass, since they emit and absorb closed strings that describe gravitons, and — in superstring theories — charge as well, since they couple to open strings that describe gauge interactions. From the point of view of open strings, D-branes are objects to which the ends of open strings are attached. The open strings attached to a D-brane are said to "live" on it, and they give rise to gauge theories "living" on it (since one of the open string modes is a gauge boson such as the photon). In the case of one D-brane there will be one type of a gauge boson and we will have an Abelian gauge theory (with the gauge boson being the photon). If there are multiple parallel D-branes there will be multiple types of gauge bosons, giving rise to a non-Abelian gauge theory. D-branes are thus gravitational sources, on which a gauge theory "lives". This gauge theory is coupled to gravity (which is said to exist in the bulk), so that normally each of these two viewpoints is incomplete.

165

String Theory

Testability and experimental predictions Several major difficulties complicate efforts to test string theory. The most significant is the extremely small size of the Planck length, which is expected to be close to the string length (the characteristic size of a string, where strings become easily distinguishable from particles). Another issue is the huge number of metastable vacua of string theory, which might be sufficiently diverse to accommodate almost any phenomena we might observe at lower energies.

Predictions String harmonics One unique prediction of string theory is the existence of string harmonics: at sufficiently high energies, the string-like nature of particles would become obvious. There should be heavier copies of all particles, corresponding to higher vibrational harmonics of the string. It is not clear how high these energies are. In most conventional string models they would be not far below the Planck energy, around 1014 times higher than the energies accessible in the newest particle accelerator, the LHC, making this prediction impossible to test with any particle accelerator in the foreseeable future. However, in models with large extra dimensions they could potentially be produced at the LHC or at energies not far above its reach. Cosmology String theory as currently understood makes a series of predictions for the structure of the universe at the largest scales. Many phases in string theory have very large, positive vacuum energy.[23] Regions of the universe that are in such a phase will inflate exponentially rapidly in a process known as eternal inflation. As such, the theory predicts that most of the universe is very rapidly expanding. However, these expanding phases are not stable, and can decay via the nucleation of bubbles of lower vacuum energy. Since our local region of the universe is not very rapidly expanding, string theory predicts we are inside such a bubble. The spatial curvature of the "universe" inside the bubbles that form by this process is negative, a testable prediction.[24] Moreover, other bubbles will eventually form in the parent vacuum outside the bubble and collide with it. These collisions lead to potentially observable imprints on cosmology.[25][26] However, it is possible that neither of these will be observed if the spatial curvature is too small and the collisions are too rare. Cosmic strings Under certain circumstances, fundamental strings produced at or near the end of inflation can be "stretched" to astronomical proportions. These cosmic strings could be observed in various ways, for instance by their gravitational lensing effects. However, certain field theories also predict cosmic strings arising from topological defects in the field configuration.[27] Strength of gravity Theories with extra dimensions predict that the strength of gravity increases much more rapidly at small distances than is the case in 3 dimensions (where it increase as r−2). Depending on the size of the dimensions, this could lead to phenomena such as the production of micro black holes at the LHC, or be detected in microgravity experiments.

166

String Theory Quantum chromodynamics String theory was originally proposed as a theory of hadrons, and its study has led to new insights on quantum chromodynamics, a gauge theory, which is the fundamental theory of the strong nuclear force. To this end, it is hoped that a gravitational theory dual to quantum chromodynamics will be found.[28] A mathematical technique from string theory (the AdS/CFT correspondence) has been used to describe qualitative features of quark–gluon plasma behavior in relativistic heavy-ion collisions;[9][10][11][12] the physics, however, is strictly that of standard quantum chromodynamics, which has been quantitatively modeled by lattice QCD methods with good results.[29] Supersymmetry If confirmed experimentally, supersymmetry could also be considered evidence, because it was discovered in the context of string theory, and all consistent string theories are supersymmetric. However, the absence of supersymmetric particles at energies accessible to the LHC would not necessarily disprove string theory, since the energy scale at which supersymmetry is broken could be well above the accelerator's range. A central problem for applications is that the best-understood backgrounds of string theory preserve much of the supersymmetry of the underlying theory, which results in time-invariant spacetimes: At present, string theory cannot deal well with time-dependent, cosmological backgrounds. However, several models have been proposed to predict supersymmetry breaking, the most notable one being the KKLT model,[23] which incorporates branes and fluxes to make a metastable compactification. AdS/CFT correspondence AdS/CFT relates string theory to gauge theory, and allows contact with low energy experiments in quantum chromodynamics. This type of string theory, which describes only the strong interactions, is much less controversial today than string theories of everything (although two decades ago, it was the other way around).[30] Coupling constant unification Grand unification natural in string theories of everything requires that the coupling constants of the four forces meet at one point under renormalization group rescaling. This is also a falsifiable statement, but it is not restricted to string theory, but is shared by grand unified theories.[31] The LHC will be used both for testing AdS/CFT, and to check if the electroweakstrong unification does happen as predicted.[32]

Gauge/gravity duality Gauge/gravity duality is a conjectured duality between a quantum theory of gravity in certain cases and gauge theory in a lower number of dimensions. This means that each predicted phenomenon and quantity in one theory has an analogue in the other theory, with a "dictionary" translating from one theory to the other.

Description of the duality In certain cases the gauge theory on the D-branes is decoupled from the gravity living in the bulk; thus open strings attached to the D-branes are not interacting with closed strings. Such a situation is termed a decoupling limit. In those cases, the D-branes have two independent alternative descriptions. As discussed above, from the point of view of closed strings, the D-branes are gravitational sources, and thus we have a gravitational theory on spacetime with some background fields. From the point of view of open strings, the physics of the D-branes is described by the appropriate gauge theory. Therefore in such cases it is often conjectured that the gravitational theory on spacetime with the appropriate background fields is dual (i.e. physically equivalent) to the gauge theory on the boundary of this spacetime (since the subspace filled by the D-branes is the boundary of this spacetime). So far, this duality has not been proven in any cases, so there is also disagreement among string theorists regarding how strong the duality

167

String Theory applies to various models.

Examples and intuition The best known example and the first one to be studied is the duality between Type IIB superstring on AdS5 × S5 (a product space of a five-dimensional Anti de Sitter space and a five-sphere) on one hand, and N = 4 supersymmetric Yang–Mills theory on the four-dimensional boundary of the Anti de Sitter space (either a flat four-dimensional spacetime R3,1 or a three-sphere with time S3 × R). This is known as the AdS/CFT correspondence,[33][34][35][36] a name often used for Gauge / gravity duality in general. This duality can be thought of as follows: suppose there is a spacetime with a gravitational source, for example an extremal black hole.[37] When particles are far away from this source, they are described by closed strings (i.e., a gravitational theory, or usually supergravity). As the particles approach the gravitational source, they can still be described by closed strings; also, they can be described by objects similar to QCD strings,[38][39][40] which are made of gauge bosons (gluons) and other gauge theory degrees of freedom.[41] So if one is able (in a decoupling limit) to describe the gravitational system as two separate regions — one (the bulk) far away from the source, and the other close to the source — then the latter region can also be described by a gauge theory on D-branes. This latter region (close to the source) is termed the near-horizon limit, since usually there is an event horizon around (or at) the gravitational source. In the gravitational theory, one of the directions in spacetime is the radial direction, going from the gravitational source and away (toward the bulk). The gauge theory lives only on the D-brane itself, so it does not include the radial direction: it lives in a spacetime with one less dimension compared to the gravitational theory (in fact, it lives on a spacetime identical to the boundary of the near-horizon gravitational theory). Let us understand how the two theories are still equivalent: The physics of the near-horizon gravitational theory involves only on-shell states (as usual in string theory), while the field theory includes also off-shell correlation function. The on-shell states in the near-horizon gravitational theory can be thought of as describing only particles arriving from the bulk to the near-horizon region and interacting there between themselves. In the gauge theory, these are "projected" onto the boundary, so that particles that arrive at the source from different directions will be seen in the gauge theory as (off-shell) quantum fluctuations far apart from each other, while particles arriving at the source from almost the same direction in space will be seen in the gauge theory as (off-shell) quantum fluctuations close to each other. Thus the angle between the arriving particles in the gravitational theory translates to the distance scale between quantum fluctuations in the gauge theory. The angle between arriving particles in the gravitational theory is related to the radial distance from the gravitational source at which the particles interact: The larger the angle the closer the particles have to get to the source to interact with each other. On the other hand, the scale of the distance between quantum fluctuations in a quantum field theory is related (inversely) to the energy scale in this theory, so small radius in the gravitational theory translates to low energy scale in the gauge theory (i.e., the IR regime of the field theory), while large radius in the gravitational theory translates to high energy scale in the gauge theory (i.e., the UV regime of the field theory). A simple example to this principle is that if in the gravitational theory there is a setup in which the dilaton field (which determines the strength of the coupling) is decreasing with the radius, then its dual field theory will be asymptotically free, i.e. its coupling will grow weaker in high energies.

168

String Theory

History Some of the structures reintroduced by string theory arose for the first time much earlier as part of the program of classical unification started by Albert Einstein. The first person to add a fifth dimension to general relativity was German mathematician Theodor Kaluza in 1919, who noted that gravity in five dimensions describes both gravity and electromagnetism in four. In 1926, the Swedish physicist Oskar Klein gave a physical interpretation of the unobservable extra dimension — it is wrapped into a small circle. Einstein introduced a non-symmetric metric tensor, while much later Brans and Dicke added a scalar component to gravity. These ideas would be revived within string theory, where they are demanded by consistency conditions. String theory was originally developed during the late 1960s and early 1970s as a never completely successful theory of hadrons, the subatomic particles like the proton and neutron that feel the strong interaction. In the 1960s, Geoffrey Chew and Steven Frautschi discovered that the mesons make families called Regge trajectories with masses related to spins in a way that was later understood by Yoichiro Nambu, Holger Bech Nielsen and Leonard Susskind to be the relationship expected from rotating strings. Chew advocated making a theory for the interactions of these trajectories that did not presume that they were composed of any fundamental particles, but would construct their interactions from self-consistency conditions on the S-matrix. The S-matrix approach was started by Werner Heisenberg in the 1940s as a way of constructing a theory that did not rely on the local notions of space and time, which Heisenberg believed break down at the nuclear scale. While the scale was off by many orders of magnitude, the approach he advocated was ideally suited for a theory of quantum gravity. Working with experimental data, R. Dolen, D. Horn and C. Schmid[42] developed some sum rules for hadron exchange. When a particle and antiparticle scatter, virtual particles can be exchanged in two qualitatively different ways. In the s-channel, the two particles annihilate to make temporary intermediate states that fall apart into the final state particles. In the t-channel, the particles exchange intermediate states by emission and absorption. In field theory, the two contributions add together, one giving a continuous background contribution, the other giving peaks at certain energies. In the data, it was clear that the peaks were stealing from the background — the authors interpreted this as saying that the t-channel contribution was dual to the s-channel one, meaning both described the whole amplitude and included the other. The result was widely advertised by Murray Gell-Mann, leading Gabriele Veneziano to construct a scattering amplitude that had the property of Dolen-Horn-Schmid duality, later renamed world-sheet duality. The amplitude needed poles where the particles appear, on straight line trajectories, and there is a special mathematical function whose poles are evenly spaced on half the real line— the Gamma function— which was widely used in Regge theory. By manipulating combinations of Gamma functions, Veneziano was able to find a consistent scattering amplitude with poles on straight lines, with mostly positive residues, which obeyed duality and had the appropriate Regge scaling at high energy. The amplitude could fit near-beam scattering data as well as other Regge type fits, and had a suggestive integral representation that could be used for generalization. Over the next years, hundreds of physicists worked to complete the bootstrap program for this model, with many surprises. Veneziano himself discovered that for the scattering amplitude to describe the scattering of a particle that appears in the theory, an obvious self-consistency condition, the lightest particle must be a tachyon. Miguel Virasoro and Joel Shapiro found a different amplitude now understood to be that of closed strings, while Ziro Koba and Holger Nielsen generalized Veneziano's integral representation to multiparticle scattering. Veneziano and Sergio Fubini introduced an operator formalism for computing the scattering amplitudes that was a forerunner of world-sheet conformal theory, while Virasoro understood how to remove the poles with wrong-sign residues using a constraint on the states. Claud Lovelace calculated a loop amplitude, and noted that there is an inconsistency unless the dimension of the theory is 26. Charles Thorn, Peter Goddard and Richard Brower went on to prove that there are no wrong-sign propagating states in dimensions less than or equal to 26. In 1969, Yoichiro Nambu, Holger Bech Nielsen, and Leonard Susskind recognized that the theory could be given a description in space and time in terms of strings. The scattering amplitudes were derived systematically from the

169

String Theory action principle by Peter Goddard, Jeffrey Goldstone, Claudio Rebbi, and Charles Thorn, giving a space-time picture to the vertex operators introduced by Veneziano and Fubini and a geometrical interpretation to the Virasoro conditions. In 1970, Pierre Ramond added fermions to the model, which led him to formulate a two-dimensional supersymmetry to cancel the wrong-sign states. John Schwarz and André Neveu added another sector to the fermi theory a short time later. In the fermion theories, the critical dimension was 10. Stanley Mandelstam formulated a world sheet conformal theory for both the bose and fermi case, giving a two-dimensional field theoretic path-integral to generate the operator formalism. Michio Kaku and Keiji Kikkawa gave a different formulation of the bosonic string, as a string field theory, with infinitely many particle types and with fields taking values not on points, but on loops and curves. In 1974, Tamiaki Yoneya discovered that all the known string theories included a massless spin-two particle that obeyed the correct Ward identities to be a graviton. John Schwarz and Joel Scherk came to the same conclusion and made the bold leap to suggest that string theory was a theory of gravity, not a theory of hadrons. They reintroduced Kaluza–Klein theory as a way of making sense of the extra dimensions. At the same time, quantum chromodynamics was recognized as the correct theory of hadrons, shifting the attention of physicists and apparently leaving the bootstrap program in the dustbin of history. String theory eventually made it out of the dustbin, but for the following decade all work on the theory was completely ignored. Still, the theory continued to develop at a steady pace thanks to the work of a handful of devotees. Ferdinando Gliozzi, Joel Scherk, and David Olive realized in 1976 that the original Ramond and Neveu Schwarz-strings were separately inconsistent and needed to be combined. The resulting theory did not have a tachyon, and was proven to have space-time supersymmetry by John Schwarz and Michael Green in 1981. The same year, Alexander Polyakov gave the theory a modern path integral formulation, and went on to develop conformal field theory extensively. In 1979, Daniel Friedan showed that the equations of motions of string theory, which are generalizations of the Einstein equations of General Relativity, emerge from the Renormalization group equations for the two-dimensional field theory. Schwarz and Green discovered T-duality, and constructed two superstring theories — IIA and IIB related by T-duality, and type I theories with open strings. The consistency conditions had been so strong, that the entire theory was nearly uniquely determined, with only a few discrete choices. In the early 1980s, Edward Witten discovered that most theories of quantum gravity could not accommodate chiral fermions like the neutrino. This led him, in collaboration with Luis Alvarez-Gaumé to study violations of the conservation laws in gravity theories with anomalies, concluding that type I string theories were inconsistent. Green and Schwarz discovered a contribution to the anomaly that Witten and Alvarez-Gaumé had missed, which restricted the gauge group of the type I string theory to be SO(32). In coming to understand this calculation, Edward Witten became convinced that string theory was truly a consistent theory of gravity, and he became a high-profile advocate. Following Witten's lead, between 1984 and 1986, hundreds of physicists started to work in this field, and this is sometimes called the first superstring revolution. During this period, David Gross, Jeffrey Harvey, Emil Martinec, and Ryan Rohm discovered heterotic strings. The gauge group of these closed strings was two copies of E8, and either copy could easily and naturally include the standard model. Philip Candelas, Gary Horowitz, Andrew Strominger and Edward Witten found that the Calabi-Yau manifolds are the compactifications that preserve a realistic amount of supersymmetry, while Lance Dixon and others worked out the physical properties of orbifolds, distinctive geometrical singularities allowed in string theory. Cumrun Vafa generalized T-duality from circles to arbitrary manifolds, creating the mathematical field of mirror symmetry. Daniel Friedan, Emil Martinec and Stephen Shenker further developed the covariant quantization of the superstring using conformal field theory techniques. David Gross and Vipul Periwal discovered that string perturbation theory was divergent. Stephen Shenker showed it diverged much faster than in field theory suggesting that new non-perturbative objects were missing. In the 1990s, Joseph Polchinski discovered that the theory requires higher-dimensional objects, called D-branes and identified these with the black-hole solutions of supergravity. These were understood to be the new objects suggested

170

String Theory by the perturbative divergences, and they opened up a new field with rich mathematical structure. It quickly became clear that D-branes and other p-branes, not just strings, formed the matter content of the string theories, and the physical interpretation of the strings and branes was revealed — they are a type of black hole. Leonard Susskind had incorporated the holographic principle of Gerardus 't Hooft into string theory, identifying the long highly excited string states with ordinary thermal black hole states. As suggested by 't Hooft, the fluctuations of the black hole horizon, the world-sheet or world-volume theory, describes not only the degrees of freedom of the black hole, but all nearby objects too. In 1995, at the annual conference of string theorists at the University of Southern California (USC), Edward Witten gave a speech on string theory that in essence united the five string theories that existed at the time, and giving birth to a new 11-dimensional theory called M-theory. M-theory was also foreshadowed in the work of Paul Townsend at approximately the same time. The flurry of activity that began at this time is sometimes called the second superstring revolution. During this period, Tom Banks, Willy Fischler, Stephen Shenker and Leonard Susskind formulated matrix theory, a full holographic description of M-theory using IIA D0 branes.[43] This was the first definition of string theory that was fully non-perturbative and a concrete mathematical realization of the holographic principle. It is an example of a gauge-gravity duality and is now understood to be a special case of the AdS/CFT correspondence. Andrew Strominger and Cumrun Vafa calculated the entropy of certain configurations of D-branes and found agreement with the semi-classical answer for extreme charged black holes. Petr Hořava and Edward Witten found the eleven-dimensional formulation of the heterotic string theories, showing that orbifolds solve the chirality problem. Witten noted that the effective description of the physics of D-branes at low energies is by a supersymmetric gauge theory, and found geometrical interpretations of mathematical structures in gauge theory that he and Nathan Seiberg had earlier discovered in terms of the location of the branes. In 1997, Juan Maldacena noted that the low energy excitations of a theory near a black hole consist of objects close to the horizon, which for extreme charged black holes looks like an anti de Sitter space. He noted that in this limit the gauge theory describes the string excitations near the branes. So he hypothesized that string theory on a near-horizon extreme-charged black-hole geometry, an anti-deSitter space times a sphere with flux, is equally well described by the low-energy limiting gauge theory, the N=4 supersymmetric Yang-Mills theory. This hypothesis, which is called the AdS/CFT correspondence, was further developed by Steven Gubser, Igor Klebanov and Alexander Polyakov, and by Edward Witten, and it is now well-accepted. It is a concrete realization of the holographic principle, which has far-reaching implications for black holes, locality and information in physics, as well as the nature of the gravitational interaction. Through this relationship, string theory has been shown to be related to gauge theories like quantum chromodynamics and this has led to more quantitative understanding of the behavior of hadrons, bringing string theory back to its roots.

Criticisms Some critics of string theory say that it is a failure as a theory of everything.[44][45][46][47][48][49] Notable critics include Peter Woit, Lee Smolin, Philip Warren Anderson,[50] Sheldon Glashow,[51] Lawrence Krauss,[52] and Carlo Rovelli.[53] Some common criticisms include: 1. Very high energies needed to test quantum gravity. 2. Lack of uniqueness of predictions due to the large number of solutions. 3. Lack of background independence.

171

String Theory

High energies It is widely believed that any theory of quantum gravity would require extremely high energies to probe directly, higher by orders of magnitude than those that current experiments such as the Large Hadron Collider[54] can attain. This is because strings themselves are expected to be only slightly larger than the Planck length, which is twenty orders of magnitude smaller than the radius of a proton, and high energies are required to probe small length scales. Generally speaking, quantum gravity is difficult to test because the gravity is much weaker than the other forces, and because quantum effects are controlled by Planck's constant h, a very small quantity. As a result, the effects of quantum gravity are extremely weak.

Number of solutions String theory as it is currently understood has a huge number of solutions, called string vacua,[23] and these vacua might be sufficiently diverse to accommodate almost any phenomena we might observe at lower energies. The vacuum structure of the theory, called the string theory landscape (or the anthropic portion of string theory vacua), is not well understood. String theory contains an infinite number of distinct meta-stable vacua, and perhaps 10520 of these or more correspond to a universe roughly similar to ours — with four dimensions, a high planck scale, gauge groups, and chiral fermions. Each of these corresponds to a different possible universe, with a different collection of particles and forces.[23] What principle, if any, can be used to select among these vacua is an open issue. While there are no continuous parameters in the theory, there is a very large set of possible universes, which may be radically different from each other. It is also suggested that the landscape is surrounded by an even more vast swampland of consistent-looking semiclassical effective field theories, which are actually inconsistent. Some physicists believe this is a good thing, because it may allow a natural anthropic explanation of the observed values of physical constants, in particular the small value of the cosmological constant.[55][56] The argument is that most universes contain values for physical constants that do not lead to habitable universes (at least for humans), and so we happen to live in the "friendliest" universe. This principle is already employed to explain the existence of life on earth as the result of a life-friendly orbit around the medium-sized sun among an infinite number of possible orbits (as well as a relatively stable location in the galaxy).

Background independence A separate and older criticism of string theory is that it is background-dependent — string theory describes perturbative expansions about fixed spacetime backgrounds. Although the theory has some background-independence — topology change is an established process in string theory, and the exchange of gravitons is equivalent to a change in the background — mathematical calculations in the theory rely on preselecting a background as a starting point. This is because, like many quantum field theories, much of string theory is still only formulated perturbatively, as a divergent series of approximations. This criticism has been addressed to some extent by the AdS/CFT duality, which is believed to provide a full, non-perturbative definition of string theory in spacetimes with anti-de Sitter space asymptotics. Nevertheless, a non-perturbative definition of the theory in arbitrary spacetime backgrounds is still lacking. Some hope that M-theory, or a non-perturbative treatment of string theory (such as "background independent open string field theory") will have a background-independent formulation.

172

String Theory

References [1] Polchinski, Joseph (January–February 2007). "All Strung Out?" (http:/ / www. americanscientist. org/ bookshelf/ pub/ all-strung-out). American Scientist 95 (1). . [2] On the right track. Interview with Professor Edward Witten (http:/ / web. archive. org/ web/ 20050327050806/ http:/ / www. hinduonnet. com/ fline/ fl1803/ 18030830. htm). Frontline, Volume 18 – Issue 3, February 3–16, 2001. Hinduonnet.com. Retrieved on 2012-07-11. [3] Leonard Susskind, "Hold fire! This epic vessel has only just set sail...", ''Times Higher Education'', 25 August 2006 (http:/ / www. timeshighereducation. co. uk/ story. asp?storyCode=204991& sectioncode=26). Timeshighereducation.co.uk. Retrieved on 2012-07-11. [4] Brumfiel, Geoff (2006). "Our Universe: Outrageous fortune". Nature 439 (7072): 10–2. Bibcode 2006Natur.439...10B. doi:10.1038/439010a. PMID 16397473. [5] Hawking, Stephen (2010). The Grand Design. Bantam Books. ISBN 055338466X. [6] Sheldon Glashow. "NOVA – The elegant Universe" (http:/ / www. pbs. org/ wgbh/ nova/ elegant/ view-glashow. html). Pbs.org. Retrieved on 2012-07-11. [7] Sheldon Glashow. NOVA | The Elegant Universe: Series (http:/ / www. pbs. org/ wgbh/ nova/ elegant/ ). Pbs.org. Retrieved on 2012-07-11. [8] Burt A. Ovrut (2006). "A Heterotic Standard Model". Fortschritte der Physik 54-(2–3): 160–164. Bibcode 2006ForPh..54..160O. doi:10.1002/prop.200510264. [9] H. Nastase, More on the RHIC fireball and dual black holes, BROWN-HET-1466, arXiv:hep-th/0603176, March 2006, [10] Liu, Hong; Rajagopal, Krishna; Wiedemann, Urs (2007). "Anti–de Sitter/Conformal-Field-Theory Calculation of Screening in a Hot Wind". Physical Review Letters 98 (18). arXiv:hep-ph/0607062. Bibcode 2007PhRvL..98r2301L. doi:10.1103/PhysRevLett.98.182301. [11] Liu, Hong; Rajagopal, Krishna; Wiedemann, Urs Achim (2006). "Calculating the Jet Quenching Parameter". Physical Review Letters 97 (18). arXiv:hep-ph/0605178. Bibcode 2006PhRvL..97r2301L. doi:10.1103/PhysRevLett.97.182301. [12] H. Nastase, The RHIC fireball as a dual black hole, BROWN-HET-1439, arXiv:hep-th/0501068, January 2005, [13] S. James Gates, Jr., Ph.D., Superstring Theory: The DNA of Reality (http:/ / www. teach12. com/ ttcx/ coursedesclong2. aspx?cid=1284) "Lecture 23 – Can I Have That Extra Dimension in the Window?", 0:04:54, 0:21:00. [14] Hellerman, Simeon; Swanson, Ian (2007). "Dimension-changing exact solutions of string theory". Journal of High Energy Physics 2007 (9): 096. arXiv:hep-th/0612051v3. Bibcode 2007JHEP...09..096H. doi:10.1088/1126-6708/2007/09/096. [15] Aharony, Ofer; Silverstein, Eva (2007). "Supercritical stability, transitions, and (pseudo)tachyons". Physical Review D 75 (4). arXiv:hep-th/0612031v2. Bibcode 2007PhRvD..75d6003A. doi:10.1103/PhysRevD.75.046003. [16] M. J. Duff, James T. Liu and R. Minasian Eleven Dimensional Origin of String/String Duality: A One Loop Test (http:/ / arxiv. org/ abs/ hep-th/ 9506126v2) Center for Theoretical Physics, Department of Physics, Texas A&M University [17] Polchinski, Joseph (1998). String Theory, Cambridge University Press ISBN 0521672295. [18] The calculation of the number of dimensions can be circumvented by adding a degree of freedom, which compensates for the "missing" quantum fluctuations. However, this degree of freedom behaves similar to spacetime dimensions only in some aspects, and the produced theory is not Lorentz invariant, and has other characteristics that do not appear in nature. This is known as the linear dilaton or non-critical string. [19] "Quantum Geometry of Bosonic Strings – Revisited" (ftp:/ / ftp2. biblioteca. cbpf. br/ pub/ apub/ 1999/ nf/ nf_zip/ nf04299. pdf) [20] See, for example, T. Hübsch, " A Hitchhiker’s Guide to Superstring Jump Gates and Other Worlds (http:/ / homepage. mac. com/ thubsch/ HSProc. pdf)", in Proc. SUSY 96 Conference, R. Mohapatra and A. Rasin (eds.), Nucl. Phys. (Proc. Supl.) 52A (1997) 347–351 [21] Randall, Lisa (1999). "An Alternative to Compactification". Physical Review Letters 83 (23): 4690. arXiv:hep-th/9906064. Bibcode 1999PhRvL..83.4690R. doi:10.1103/PhysRevLett.83.4690. [22] Aspinwall, Paul S.; Greene, Brian R.; Morrison, David R. (1994). "Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory". Nuclear Physics B 416 (2): 414. arXiv:hep-th/9309097. Bibcode 1994NuPhB.416..414A. doi:10.1016/0550-3213(94)90321-2. [23] + Kachru, Shamit; Kallosh, Renata; Linde, Andrei; Trivedi, Sandip (2003). "De Sitter vacua in string theory". Physical Review D 68 (4). arXiv:hep-th/0301240. Bibcode 2003PhRvD..68d6005K. doi:10.1103/PhysRevD.68.046005. [24] Freivogel, Ben; Kleban, Matthew; Martínez, María Rodríguez; Susskind, Leonard (2006). "Observational consequences of a landscape". Journal of High Energy Physics 2006 (3): 039. arXiv:hep-th/0505232. Bibcode 2006JHEP...03..039F. doi:10.1088/1126-6708/2006/03/039. [25] M. Kleban, T. Levi, and K. Sigurdson, Observing the landscape with cosmic wakes, arXiv:1109.3473 [26] S. Nadis, "How we could see another universe," (http:/ / www. astronomy. com/ en/ sitecore/ content/ Magazine Issues/ 2009/ June 2009. aspx), June 2009. Astronomy.com. Retrieved on 2012-07-11. [27] Polchinski, Joseph. "Introduction to Cosmic F- and D-Strings". arXiv:hep-th/0412244. [28] Sakai, Tadakatsu; Sugimoto, Shigeki (2005). "Low Energy Hadron Physics in Holographic QCD". Progress of Theoretical Physics 113 (4): 843. arXiv:hep-th/0412141. Bibcode 2005PThPh.113..843S. doi:10.1143/PTP.113.843. [29] Recent Results (http:/ / www. physics. indiana. edu/ ~sg/ milc/ results. pdf) of the MILC research program, taken from the MILC Collaboration homepage (http:/ / www. physics. indiana. edu/ ~sg/ milc. html) [30] S. James Gates, Jr., Ph.D., Superstring Theory: The DNA of Reality "Lecture 21 – Can 4D Forces (without Gravity) Love Strings?", 0:26:06-0:26:21, cf. 0:24:05-0:26-24. [31] Idem, "Lecture 19 – Do-See-Do and Swing your Superpartner Part II" 0:16:05-0:24:29. [32] Idem, Lecture 21, 0:20:10-0:21:20.

173

String Theory [33] J. Maldacena, The Large N Limit of Superconformal Field Theories and Supergravity, arXiv:hep-th/9711200 [34] S. S. Gubser, I. R. Klebanov and A. M. Polyakov (1998). "Gauge theory correlators from non-critical string theory". Physics Letters B428: 105–114. arXiv:hep-th/9802109. Bibcode 1998PhLB..428..105G. doi:10.1016/S0370-2693(98)00377-3. [35] Edward Witten (1998). "Anti-de Sitter space and holography". Advances in Theoretical and Mathematical Physics 2: 253–291. arXiv:hep-th/9802150. Bibcode 1998hep.th....2150W. [36] Aharony, O.; S.S. Gubser, J. Maldacena, H. Ooguri, Y. Oz (2000). "Large N Field Theories, String Theory and Gravity". Phys. Rept. 323 (3–4): 183–386. arXiv:hep-th/9905111. doi:10.1016/S0370-1573(99)00083-6. [37] Dijkgraaf, Robbert; Verlinde, Erik; Verlinde, Herman (1997). "5D black holes and matrix strings". Nuclear Physics B 506: 121. arXiv:hep-th/9704018v2. Bibcode 1997NuPhB.506..121D. doi:10.1016/S0550-3213(97)00478-1. [38] Eto, Minoru; Hashimoto, Koji; Terashima, Seiji (2007). "QCD string as vortex string in Seiberg-dual theory". Journal of High Energy Physics 2007 (9): 036. arXiv:0706.2005v1. Bibcode 2007JHEP...09..036E. doi:10.1088/1126-6708/2007/09/036. [39] Meyer, Harvey B. (2005). "Vortices on the worldsheet of the QCD string". Nuclear Physics B 724: 432. arXiv:hep-th/0506034v1. Bibcode 2005NuPhB.724..432M. doi:10.1016/j.nuclphysb.2005.07.001. [40] Koji Hashimoto (2007) Cosmic Strings, Ｑ Ｃ Ｄ Strings and D-branes (http:/ / www2. yukawa. kyoto-u. ac. jp/ ~kiasyk2/ slides/ hashimoto. pdf) [41] Piljin Yi (2007) " Story of baryons in a gravity dual of QCD (http:/ / www2. yukawa. kyoto-u. ac. jp/ ~gc2007/ pdf/ yi. pdf)" [42] Dolen, R.; Horn, D.; Schmid, C. (1968). "Finite-Energy Sum Rules and Their Application to πN Charge Exchange". Physical Review 166 (5): 1768. Bibcode 1968PhRv..166.1768D. doi:10.1103/PhysRev.166.1768. [43] Banks, T.; Fischler, W.; Shenker, S. H.; Susskind, L. (1997). "M theory as a matrix model: A conjecture". Physical Review D 55 (8): 5112. arXiv:hep-th/9610043v3. Bibcode 1997PhRvD..55.5112B. doi:10.1103/PhysRevD.55.5112. [44] Peter Woit Not Even Wrong (http:/ / www. math. columbia. edu/ ~woit/ wordpress/ ?cat=2). Math.columbia.edu. Retrieved on 2012-07-11. [45] Lee Smolin. The Trouble With Physics (http:/ / www. thetroublewithphysics. com). Thetroublewithphysics.com. Retrieved on 2012-07-11. [46] The n-Category Cafe (http:/ / golem. ph. utexas. edu/ category/ 2007/ 02/ this_weeks_finds_in_mathematic_7. html). Golem.ph.utexas.edu (2007-02-25). Retrieved on 2012-07-11. [47] John Baez weblog (http:/ / math. ucr. edu/ home/ baez/ week246. html). Math.ucr.edu (2007-02-25). Retrieved on 2012-07-11. [48] P. Woit (Columbia University), String theory: An Evaluation,February 2001, arXiv:physics/0102051 [49] P. Woit, Is String Theory Testable? (http:/ / www. math. columbia. edu/ ~woit/ testable. pdf) INFN Rome March 2007 [50] "String theory is the first science in hundreds of years to be pursued in pre-Baconian fashion, without any adequate experimental guidance", New York Times, 4 January 2005 [51] "there ain't no experiment that could be done nor is there any observation that could be made that would say, `You guys are wrong.' The theory is safe, permanently safe" NOVA interview (http:/ / pbs. org/ wgbh/ nova/ elegant/ view-glashow. html)) [52] "String theory [is] yet to have any real successes in explaining or predicting anything measurable" New York Times, 8 November 2005) [53] Rovelli, Carlo (2003). International Journal of Modern Physics D [Gravitation; Astrophysics and Cosmology] 12 (9): 1509. arXiv:hep-th/0310077. Bibcode 2003IJMPD..12.1509R. doi:10.1142/S0218271803004304. [54] Elias Kiritsis (2007) " String Theory in a Nutshell (http:/ / press. princeton. edu/ chapters/ s8456. pdf)" [55] N. Arkani-Hamed, S. Dimopoulos and S. Kachru, Predictive Landscapes and New Physics at a TeV, arXiv:hep-th/0501082, SLAC-PUB-10928, HUTP-05-A0001, SU-ITP-04-44, January 2005 [56] L. Susskind The Anthropic Landscape of String Theory, arXiv:hep-th/0302219, February 2003

Further reading Popular books and articles • Davies, Paul; Julian R. Brown (Eds.) (1992). Superstrings: A Theory of Everything?. Cambridge: Cambridge University Press. p. 244. ISBN 0-521-43775-X. • Gefter, Amanda (December 2005). "Is string theory in trouble?" (http://www.newscientist.com/article/ mg18825305.800-is-string-theory-in-trouble.html?full=true). New Scientist. Retrieved December 19, 2005. – An interview with Leonard Susskind, the theoretical physicist who discovered that string theory is based on one-dimensional objects and now is promoting the idea of multiple universes. • Green, Michael (September 1986). "Superstrings" (http://www.damtp.cam.ac.uk/user/mbg15/superstrings/ superstrings.html). Scientific American. Retrieved December 19, 2005. • Greene, Brian (2003). The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory. New York: W.W. Norton & Company. p. 464. ISBN 0-393-05858-1. • Greene, Brian (2004). The Fabric of the Cosmos: Space, Time, and the Texture of Reality. New York: Alfred A. Knopf. p. 569. ISBN 0-375-41288-3.

174

String Theory • Gribbin, John (1998). The Search for Superstrings, Symmetry, and the Theory of Everything. London: Little Brown and Company. p. 224. ISBN 0-316-32975-4. • Halpern, Paul (2004). The Great Beyond: Higher Dimensions, Parallel Universes, and the Extraordinary Search for a Theory of Everything. Hoboken, New Jersey: John Wiley & Sons, Inc.. p. 326. ISBN 0-471-46595-X. • Hooper, Dan (2006). Dark Cosmos: In Search of Our Universe's Missing Mass and Energy. New York: HarperCollins. p. 240. ISBN 978-0-06-113032-8. • Kaku, Michio (1994). Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension. Oxford: Oxford University Press. p. 384. ISBN 0-19-508514-0. • Klebanov, Igor and Maldacena, Juan (January 2009). Solving Quantum Field Theories via Curved Spacetimes (http://ptonline.aip.org/journals/doc/PHTOAD-ft/vol_62/iss_1/28_1.shtml). Physics Today. • Musser, George (2008). The Complete Idiot's Guide to String Theory. Indianapolis: Alpha. p. 368. ISBN 978-1-59257-702-6. • Randall, Lisa (2005). Warped Passages: Unraveling the Mysteries of the Universe's Hidden Dimensions. New York: Ecco Press. p. 512. ISBN 0-06-053108-8. • Susskind, Leonard (2006). The Cosmic Landscape: String Theory and the Illusion of Intelligent Design. New York: Hachette Book Group/Back Bay Books. p. 403. ISBN 0-316-01333-1. • Taubes, Gary (November 1986). "Everything's Now Tied to Strings" Discover Magazine vol 7, #11. (Popular article, probably the first ever written, on the first superstring revolution.) • Vilenkin, Alex (2006). Many Worlds in One: The Search for Other Universes. New York: Hill and Wang. p. 235. ISBN 0-8090-9523-8. • Witten, Edward (June 2002). "The Universe on a String" (http://www.sns.ias.edu/~witten/papers/string.pdf) (PDF). Astronomy Magazine. Retrieved December 19, 2005. – An easy nontechnical article on the very basics of the theory. Two nontechnical books that are critical of string theory: • Smolin, Lee (2006). The Trouble with Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next. New York: Houghton Mifflin Co.. p. 392. ISBN 0-618-55105-0. • Woit, Peter (2006). Not Even Wrong – The Failure of String Theory And the Search for Unity in Physical Law. London: Jonathan Cape &: New York: Basic Books. p. 290. ISBN 978-0-465-09275-8.

Textbooks • Becker, Katrin, Becker, Melanie, and John H. Schwarz (2007) String Theory and M-Theory: A Modern Introduction . Cambridge University Press. ISBN 0-521-86069-5 • Binétruy, Pierre (2007) Supersymmetry: Theory, Experiment, and Cosmology. Oxford University Press. ISBN 978-0-19-850954-7. • Dine, Michael (2007) Supersymmetry and String Theory: Beyond the Standard Model. Cambridge University Press. ISBN 0-521-85841-0. • Paul H. Frampton (1974). Dual Resonance Models. Frontiers in Physics. ISBN 0-8053-2581-6. • Gasperini, Maurizio (2007) Elements of String Cosmology. Cambridge University Press. ISBN 978-0-521-86875-4. • Michael Green, John H. Schwarz and Edward Witten (1987) Superstring theory. Cambridge University Press. The original textbook. • Vol. 1: Introduction. ISBN 0-521-35752-7. • Vol. 2: Loop amplitudes, anomalies and phenomenology. ISBN 0-521-35753-5. • Kiritsis, Elias (2007) String Theory in a Nutshell. Princeton University Press. ISBN 978-0-691-12230-4. • Johnson, Clifford (2003). D-branes. Cambridge: Cambridge University Press. ISBN 0-521-80912-6. • Vol. 1: An introduction to the bosonic string. ISBN 0-521-63303-6.

175

String Theory • Vol. 2: Superstring theory and beyond. ISBN 0-521-63304-4. • Szabo, Richard J. (Reprinted 2007) An Introduction to String Theory and D-brane Dynamics. Imperial College Press. ISBN 978-1-86094-427-7. • Zwiebach, Barton (2004) A First Course in String Theory. Cambridge University Press. ISBN 0-521-83143-1. Contact author for errata. Technical and critical: • Penrose, Roger (2005). The Road to Reality: A Complete Guide to the Laws of the Universe. Knopf. p. 1136. ISBN 0-679-45443-8.

Online material • David Tong. "Lectures on String Theory". arXiv:0908.0333. – This is a one semester course on bosonic string theory aimed at beginning graduate students. The lectures assume a working knowledge of quantum field theory and general relativity. • Schwarz, John H.. "Introduction to Superstring Theory". arXiv:hep-ex/0008017. – Four lectures, presented at the NATO Advanced Study Institute on Techniques and Concepts of High Energy Physics, St. Croix, Virgin Islands, in June 2000, and addressed to an audience of graduate students in experimental high energy physics, survey basic concepts in string theory. • Witten, Edward (1998). "Duality, Spacetime and Quantum Mechanics" (http://online.itp.ucsb.edu/online/ plecture/witten/). Kavli Institute for Theoretical Physics. Retrieved December 16, 2005. – Slides and audio from an Ed Witten lecture where he introduces string theory and discusses its challenges. • Kibble, Tom. "Cosmic strings reborn?". arXiv:astro-ph/0410073. – Invited Lecture at COSLAB 2004, held at Ambleside, Cumbria, United Kingdom, from 10 to 17 September 2004. • Marolf, Don. "Resource Letter NSST-1: The Nature and Status of String Theory". arXiv:hep-th/0311044. – A guide to the string theory literature. • Ajay, Shakeeb, Wieland et al. (2004). "The nth dimension" (http://thenthdimension.com/). Retrieved December 16, 2005. – A comprehensive compilation of materials concerning string theory. Created by an international team of students. • Woit, Peter (2002). "Is string theory even wrong?" (http://www.americanscientist.org/issues/pub/ is-string-theory-even-wrong). American Scientist. Retrieved December 16, 2005. – A criticism of string theory. • Veneziano, Gabriele (May 2004). "The Myth of the Beginning of Time" (http://www.sciam.com/article. cfm?chanID=sa006&articleID=00042F0D-1A0E-1085-94F483414B7F0000). Scientific American. • Krauss, Lawrence (2005-11-23). "Theory of Anything?" (http://www.slate.com/articles/health_and_science/ science/2005/11/theory_of_anything.html). Slate (http://www.slate.com). – A criticism of string theory. • Harris, Richard (2006-11-07). "Short of 'All,' String Theorists Accused of Nothing" (http://www.npr.org/ templates/story/story.php?storyId=6377252). National Public Radio. Retrieved 2007-03-05. • A website dedicated to creative writing inspired by string theory. (http://banyancollege.org/scriblerus/) • An Italian Website with various papers in English language concerning the mathematical connections between String Theory and Number Theory. (http://nardelli.xoom.it/virgiliowizard/) • George Gardner (2007-01-24). " Theory of everything put to the test (news:ID109828243)". [news:tech.blorge.com tech.blorge.com]. Web link (http://tech.blorge.com/Structure: /2007/01/24/ theory-of-everything-put-to-the-test/). Retrieved 2007-03-03. • Minkel, J. R. (2006-03-02). "A Prediction from String Theory, with Strings Attached" (http://www.sciam.com/ article.cfm?chanId=sa003&articleId=1475A684-E7F2-99DF-355B95296BE6031C). Scientific American. • Chalmers, Matthew (2007-09-03). "Stringscape" (http://physicsworld.com/cws/article/indepth/30940). Physics World (http://physicsworld.com). Retrieved September 6, 2007. — An up-to-date and thorough review of string theory in a popular way.

176

String Theory • Woit, Peter. Not Even Wrong: The Failure of String Theory & the Continuing Challenge to Unify the Laws of Physics, 2006. ISBN 0-224-07605-1 (Jonathan Cape), ISBN 0-465-09275-6 (Basic Books) • Schwarz, John (2001). "Early History of String Theory: A Personal Perspective" (http://online.itp.ucsb.edu/ online/colloq/schwarz1/). Retrieved July 17, 2009. • Zidbits (2011-03-27). "A Layman's Explanation For String Theory?" (http://zidbits.com/2011/03/ a-laymans-explanation-for-string-theory/).

External links • Why String Theory (http://whystringtheory.com/) – an introduction to string theory • Dialogue on the Foundations of String Theory (http://www.mathpages.com/home/kmath632/kmath632.htm) at MathPages • Superstrings! String Theory Home Page (http://www.sukidog.com/jpierre/strings/) – Online tutorial • CI.physics. STRINGS newsgroup (http://schwinger.harvard.edu/~sps/) – A moderated newsgroup for discussion of string theory (a theory of quantum gravity and unification of forces) and related fields of high-energy physics. • Not Even Wrong (http://www.math.columbia.edu/~woit/blog/) – A blog critical of string theory. • Superstring Theory (http://www.perimeterinstitute.ca/en/Outreach/What_We_Research/Superstring_Theory/ ) Perimeter Institute for Theoretical Physics • The Official String Theory Web Site (http://superstringtheory.com/) • The Elegant Universe (http://www.pbs.org/wgbh/nova/elegant/) – A Three-Hour Miniseries with Brian Greene by NOVA (original PBS Broadcast Dates: October 28, 8–10 p.m. and November 4, 8–9 p.m., 2003). Various images, texts, videos and animations explaining string theory. • Beyond String Theory (http://www.phys.ens.fr/~troost/beyondstringtheory/) – A project by a string physicist explaining aspects of string theory to a broad audience. • Spinning the Superweb: Essays on the History of Superstring Theory (http://www.spinningthesuperweb. blogspot.com) – A Science Studies' approach to the history of string theory (an elementary knowledge of string theory is required).

177

Quantum Gravity

Quantum Gravity Quantum gravity (QG) is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics (describing three of the four known fundamental interactions) with general relativity (describing the fourth, gravity). It is hoped that development of such a theory would unify all fundamental interactions into a single mathematical framework and describe all known observable interactions in the universe, at both subatomic and cosmological scales. Such a theory of quantum gravity would yield the same experimental results as ordinary quantum mechanics in conditions of weak gravity (gravitational potentials much less than c2) and the same results as Einsteinian general relativity in phenomena at scales much larger than individual molecules (action much larger than reduced Planck's constant), but moreover be able to predict the outcome of situations where both quantum effects and strong-field gravity are important (at the Planck scale, unless large extra dimension conjectures are correct). If the theory of quantum gravity also achieves a grand unification of the other known interactions, it is referred to as a theory of everything (TOE). Motivation for quantizing gravity comes from the remarkable success of the quantum theories of the other three fundamental interactions, and from experimental evidence suggesting that gravity can be made to show quantum effects.[1][2][3] Although some quantum gravity theories such as string theory and other unified field theories (or 'theories of everything') attempt to unify gravity with the other fundamental forces, others such as loop quantum gravity make no such attempt; they simply quantize the gravitational field while keeping it separate from the other forces. Most observed physical phenomena can be described well by quantum mechanics or general relativity, without needing both. This can be thought of as due to an extreme separation of mass scales at which they are important. Quantum effects are usually important only for the "very small", that is, for objects no larger than typical molecules. General relativistic effects, on the other hand, show up mainly for the "very large" bodies such as collapsed stars. (Planets' gravitational fields, as of 2011, are well-described by linearized gravity except for Mercury's perihelion precession; so strong-field effects—any effects of gravity beyond lowest nonvanishing order in φ/c2—have not been observed even in the gravitational fields of planets and main sequence stars). There is a lack of experimental evidence relating to quantum gravity, and classical physics adequately describes the observed effects of gravity over a range of 50 orders of magnitude of mass, i.e., for masses of objects from about 10−23 to 1030 kg. However, certain physical phenomena, such as singularities, are "very small" spatially yet are "very large" from a mass or energy perspective; such objects cannot be understood with current theories of quantum mechanics or general relativity, thus motivating the search for a quantum theory of gravity.

178

Quantum Gravity

Overview Much of the difficulty in meshing these theories at all energy scales comes from the different assumptions that these theories make on how the universe works. Quantum field theory depends on particle fields embedded in the flat space-time of special relativity. General relativity models gravity as a curvature within space-time that changes as a gravitational mass moves. Historically, the most obvious way of combining the two (such as treating gravity as simply another particle field) ran quickly into what is known as the Diagram showing where quantum gravity sits in the hierarchy of physics theories renormalization problem. In the old-fashioned understanding of renormalization, gravity particles would attract each other and adding together all of the interactions results in many infinite values which cannot easily be cancelled out mathematically to yield sensible, finite results. This is in contrast with quantum electrodynamics where, while the series still do not converge, the interactions sometimes evaluate to infinite results, but those are few enough in number to be removable via renormalization.

Effective field theories Quantum gravity can be treated as an effective field theory. Effective quantum field theories come with some high-energy cutoff, beyond which we do not expect that the theory provides a good description of nature. The "infinities" then become large but finite quantities proportional to this finite cutoff scale, and correspond to processes that involve very high energies near the fundamental cutoff. These quantities can then be absorbed into an infinite collection of coupling constants, and at energies well below the fundamental cutoff of the theory, to any desired precision; only a finite number of these coupling constants need to be measured in order to make legitimate quantum-mechanical predictions. This same logic works just as well for the highly successful theory of low-energy pions as for quantum gravity. Indeed, the first quantum-mechanical corrections to graviton-scattering and Newton's law of gravitation have been explicitly computed[4] (although they are so astronomically small that we may never be able to measure them). In fact, gravity is in many ways a much better quantum field theory than the Standard Model, since it appears to be valid all the way up to its cutoff at the Planck scale. (By comparison, the Standard Model is expected to start to break down above its cutoff at the much smaller scale of around 1000 GeV.) While confirming that quantum mechanics and gravity are indeed consistent at reasonable energies, it is clear that near or above the fundamental cutoff of our effective quantum theory of gravity (the cutoff is generally assumed to be of the order of the Planck scale), a new model of nature will be needed. Specifically, the problem of combining quantum mechanics and gravity becomes an issue only at very high energies, and may well require a totally new kind of model.

179

Quantum Gravity

180

Quantum gravity theory for the highest energy scales The general approach to deriving a quantum gravity theory that is valid at even the highest energy scales is to assume that such a theory will be simple and elegant and, accordingly, to study symmetries and other clues offered by current theories that might suggest ways to combine them into a comprehensive, unified theory. One problem with this approach is that it is unknown whether quantum gravity will actually conform to a simple and elegant theory, as it should resolve the dual conundrums of special relativity with regard to the uniformity of acceleration and gravity, and general relativity with regard to spacetime curvature. Such a theory is required in order to understand problems involving the combination of very high energy and very small dimensions of space, such as the behavior of black holes, and the origin of the universe.

Quantum mechanics and general relativity The graviton At present, one of the deepest problems in theoretical physics is harmonizing the theory of general relativity, which describes gravitation, and applies to large-scale structures (stars, planets, galaxies), with quantum mechanics, which describes the other three fundamental forces acting on the atomic scale. This problem must be put in the proper context, however. In particular, contrary to the popular claim that quantum mechanics and general relativity are fundamentally incompatible, one can demonstrate that the structure of general relativity essentially follows inevitably from the quantum mechanics of interacting theoretical spin-2 massless particles [5][6][7][8][9] (called gravitons). While there is no concrete proof of the existence of gravitons, quantized theories of matter may necessitate their existence. Supporting this theory is the observation that all fundamental forces except gravity have one or more known messenger particles, leading researchers to believe that at least one most likely does exist; they have dubbed these hypothetical particles gravitons. Many of the accepted notions of a unified theory of physics since the 1970s, including string theory, superstring theory, M-theory, loop quantum gravity, all assume, and to some degree depend upon, the existence of the graviton. Many researchers view the detection of the graviton as vital to validating their work.

Gravity Probe B (GP-B) has measured spacetime curvature near Earth to test related models in application of Einstein's general theory of relativity.

The dilaton The dilaton made its first appearance in Kaluza–Klein theory, a five-dimensional theory that combined gravitation and electromagnetism. Generally, it appears in string theory. More recently, it has appeared in the lower-dimensional many-bodied gravity problem[10] based on the field theoretic approach of Roman Jackiw. The impetus arose from the fact that complete analytical solutions for the metric of a covariant N-body system have proven elusive in General Relativity. To simplify the problem, the number of dimensions was lowered to (1+1) namely one spatial dimension and one temporal dimension. This model problem, known as R=T theory[11] (as opposed to the general G=T theory) was amenable to exact solutions in terms of a generalization of the Lambert W function. It was also found that the field equation governing the dilaton (derived from differential geometry) was the Schrödinger equation and consequently amenable to quantization.[12] Thus, one had a theory which combined gravity, quantization and even

Quantum Gravity the electromagnetic interaction, promising ingredients of a fundamental physical theory. It is worth noting that the outcome revealed a previously unknown and already existing natural link between general relativity and quantum mechanics. However, this theory needs to be generalized in (2+1) or (3+1) dimensions although, in principle, the field equations are amenable to such generalization as shown with the inclusion of a one-graviton process[13] and yielding the correct Newtonian limit in d dimensions if a dilaton is included. However, it is not yet clear what the full field equation will govern the dilaton in higher dimensions. This is further complicated by the fact that gravitons can propagate in (3+1) dimensions and consequently that would imply gravitons and dilatons exist in the real world. Moreover, detection of the dilaton is expected to be even more elusive than the graviton. However, since this approach allows for the combination of gravitational, electromagnetic and quantum effects, their coupling could potentially lead to a means of vindicating the theory, through cosmology and perhaps even experimentally.

Nonrenormalizability of gravity General relativity, like electromagnetism, is a classical field theory. One might expect that, as with electromagnetism, there should be a corresponding quantum field theory. However, gravity is perturbatively nonrenormalizable.[14] For a quantum field theory to be well-defined according to this understanding of the subject, it must be asymptotically free or asymptotically safe. The theory must be characterized by a choice of finitely many parameters, which could, in principle, be set by experiment. For example, in quantum electrodynamics, these parameters are the charge and mass of the electron, as measured at a particular energy scale. On the other hand, in quantizing gravity, there are infinitely many independent parameters (counterterm coefficients) needed to define the theory. For a given choice of those parameters, one could make sense of the theory, but since we can never do infinitely many experiments to fix the values of every parameter, we do not have a meaningful physical theory: • At low energies, the logic of the renormalization group tells us that, despite the unknown choices of these infinitely many parameters, quantum gravity will reduce to the usual Einstein theory of general relativity. • On the other hand, if we could probe very high energies where quantum effects take over, then every one of the infinitely many unknown parameters would begin to matter, and we could make no predictions at all. As explained below, there is a way around this problem by treating QG as an effective field theory. Any meaningful theory of quantum gravity that makes sense and is predictive at all energy scales must have some deep principle that reduces the infinitely many unknown parameters to a finite number that can then be measured. • One possibility is that normal perturbation theory is not a reliable guide to the renormalizability of the theory, and that there really is a UV fixed point for gravity. Since this is a question of non-perturbative quantum field theory, it is difficult to find a reliable answer, but some people still pursue this option. • Another possibility is that there are new symmetry principles that constrain the parameters and reduce them to a finite set. This is the route taken by string theory, where all of the excitations of the string essentially manifest themselves as new symmetries.

QG as an effective field theory In an effective field theory, all but the first few of the infinite set of parameters in a non-renormalizable theory are suppressed by huge energy scales and hence can be neglected when computing low-energy effects. Thus, at least in the low-energy regime, the model is indeed a predictive quantum field theory.[4] (A very similar situation occurs for the very similar effective field theory of low-energy pions.) Furthermore, many theorists agree that even the Standard Model should really be regarded as an effective field theory as well, with "nonrenormalizable" interactions suppressed by large energy scales and whose effects have consequently not been observed experimentally.

181

Quantum Gravity

182

Recent work[4] has shown that by treating general relativity as an effective field theory, one can actually make legitimate predictions for quantum gravity, at least for low-energy phenomena. An example is the well-known calculation of the tiny first-order quantum-mechanical correction to the classical Newtonian gravitational potential between two masses.

Spacetime background dependence A fundamental lesson of general relativity is that there is no fixed spacetime background, as found in Newtonian mechanics and special relativity; the spacetime geometry is dynamic. While easy to grasp in principle, this is the hardest idea to understand about general relativity, and its consequences are profound and not fully explored, even at the classical level. To a certain extent, general relativity can be seen to be a relational theory,[15] in which the only physically relevant information is the relationship between different events in space-time. On the other hand, quantum mechanics has depended since its inception on a fixed background (non-dynamic) structure. In the case of quantum mechanics, it is time that is given and not dynamic, just as in Newtonian classical mechanics. In relativistic quantum field theory, just as in classical field theory, Minkowski spacetime is the fixed background of the theory. String theory String theory can be seen as a generalization of quantum field theory where instead of point particles, string-like objects propagate in a fixed spacetime background, although the interactions among closed strings give rise to space-time in a dynamical way. Although string theory had its origins in the study of quark confinement and not of quantum gravity, it was soon discovered that the string spectrum contains the graviton, and that "condensation" of certain vibration modes of strings is equivalent to a modification of the original background. In this sense, string perturbation theory exhibits exactly the features one would expect of a perturbation theory that may exhibit a strong dependence on asymptotics (as seen, for example, in the AdS/CFT correspondence) which is a weak form of background dependence.

Interaction in the subatomic world: world lines of point-like particles in the Standard Model or a world sheet swept up by closed strings in string theory

Background independent theories Loop quantum gravity is the fruit of an effort to formulate a background-independent quantum theory. Topological quantum field theory provided an example of background-independent quantum theory, but with no local degrees of freedom, and only finitely many degrees of freedom globally. This is inadequate to describe gravity in 3+1 dimensions which has local degrees of freedom according to general relativity. In 2+1 dimensions, however, gravity is a topological field theory, and it has been successfully quantized in several different ways, including spin networks.

Quantum Gravity

183

Semi-classical quantum gravity Quantum field theory on curved (non-Minkowskian) backgrounds, while not a full quantum theory of gravity, has shown many promising early results. In an analogous way to the development of quantum electrodynamics in the early part of the 20th century (when physicists considered quantum mechanics in classical electromagnetic fields), the consideration of quantum field theory on a curved background has led to predictions such as black hole radiation. Phenomena such as the Unruh effect, in which particles exist in certain accelerating frames but not in stationary ones, do not pose any difficulty when considered on a curved background (the Unruh effect occurs even in flat Minkowskian backgrounds). The vacuum state is the state with least energy (and may or may not contain particles). See Quantum field theory in curved spacetime for a more complete discussion.

Points of tension There are other points of tension between quantum mechanics and general relativity. • First, classical general relativity breaks down at singularities, and quantum mechanics becomes inconsistent with general relativity in the neighborhood of singularities (however, no one is certain that classical general relativity applies near singularities in the first place). • Second, it is not clear how to determine the gravitational field of a particle, since under the Heisenberg uncertainty principle of quantum mechanics its location and velocity cannot be known with certainty. The resolution of these points may come from a better understanding of general relativity.[16] • Third, there is the Problem of Time in quantum gravity. Time has a different meaning in quantum mechanics and general relativity and hence there are subtle issues to resolve when trying to formulate a theory which combines the two.[17]

Candidate theories There are a number of proposed quantum gravity theories.[18] Currently, there is still no complete and consistent quantum theory of gravity, and the candidate models still need to overcome major formal and conceptual problems. They also face the common problem that, as yet, there is no way to put quantum gravity predictions to experimental tests, although there is hope for this to change as future data from cosmological observations and particle physics experiments becomes available.[19][20]

String theory One suggested starting point is ordinary quantum field theories which, after all, are successful in describing the other three basic fundamental forces in the context of the standard model of elementary particle physics. However, while this leads to an acceptable effective (quantum) field theory of gravity at low energies,[21] gravity turns out to be much more problematic at higher energies. Where, for ordinary field theories such as quantum electrodynamics, a technique known as renormalization is an integral part of deriving predictions which take into account higher-energy contributions,[22] gravity turns out to be nonrenormalizable: at high energies, applying the recipes of ordinary quantum field theory yields models that are devoid of all predictive power.[23]

Projection of a Calabi-Yau manifold, one of the ways of compactifying the extra dimensions posited by string theory

Quantum Gravity

184

One attempt to overcome these limitations is to replace ordinary quantum field theory, which is based on the classical concept of a point particle, with a quantum theory of one-dimensional extended objects: string theory.[24] At the energies reached in current experiments, these strings are indistinguishable from point-like particles, but, crucially, different modes of oscillation of one and the same type of fundamental string appear as particles with different (electric and other) charges. In this way, string theory promises to be a unified description of all particles and interactions.[25] The theory is successful in that one mode will always correspond to a graviton, the messenger particle of gravity; however, the price to pay are unusual features such as six extra dimensions of space in addition to the usual three for space and one for time.[26] In what is called the second superstring revolution, it was conjectured that both string theory and a unification of general relativity and supersymmetry known as supergravity[27] form part of a hypothesized eleven-dimensional model known as M-theory, which would constitute a uniquely defined and consistent theory of quantum gravity.[28][29] As presently understood, however, string theory admits a very large number (10500 by some estimates) of consistent vacua, comprising the so-called "string landscape". Sorting through this large family of solutions remains one of the major challenges.

Loop quantum gravity Another approach to quantum gravity starts with the canonical quantization procedures of quantum theory. Starting with the initial-value-formulation of general relativity (cf. the section on evolution equations, above), the result is an analogue of the Schrödinger equation: the Wheeler–DeWitt equation, which some argue is ill-defined.[30] A major break-through came with the introduction of what are now known as Ashtekar variables, which represent geometric gravity using mathematical analogues of electric and magnetic fields.[31][32] The resulting candidate for a theory of quantum gravity is Loop quantum gravity, in which space is represented by a network structure called a spin network, evolving over time in discrete steps.[33][34][35][36]

Simple spin network of the type used in loop quantum gravity

Other approaches There are a number of other approaches to quantum gravity. The approaches differ depending on which features of general relativity and quantum theory are accepted unchanged, and which features are modified.[37][38] Examples include: • • • • • • •

Acoustic metric and other analog models of gravity Algebraic Graviton Quantizing Asymptotic safety Causal Dynamical Triangulation[39] Causal sets[40] Group field theory[41] MacDowell–Mansouri action

• Noncommutative geometry. • Path-integral based models of quantum cosmology[42]

Quantum Gravity • • • • •

Regge calculus String-nets giving rise to gapless helicity ±2 excitations with no other gapless excitations[43] Superfluid vacuum theory a.k.a. theory of BEC vacuum Supergravity Twistor models[44]

Weinberg–Witten theorem In quantum field theory, the Weinberg–Witten theorem places some constraints on theories of composite gravity/emergent gravity. However, recent developments attempt to show that if locality is only approximate and the holographic principle is correct, the Weinberg–Witten theorem would not be valid.

Experimental Tests As was emphasized above, quantum gravitational effects are extremely weak and therefore difficult to test. For this reason, the possibility of experimentally testing quantum gravity had not received much attention prior to the late 1990s. However, in the past decade, physicists have realized that evidence for quantum gravitational effects can guide the development of the theory. Since the theoretical development has been slow, the phenomenology of quantum gravity which studies the possibility of experimental tests, has obtained increased attention.[45][46] There is presently no confirmed experimental signature of quantum gravitational effects. The most widely pursued possibilities for quantum gravity phenomenology include violations of Lorentz invariance, imprints of quantum gravitational effects in the Cosmic Microwave Background (in particular its polarization), and decoherence induced by fluctuations in the space-time foam.

References [1] Nesvizhevsky, Nesvizhevsky et al. (2002-01-17). "Quantum states of neutrons in the Earth's gravitational field" (http:/ / www. nature. com/ nature/ journal/ v415/ n6869/ abs/ 415297a. html). Nature 415 (6869): 297–299. Bibcode 2002Natur.415..297N. doi:10.1038/415297a. . Retrieved 2011-04-21. [2] Jenke, Geltenbort, Lemmel & Abele; Geltenbort, Peter; Lemmel, Hartmut; Abele, Hartmut (2011-04-17). "Realization of a gravity-resonance-spectroscopy technique" (http:/ / www. nature. com/ nphys/ journal/ vaop/ ncurrent/ full/ nphys1970. html). Nature 7 (6): 468–472. Bibcode 2011NatPh...7..468J. doi:10.1038/nphys1970. . Retrieved 2011-04-21. [3] Palmer, Jason (2011-04-18). "Neutrons could test Newton's gravity and string theory" (http:/ / www. bbc. co. uk/ news/ science-environment-13097370). BBC News. . Retrieved 2011-04-21. [4] Donoghue (1995). "Introduction to the Effective Field Theory Description of Gravity". arXiv:gr-qc/9512024 [gr-qc]. [5] Kraichnan, R. H. (1955). "Special-Relativistic Derivation of Generally Covariant Gravitation Theory". Physical Review 98 (4): 1118–1122. Bibcode 1955PhRv...98.1118K. doi:10.1103/PhysRev.98.1118. [6] Gupta, S. N. (1954). "Gravitation and Electromagnetism". Physical Review 96 (6): 1683–1685. Bibcode 1954PhRv...96.1683G. doi:10.1103/PhysRev.96.1683. [7] Gupta, S. N. (1957). "Einstein's and Other Theories of Gravitation". Reviews of Modern Physics 29 (3): 334–336. Bibcode 1957RvMP...29..334G. doi:10.1103/RevModPhys.29.334. [8] Gupta, S. N. (1962). "Quantum Theory of Gravitation". Recent Developments in General Relativity. Pergamon Press. pp. 251–258. [9] Deser, S. (1970). "Self-Interaction and Gauge Invariance". General Relativity and Gravitation 1: 9–18. arXiv:gr-qc/0411023. Bibcode 1970GReGr...1....9D. doi:10.1007/BF00759198. [10] Ohta, Tadayuki; Mann, Robert (1996). "Canonical reduction of two-dimensional gravity for particle dynamics". Classical and Quantum Gravity 13 (9): 2585–2602. arXiv:gr-qc/9605004. Bibcode 1996CQGra..13.2585O. doi:10.1088/0264-9381/13/9/022. [11] Sikkema, A E; Mann, R B (1991). "Gravitation and cosmology in (1+1) dimensions". Classical and Quantum Gravity 8: 219–235. Bibcode 1991CQGra...8..219S. doi:10.1088/0264-9381/8/1/022. [12] Farrugia; Mann; Scott (2007). "N-body Gravity and the Schroedinger Equation". Classical and Quantum Gravity 24 (18): 4647–4659. arXiv:gr-qc/0611144. Bibcode 2007CQGra..24.4647F. doi:10.1088/0264-9381/24/18/006. [13] Mann, R B; Ohta, T (1997). "Exact solution for the metric and the motion of two bodies in (1+1)-dimensional gravity". Phys. Rev. D. 55 (8): 4723–4747. arXiv:gr-qc/9611008. Bibcode 1997PhRvD..55.4723M. doi:10.1103/PhysRevD.55.4723. [14] Feynman, R. P.; Morinigo, F. B., Wagner, W. G., & Hatfield, B. (1995). Feynman lectures on gravitation. Addison-Wesley. ISBN 0-201-62734-5.

185

Quantum Gravity [15] Smolin, Lee (2001). Three Roads to Quantum Gravity. Basic Books. pp. 20–25. ISBN 0-465-07835-4. Pages 220–226 are annotated references and guide for further reading. [16] Hunter Monroe (2005). "Singularity-Free Collapse through Local Inflation". arXiv:astro-ph/0506506 [astro-ph]. [17] Edward Anderson (2010). "The Problem of Time in Quantum Gravity". arXiv:1009.2157 [gr-qc]. [18] A timeline and overview can be found in Rovelli, Carlo (2000). "Notes for a brief history of quantum gravity". arXiv:gr-qc/0006061 [gr-qc]. [19] Ashtekar, Abhay (2007). "Loop Quantum Gravity: Four Recent Advances and a Dozen Frequently Asked Questions". 11th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity. p. 126. arXiv:0705.2222. Bibcode 2008mgm..conf..126A. doi:10.1142/9789812834300_0008. [20] Schwarz, John H. (2007). "String Theory: Progress and Problems". Progress of Theoretical Physics Supplement 170: 214–226. arXiv:hep-th/0702219. Bibcode 2007PThPS.170..214S. doi:10.1143/PTPS.170.214. [21] Donoghue, John F.(editor), (1995). "Introduction to the Effective Field Theory Description of Gravity". In Cornet, Fernando. Effective Theories: Proceedings of the Advanced School, Almunecar, Spain, 26 June–1 July 1995. Singapore: World Scientific. arXiv:gr-qc/9512024. ISBN 981-02-2908-9. [22] Weinberg, Steven (1996). "17–18". The Quantum Theory of Fields II: Modern Applications. Cambridge University Press. ISBN 0-521-55002-5. [23] Goroff, Marc H.; Sagnotti, Augusto (1985). "Quantum gravity at two loops". Physics Letters B 160: 81–86. Bibcode 1985PhLB..160...81G. doi:10.1016/0370-2693(85)91470-4. [24] An accessible introduction at the undergraduate level can be found in Zwiebach, Barton (2004). A First Course in String Theory. Cambridge University Press. ISBN 0-521-83143-1., and more complete overviews in Polchinski, Joseph (1998). String Theory Vol. I: An Introduction to the Bosonic String. Cambridge University Press. ISBN 0-521-63303-6. and Polchinski, Joseph (1998b). String Theory Vol. II: Superstring Theory and Beyond. Cambridge University Press. ISBN 0-521-63304-4. [25] Ibanez, L. E. (2000). "The second string (phenomenology) revolution". Classical & Quantum Gravity 17 (5): 1117–1128. arXiv:hep-ph/9911499. Bibcode 2000CQGra..17.1117I. doi:10.1088/0264-9381/17/5/321. [26] For the graviton as part of the string spectrum, e.g. Green, Schwarz & Witten 1987, sec. 2.3 and 5.3; for the extra dimensions, ibid sec. 4.2. [27] Weinberg, Steven (2000). "31" (http:/ / books. google. com/ books?id=aYDDRKqODpUC& printsec=frontcover). The Quantum Theory of Fields II: Modern Applications. Cambridge University Press. ISBN 0-521-55002-5. . [28] Townsend, Paul K. (1996). Four Lectures on M-Theory. ICTP Series in Theoretical Physics. p. 385. arXiv:hep-th/9612121. Bibcode 1997hepcbconf..385T. [29] Duff, Michael (1996). "M-Theory (the Theory Formerly Known as Strings)". International Journal of Modern Physics A 11 (32): 5623–5642. arXiv:hep-th/9608117. Bibcode 1996IJMPA..11.5623D. doi:10.1142/S0217751X96002583. [30] Kuchař, Karel (1973). "Canonical Quantization of Gravity". In Israel, Werner. Relativity, Astrophysics and Cosmology. D. Reidel. pp. 237–288 (section 3). ISBN 90-277-0369-8. [31] Ashtekar, Abhay (1986). "New variables for classical and quantum gravity". Physical Review Letters 57 (18): 2244–2247. Bibcode 1986PhRvL..57.2244A. doi:10.1103/PhysRevLett.57.2244. PMID 10033673. [32] Ashtekar, Abhay (1987). "New Hamiltonian formulation of general relativity". Physical Review D 36 (6): 1587–1602. Bibcode 1987PhRvD..36.1587A. doi:10.1103/PhysRevD.36.1587. [33] Thiemann, Thomas (2006). "Loop Quantum Gravity: An Inside View". Approaches to Fundamental Physics 721: 185. arXiv:hep-th/0608210. Bibcode 2007LNP...721..185T. [34] Rovelli, Carlo (1998). "Loop Quantum Gravity" (http:/ / www. livingreviews. org/ lrr-1998-1). Living Reviews in Relativity 1. . Retrieved 2008-03-13. [35] Ashtekar, Abhay; Lewandowski, Jerzy (2004). "Background Independent Quantum Gravity: A Status Report". Classical & Quantum Gravity 21 (15): R53–R152. arXiv:gr-qc/0404018. Bibcode 2004CQGra..21R..53A. doi:10.1088/0264-9381/21/15/R01. [36] Thiemann, Thomas (2003). "Lectures on Loop Quantum Gravity". Lecture Notes in Physics 631: 41–135. arXiv:gr-qc/0210094. Bibcode 2003LNP...631...41T. doi:10.1007/978-3-540-45230-0_3. [37] Isham, Christopher J. (1994). "Prima facie questions in quantum gravity". In Ehlers, Jürgen; Friedrich, Helmut. Canonical Gravity: From Classical to Quantum. Springer. arXiv:gr-qc/9310031. ISBN 3-540-58339-4. [38] Sorkin, Rafael D. (1997). "Forks in the Road, on the Way to Quantum Gravity". International Journal of Theoretical Physics 36 (12): 2759–2781. arXiv:gr-qc/9706002. Bibcode 1997IJTP...36.2759S. doi:10.1007/BF02435709. [39] Loll, Renate (1998). "Discrete Approaches to Quantum Gravity in Four Dimensions" (http:/ / www. livingreviews. org/ lrr-1998-13). Living Reviews in Relativity 1: 13. arXiv:gr-qc/9805049. Bibcode 1998LRR.....1...13L. . Retrieved 2008-03-09. [40] Sorkin, Rafael D. (2005). "Causal Sets: Discrete Gravity". In Gomberoff, Andres; Marolf, Donald. Lectures on Quantum Gravity. Springer. arXiv:gr-qc/0309009. ISBN 0-387-23995-2. [41] See Daniele Oriti and references therein. [42] Hawking, Stephen W. (1987). "Quantum cosmology". In Hawking, Stephen W.; Israel, Werner. 300 Years of Gravitation. Cambridge University Press. pp. 631–651. ISBN 0-521-37976-8. [43] Wen 2006 [44] See ch. 33 in Penrose 2004 and references therein. [45] Hossenfelder, Sabine (2011). "Experimental Search for Quantum Gravity" (https:/ / www. novapublishers. com/ catalog/ product_info. php?products_id=15903). In V. R. Frignanni. Classical and Quantum Gravity: Theory, Analysis and Applications. Chapter 5: Nova

186

Quantum Gravity Publishers. ISBN 978-1-61122-957-8. . [46] "1010.3420] Experimental Search for Quantum Gravity" (http:/ / arxiv. org/ abs/ 1010. 3420). Arxiv.org. 2010-10-17. . Retrieved 2012-04-08.

Further reading • Ahluwalia, D. V. (2002). "Interface of Gravitational and Quantum Realms". Modern Physics Letters A 17 (15–17): 1135. arXiv:gr-qc/0205121. Bibcode 2002MPLA...17.1135A. doi:10.1142/S021773230200765X. • Ashtekar, Abhay (2005). "The winding road to quantum gravity" (http://www.ias.ac.in/currsci/dec252005/ 2064.pdf). Current Science 89: 2064–2074. • Carlip, Steven (2001). "Quantum Gravity: a Progress Report". Reports on Progress in Physics 64 (8): 885–942. arXiv:gr-qc/0108040. Bibcode 2001RPPh...64..885C. doi:10.1088/0034-4885/64/8/301. • Kiefer, Claus (2007). Quantum Gravity. Oxford University Press. ISBN 0-19-921252-X. • Kiefer, Claus (2005). "Quantum Gravity: General Introduction and Recent Developments". Annalen der Physik 15: 129–148. arXiv:gr-qc/0508120. Bibcode 2006AnP...518..129K. doi:10.1002/andp.200510175. • Lämmerzahl, Claus, ed. (2003). Quantum Gravity: From Theory to Experimental Search. Lecture Notes in Physics. Springer. ISBN 3-540-40810-X. • Rovelli, Carlo (2004). Quantum Gravity. Cambridge University Press. ISBN 0-521-83733-2. • Trifonov, Vladimir (2008). "GR-friendly description of quantum systems". International Journal of Theoretical Physics 47 (2): 492–510. arXiv:math-ph/0702095. Bibcode 2008IJTP...47..492T. doi:10.1007/s10773-007-9474-3.

187

188

Appendix Quantum In physics, a quantum (plural: quanta) is the minimum amount of any physical entity involved in an interaction. Behind this, one finds the fundamental notion that a physical property may be "quantized," referred to as "the hypothesis of quantization".[1] This means that the magnitude can take on only certain discrete values. There is a related term of quantum number. An example of an entity that is quantized is the energy transfer of elementary particles of matter (called fermions) and of photons and other bosons. A photon is a single quantum of light, and is referred to as a "light quantum". The energy of an electron bound to an atom (at rest) is said to be quantized, which results in the stability of atoms, and of matter in general. As incorporated into the theory of quantum mechanics, this is regarded by physicists as part of the fundamental framework for understanding and describing nature at the infinitesimal level. Normally quanta are considered to be discrete packets with energy stored in them. Max Planck considered these quanta to be particles that can change their form (meaning that they can be absorbed and released). This phenomenon can be observed in the case of black body radiation, when it is being heated and cooled.

Etymology and discovery The word "quantum" comes from the Latin "quantus," for "how much." "Quanta" meaning short for "quanta of electricity" (or electron) was used in a 1902 article on the photoelectric effect by Philipp Lenard, who credited Hermann von Helmholtz for using the word in the area of electricity. However, the word quantum in general was well known before 1900.[2] It was often used by physicians, such as the term quantum satis. Both Helmholtz and Julius von Mayer were physicians as well as physicists. Helmholtz used quantum with reference to heat in his article [3] on Mayer's work, and indeed, the word quantum can be found in the formulation of the first law of thermodynamics by Mayer in his letter [4] dated July 24, 1841. Max Planck used "quanta" to mean "quanta of matter and electricity",[5] gas, and heat.[6] In 1905, in response to Planck's work and the experimental work of Lenard, who explained his results by using the term "quanta of electricity", Albert Einstein suggested that radiation existed in spatially localized packets which he called "quanta of light" ("Lightquanta").[7] The concept of quantization of radiation was discovered in 1900 by Max Planck, who had been trying to understand the emission of radiation from heated objects, known as black-body radiation. By assuming that energy can only be absorbed or released in tiny, differential, discrete packets he called "bundles" or "energy elements",[8] Planck accounted for the fact that certain objects change colour when heated.[9] On December 14, 1900, Planck reported his revolutionary findings to the German Physical Society and introduced the idea of quantization for the first time as a part of his research on black body radiation.[10] As a result of his experiments, Planck deduced the numerical value of h, known as the Planck constant, and could also report a more precise value for the Avogadro–Loschmidt number, the number of real molecules in a mole and the unit of electrical charge, to the German Physical Society. After his theory was validated, Planck was awarded the Nobel Prize in Physics in 1918 for his discovery.

Quantum

Beyond electromagnetic radiation While quantization was first discovered in electromagnetic radiation, it describes a fundamental aspect of energy not just restricted to photons.[11] In the attempt to bring experiment into agreement with theory, Max Planck postulated that electromagnetic energy is absorbed or emitted in discrete packets, or quanta.[12]

References [1] Wiener, N. (1966). Differential Space, Quantum Systems, and Prediction. Cambridge: The Massachusetts Institute of Technology Press [2] E. Cobham Brewer 1810–1897. Dictionary of Phrase and Fable. 1898. (http:/ / www. bartleby. com/ 81/ 13830. html) [3] E. Helmholtz, Robert Mayer's Priorität (http:/ / www. ub. uni-heidelberg. de/ helios/ fachinfo/ www/ math/ edd/ helmholtz/ R-Mayer. pdf)

(German) [4] Herrmann,A. Weltreich der Physik, GNT-Verlag (1991) (http:/ / wayback. archive. org/ web/ */ http:/ / fs. math. uni-frankfurt. de/ fsmath/ misc/ RobertMayer. html) (German) [5] Planck, M. (1901). "Ueber die Elementarquanta der Materie und der Elektricität". Annalen der Physik 309 (3): 564–566. Bibcode 1901AnP...309..564P. doi:10.1002/andp.19013090311. (German) [6] Planck, Max (1883). "Ueber das thermodynamische Gleichgewicht von Gasgemengen". Annalen der Physik 255 (6): 358. Bibcode 1883AnP...255..358P. doi:10.1002/andp.18832550612. (German) [7] Einstein, A. (1905). "Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt" (http:/ / www. physik. uni-augsburg. de/ annalen/ history/ einstein-papers/ 1905_17_132-148. pdf). Annalen der Physik 17 (6): 132–148. Bibcode 1905AnP...322..132E. doi:10.1002/andp.19053220607. . (German). A partial English translation is available from Wikisource. [8] Max Planck (1901). "Ueber das Gesetz der Energieverteilung im Normalspectrum (On the Law of Distribution of Energy in the Normal Spectrum)" (http:/ / web. archive. org/ web/ 20080418002757/ http:/ / dbhs. wvusd. k12. ca. us/ webdocs/ Chem-History/ Planck-1901/ Planck-1901. html). Annalen der Physik 309 (3): 553. Bibcode 1901AnP...309..553P. doi:10.1002/andp.19013090310. Archived from the original (http:/ / dbhs. wvusd. k12. ca. us/ webdocs/ Chem-History/ Planck-1901/ Planck-1901. html) on 2008-04-18. . [9] Brown, T., LeMay, H., Bursten, B. (2008). Chemistry: The Central Science Upper Saddle River, NJ: Pearson Education ISBN 0-13-600617-5 [10] Klein, Martin J. (1961). "Max Planck and the beginnings of the quantum theory". Archive for History of Exact Sciences 1 (5): 459. doi:10.1007/BF00327765. [11] Melville, K. (2005, February 11). Real-World Quantum Effects Demonstrated (http:/ / www. scienceagogo. com/ news/ 20050110221715data_trunc_sys. shtml) [12] Modern Applied Physics-Tippens third edition; McGraw-Hill.

Further reading • B. Hoffmann, The Strange Story of the Quantum, Pelican 1963. • Lucretius, On the Nature of the Universe, transl. from the Latin by R.E. Latham, Penguin Books Ltd., Harmondsworth 1951. There are, of course, many translations, and the translation's title varies. Some put emphasis on how things work, others on what things are found in nature. • J. Mehra and H. Rechenberg, The Historical Development of Quantum Theory, Vol.1, Part 1, Springer-Verlag New York Inc., New York 1982. • M. Planck, A Survey of Physical Theory, transl. by R. Jones and D.H. Williams, Methuen & Co., Ltd., London 1925 (Dover editions 1960 and 1993) including the Nobel lecture.

189

Quantum state

190

Quantum state In quantum physics, quantum state refers to the state of a quantum system. A quantum state is given as a vector in a vector space, called the state vector. The state vector theoretically contains statistical information about the quantum system. For example, when dealing with the energy spectrum of the electron in a hydrogen atom, the relevant state vector is given by the principal quantum number . For a more complicated case, consider Bohm formulation of EPR experiment, where the state vector

involves superposition of joint spin states

for 2 different particles.[1]:47-48 In a more general usage, a quantum state can be either "pure" or "mixed." The above example is pure. Mathematically, a pure quantum state is represented by a state vector in a vector space, which is a generalization of our more usual three dimensional space. A mixed quantum state corresponds to a probabilistic mixture of pure states; however, different distributions of pure states can generate equivalent (i.e., physically indistinguishable) mixed states. Quantum states, mixed as well as pure, are described by so-called density matrices, although these give probabilities, not densities. For example, if the spin of an electron is measured in any direction, e.g., with a Stern-Gerlach experiment, there are two possible results, up or down. The vector space for the electron's spin is therefore two-dimensional. A pure state is a two-dimensional complex vector , with a length of one. That is, . A mixed state is a matrix that is Hermitian, positive-definite, and has trace 1. Before a particular measurement is performed on a quantum system, the theory usually gives only a probability distribution for the outcome, and the form that this distribution takes is completely determined by the quantum state and the observable describing the measurement. These probability distributions arise for both mixed states and pure states: it is impossible in quantum mechanics (unlike classical mechanics) to prepare a state in which all properties of the system are fixed and certain. This is exemplified by the uncertainty principle, and reflects a core difference between classical and quantum physics. Even in quantum theory, however, for every observable there are states that determine its value exactly.[2]

Quantum state

191

Conceptual description Quantum states In the mathematical formulation of quantum mechanics, pure quantum states correspond to vectors in a vector space, while each observable quantity (such as the energy or momentum of a particle) is associated with a mathematical operator. The operator serves as a linear function which acts on the states of the system. The eigenvalues of the operator correspond to the possible values of the observable: For example, it is possible to observe a particle with a momentum of 1 kg·m/s if and only if one of the eigenvalues of the momentum operator is 1 kg·m/s. The corresponding eigenvector (which physicists call an "eigenstate") with eigenvalue 1 kg·m/s would be a quantum state with a Probability densities for the electron of a hydrogen atom in different quantum states. definite, well-defined value of momentum of 1 kg·m/s, with no quantum uncertainty. If its momentum were measured, the result is guaranteed to be 1 kg·m/s. On the other hand, a system in a linear combination of multiple different eigenstates does in general have quantum uncertainty. We can represent this linear combination of eigenstates as:

. The

coefficient which corresponds to a particular state in the linear combination is complex thus allowing interference effects between states. The coefficients are time dependent. How a quantum system changes in time is governed by the time evolution operator. Statistical mixtures of states are separate from a linear combination. A statistical mixture of states occurs with a statistical ensemble of independent systems. Statistical mixtures represent the degree of knowledge whilst the uncertainty within quantum mechanics is fundamental. Mathematically a statistical mixture is not a combination of complex coefficients but by a combination of probabilities of different states . represents the probability of a randomly selected system being in the state

. Unlike the linear combination case each system is in a definite

[3][4]

eigenstate.

In quantum theory, even pure states show statistical behaviour. Regardless of how carefully we prepare the state ρ of the system, measurement results are not repeatable in general, and we must understand the expectation value of an observable A as a statistical mean. It is this mean and the distribution of probabilities that is predicted by physical theories. There is no state which is simultaneously an eigenstate for all observables. For example, we cannot prepare a state such that both the position measurement Q(t) and the momentum measurement P(t) (at the same time t) are known exactly; at least one of them will have a range of possible values.[5] This is the content of the Heisenberg uncertainty relation.

Quantum state Moreover, in contrast to classical mechanics, it is unavoidable that performing a measurement on the system generally changes its state. More precisely: After measuring an observable A, the system will be in an eigenstate of A; thus the state has changed, unless the system was already in that eigenstate. This expresses a kind of logical consistency: If we measure A twice in the same run of the experiment, the measurements being directly consecutive in time, then they will produce the same results. This has some strange consequences however: Consider two observables, A and B, where A corresponds to a measurement earlier in time than B.[6] Suppose that the system is in an eigenstate of B. If we measure only B, we will not notice statistical behaviour. If we measure first A and then B in the same run of the experiment, the system will transfer to an eigenstate of A after the first measurement, and we will generally notice that the results of B are statistical. Thus: Quantum mechanical measurements influence one another, and it is important in which order they are performed. Another feature of quantum states becomes relevant if we consider a physical system that consists of multiple subsystems; for example, an experiment with two particles rather than one. Quantum physics allows for certain states, called entangled states, that show certain statistical correlations between measurements on the two particles which cannot be explained by classical theory. For details, see entanglement. These entangled states lead to experimentally testable properties (Bell's theorem) that allow us to distinguish between quantum theory and alternative classical (non-quantum) models.

Schrödinger picture vs. Heisenberg picture In the discussion above, we have taken the observables P(t), Q(t) to be dependent on time, while the state σ was fixed once at the beginning of the experiment. This approach is called the Heisenberg picture. One can, equivalently, treat the observables as fixed, while the state of the system depends on time; that is known as the Schrödinger picture. Conceptually (and mathematically), both approaches are equivalent; choosing one of them is a matter of convention. Both viewpoints are used in quantum theory. While non-relativistic quantum mechanics is usually formulated in terms of the Schrödinger picture, the Heisenberg picture is often preferred in a relativistic context, that is, for quantum field theory. Compare with Dirac picture.

Formalism in quantum physics Pure states as rays in a Hilbert space Quantum physics is most commonly formulated in terms of linear algebra, as follows. Any given system is identified with some finite- or infinite-dimensional Hilbert space. The pure states correspond to vectors of norm 1. Thus the set of all pure states corresponds to the unit sphere in the Hilbert space. If two unit vectors differ only by a scalar of magnitude 1, known as a "global phase factor," then they are indistinguishable. Therefore, distinct pure states can be put in correspondence with "rays" in the Hilbert space, or equivalently points in the projective Hilbert space.

Bra-ket notation Calculations in quantum mechanics make frequent use of linear operators, inner products, dual spaces and Hermitian conjugation. In order to make such calculations more straightforward, and to obviate the need (in some contexts) to fully understand the underlying linear algebra, Paul Dirac invented a notation to describe quantum states, known as bra-ket notation. Although the details of this are beyond the scope of this article (see the article Bra-ket notation), some consequences of this are: • The variable name used to denote a vector (which corresponds to a pure quantum state) is chosen to be of the form (where the " " can be replaced by any other symbols, letters, numbers, or even words). This can be

192

Quantum state

193

contrasted with the usual mathematical notation, where vectors are usually bold, lower-case letters, or letters with arrows on top. • Instead of vector, the term ket is used synonymously. • Each ket

is uniquely associated with a so-called bra, denoted

, which is also said to correspond to the

same physical quantum state. Technically, the bra is the adjoint of the ket. It is an element of the dual space, and related to the ket by the Riesz representation theorem. In a finite-dimensional space with a chosen basis, writing as a column vector, is a row vector; just take the transpose and entry-wise complex conjugate of . • Inner products (also called brackets) are written so as to look like a bra and ket next to each other:

.

(The phrase "bra-ket" is supposed to resemble "bracket".)

Spin, many-body states It is important to note that in quantum mechanics besides, e.g., the usual position variable r, a discrete variable m exists, corresponding to the value of the z-component of the spin vector. This can be thought of as a kind of intrinsic angular momentum. However, it does not appear at all in classical mechanics and arises from Dirac's relativistic generalization of the theory. As a consequence, the quantum state of a system of N particles is described by a function with four variables per particle, e.g.

Here, the variables mν assume values from the set where

(in units of Planck's reduced constant ħ = 1), is either a non-negative integer (0, 1, 2 ... for bosons), or

semi-integer (1/2, 3/2, 5/2 ... for fermions). Moreover, in the case of identical particles, the above N-particle function must either be symmetrized (in the bosonic case) or anti-symmetrized (in the fermionic case) with respect to the particle numbers. Electrons are fermions with S = 1/2, photons (quanta of light) are bosons with S = 1. Apart from the symmetrization or anti-symmetrization, N-particle states can thus simply be obtained by tensor products of one-particle states, to which we return herewith.

Basis states of one-particle systems As with any vector space, if a basis is chosen for the Hilbert space of a system, then any ket can be expanded as a linear combination of those basis elements. Symbolically, given basis kets , any ket can be written

where ci are complex numbers. In physical terms, this is described by saying that quantum superposition of the states

has been expressed as a

. If the basis kets are chosen to be orthonormal (as is often the case), then

. One property worth noting is that the normalized states

are characterized by

Expansions of this sort play an important role in measurement in quantum mechanics. In particular, if the eigenstates (with eigenvalues ki) of an observable, and that observable is measured on the normalized state

are ,

2

then the probability that the result of the measurement is ki is |ci| . (The normalization condition above mandates that the total sum of probabilities is equal to one.) A particularly important example is the position basis, which is the basis consisting of eigenstates of the observable which corresponds to measuring position. If these eigenstates are nondegenerate (for example, if the system is a

Quantum state

194

single, spinless particle), then any ket

is associated with a complex-valued function of three-dimensional space:

This function is called the wavefunction corresponding to

.

Superposition of pure states One aspect of quantum states, mentioned above, is that superpositions of them can be formed. If

and

are

two kets corresponding to quantum states, the ket is a different quantum state (possibly not normalized). Note that which quantum state it is depends on both the amplitudes and phases (arguments) of and . In other words, for example, even though and (for real θ) correspond to the same physical quantum state, they are not interchangeable, since for example and

do not (in general) correspond to the same physical state. However,

and

do correspond to the same physical state. This is sometimes described by saying that "global" phase factors are unphysical, but "relative" phase factors are physical and important. One example of a quantum interference phenomenon that arises from superposition is the double-slit experiment. The photon state is a superposition of two different states, one of which corresponds to the photon having passed through the left slit, and the other corresponding to passage through the right slit. The relative phase of those two states has a value which depends on the distance from each of the two slits. Depending on what that phase is, the interference is constructive at some locations and destructive in others, creating the interference pattern. Another example of the importance of relative phase in quantum superposition is Rabi oscillations, where the relative phase of two states varies in time due to the Schrödinger equation. The resulting superposition ends up oscillating back and forth between two different states.

Mixed states A pure quantum state is a state which can be described by a single ket vector, as described above. A mixed quantum state is a statistical ensemble of pure states (see quantum statistical mechanics). Equivalently, a mixed-quantum state on a given quantum system described by a Hilbert space naturally arises as a pure quantum state (called a purification) on a larger bipartite system

, the other half of which is inaccessible to the observer.

A mixed state cannot be described as a ket vector. Instead, it is described by its associated density matrix (or density operator), usually denoted ρ. Note that density matrices can describe both mixed and pure states, treating them on the same footing. The density matrix is defined as

where

is the fraction of the ensemble in each pure state

Here, one typically uses a one-particle formalism

to describe the average behaviour of an N-particle system. A simple criterion for checking whether a density matrix is describing a pure or mixed state is that the trace of ρ2 is equal to 1 if the state is pure, and less than 1 if the state is mixed.[7] Another, equivalent, criterion is that the von Neumann entropy is 0 for a pure state, and strictly positive for a mixed state. The rules for measurement in quantum mechanics are particularly simple to state in terms of density matrices. For example, the ensemble average (expectation value) of a measurement corresponding to an observable A is given by

where

are eigenkets and eigenvalues, respectively, for the operator A, and tr denotes trace. It is important

to note that two types of averaging are occurring, one being a quantum average over the basis kets

of the pure

Quantum state states, and the other being a statistical average with the probabilities ps of those states. With respect to these different types of averaging, i.e. to distinguish pure and/or mixed states, one often uses the expressions 'coherent' and/or 'incoherent superposition' of quantum states. For a mathematical discussion on states as positive normalized linear functionals on a C* algebra, see Gelfand–Naimark–Segal construction. There, the same objects are described in a C*-algebraic context.

Notes [1] Ballentine, Leslie (1998). Quantum Mechanics: A Modern Development (2nd, illustrated, reprint ed.). World Scientific. ISBN 9789810241056. [2] Ballentine, L. E. (1970), "The Statistical Interpretation of Quantum Mechanics" (http:/ / link. aps. org/ doi/ 10. 1103/ RevModPhys. 42. 358), Reviews of Modern Physics 42: 358-381, doi:10.1103/RevModPhys.42.358, [3] http:/ / xbeams. chem. yale. edu/ ~batista/ vaa/ node4. html [4] http:/ / electron6. phys. utk. edu/ qm1/ modules/ m6/ statistical. htm [5] To avoid misunderstandings: Here we mean that Q(t) and P(t) are measured in the same state, but not in the same run of the experiment. [6] For concreteness' sake, suppose that A = Q(t1) and B = P(t2) in the above example, with t2 > t1 > 0. [7] Blum, Density matrix theory and applications, page 39 (http:/ / books. google. com/ books?id=kl-pMd9Qx04C& pg=PA39). Note that this criterion works when the density matrix is normalized so that the trace of ρ is 1, as it is for the standard definition given in this section. Occasionally a density matrix will be normalized differently, in which case the criterion is

References Further reading The concept of quantum states, in particular the content of the section Formalism in quantum physics above, is covered in most standard textbooks on quantum mechanics. For a discussion of conceptual aspects and a comparison with classical states, see: • Isham, Chris J (1995). Lectures on Quantum Theory: Mathematical and Structural Foundations. Imperial College Press. ISBN 978-1-86094-001-9. For a more detailed coverage of mathematical aspects, see: • Bratteli, Ola; Robinson, Derek W (1987). Operator Algebras and Quantum Statistical Mechanics 1. Springer. ISBN 978-3-540-17093-8. 2nd edition. In particular, see Sec. 2.3. For a discussion of purifications of mixed quantum states, see Chapter 2 of John Preskill's lecture notes for Physics 219 (http://www.theory.caltech.edu/~preskill/ph229/) at Caltech.

195

Article Sources and Contributors

Article Sources and Contributors History of Quantum Mechanics  Source: http://en.wikipedia.org/w/index.php?oldid=523504781  Contributors: Agge1000, Alan Liefting, BD2412, Bci2, Bduke, BenRG, Betacommand, Biscuittin, Boardhead, Bookalign, Chris Howard, Coppertwig, D.H, DVdm, Demaag, Dharmaraj.S, Dratman, Dstuck, DuncanHill, Fadesga, GeorgeLouis, Geremia, GoingBatty, Graeme Bartlett, Grafen, Grimlock, Headbomb, Ignoranteconomist, InformationvsInjustice, Iohannes Animosus, Itub, JSquish, JaGa, Jac16888, Jagged 85, Jbergquist, John Vandenberg, Kbdank71, Khazar, Kinema, Kusunose, Laurascudder, Lmatt, Lottamiata, LuchoX, Matthew Fennell, Michael C Price, Mild Bill Hiccup, Modify, Moose-32, Mpkannan, Myasuda, Nagualdesign, Niceguyedc, Okedem, Ospalh, PKT, PSimeon, Rhetth, Riversider2008, Rjwilmsi, RockMagnetist, Sadi Carnot, SchreiberBike, Shadowjams, Stephen Poppitt, StradivariusTV, Strait, Topbanana, Topherwhelan, Truthanado, Wbm1058, Welsh, Wikiklrsc, Yill577, Yvwv, Yym1997, 46 anonymous edits Basic Concepts of Quantum Mechanics  Source: http://en.wikipedia.org/w/index.php?oldid=463882772  Contributors: 16@r, 1howardsr1, 2001:16D8:FF78:1:20F:3DFF:FECB:9FDF, A. di M., Accurrent, Adashiel, Adrignola, Adwaele, Alain10, Alansohn, Aliencam, Altzinn, Amihaly, Ancheta Wis, Android79, AndyZ, Anoko moonlight, Aostrander, Arion 3x3, Arjen Dijksman, Army1987, Arpingstone, AussieBand1, Avb, Avramov, Awolf002, BabyNuke, Baccyak4H, Barbara Shack, Barraki, Batmanand, Bearian, BenRG, Benhocking, Bevo, BiT, Bloodshedder, BohemianWikipedian, Brews ohare, Brina700, Burn, Butwhatdoiknow, C. Trifle, Calmer Waters, Caltas, Cantiun, ChangChienFu, Changcho, CharlesHBennett, Chris Howard, ChristopherWillis, Chzz, Clay Juicer, Closedmouth, Cmeyer1969, Colonies Chris, Comrade009, ConfuciusOrnis, Count Iblis, Craig Stuntz, Csmiller, Curtmcd, Cybercobra, D-Notice, DESiegel, DJIndica, DMacks, DVdm, Danaudick, Daniel Vollmer, Danny Rathjens, Dauto, David R. Ingham, David edwards, Deagle AP, Delirium, Demaag, Demoni, Dionyziz, Discospinster, Djr32, DrFarhadn, DrKiernan, Droll, ESkog, Easchiff, Editwiki, Edreher, Edward Z. Yang, El C, Elungtrings, EncycloPetey, Erkenbrack, EryZ, Estrellador*, Everyking, FT2, Farrant, Fg2, Fones4jenke, Gandalf61, Gareth Griffith-Jones, Gareth Jones, Garuda0001, Gary King, Gdrg22, Geffb, Gene Nygaard, George100, GeorgeLouis, Georgelulu, Gifani, Giftlite, Gill110951, Gmalivuk, Goethean, Gogo Dodo, Gommert, Gonzonoir, Grika, Gunderburg, Harmfulman, Hbar12, Headbomb, Hqb, IPSOS, Ian Pitchford, Ignoranteconomist, Ilmari Karonen, Inwind, J04n, JWSchmidt, Jaganath, Jak86, Janechii, JocK, John Vandenberg, JoseREMY, Jpkole, KTC, Kasimant, Klonimus, KnowledgeOfSelf, Lambex, Larryisgood, Legare, Likebox, LilHelpa, Linas, Lisatwo, Lonappan, Loom91, Lorne ipsum, Lotje, Lue3378, Magog the Ogre, MarSch, Mario777Zelda, Masleyko, Materialscientist, Maurice Carbonaro, Mbell, Mejor Los Indios, Michael C Price, Michael Courtney, Michael Hardy, Midnight Madness, Mike sa, Mobius Clock, Modify, Mogudo, Morten Isaksen, Mr buick, Mteorman, Mukake, Mushin, Nageh, Nxtid, OpenScience, Orton 4 President, Owl, P.wormer, Paeosl, Papa November, Patrick0Moran, Paulc1001, Pedro Fonini, Pepper, Petersburg, Petri Krohn, PhySusie, Pjacobi, Promethean, Provider uk, Proxmi, Psyche825, Ptpare, Qbnmichael, QuasiAbstract, Qwertyus, Qwyrxian, R'n'B, RG2, RJSampson, Ramanpotential, Ranjithsutari, Rich Farmbrough, Richard001, Richwil, Riley Huntley, RobertG, Robth, RockMagnetist, Rocknrollanoah, Rose Garden, Rsocol, SCZenz, Samanthaclark11, Sarbruis, SarekOfVulcan, Sciamanna, ScienceApologist, Scolobb, Sean Whitton, Sergio.ballestrero, Sheliak, Shlomi Hillel, Sitar Physics, Skizzik, Sneakums, Sokari, Spoon!, Srleffler, Stephen Poppitt, Steve Quinn, Stevertigo, StradivariusTV, Svick, Swwright, T.M.M. Dowd, T2X, Tankred, Tcnuk, Tealish, Techdawg667, TheProject, Thekidmastergx, Tide rolls, Todayishere, Tone, Trackerseal, Twas Now, Tyler R, Ujm, UncleDouggie, Ventril2009, Viraj nadkarni, Voyajer, Vyznev Xnebara, WAS 4.250, Wafulz, Wakebrdkid, Watchayakan, Welsh, Weregerbil, Wernhervonbraun, WikHead, Wikaholic, Will Gladstone, Windowlicker, Woohookitty, Xt318, Yensin, Ytrewqt, Yumpis, Zadock9, Zarniwoot, Zundark, 440 anonymous edits Introduction to Quantum Mechanics  Source: http://en.wikipedia.org/w/index.php?oldid=523980448  Contributors: 16@r, 1howardsr1, 2001:16D8:FF78:1:20F:3DFF:FECB:9FDF, A. di M., Accurrent, Adashiel, Adrignola, Adwaele, Alain10, Alansohn, Aliencam, Altzinn, Amihaly, Ancheta Wis, Android79, AndyZ, Anoko moonlight, Aostrander, Arion 3x3, Arjen Dijksman, Army1987, Arpingstone, AussieBand1, Avb, Avramov, Awolf002, BabyNuke, Baccyak4H, Barbara Shack, Barraki, Batmanand, Bearian, BenRG, Benhocking, Bevo, BiT, Bloodshedder, BohemianWikipedian, Brews ohare, Brina700, Burn, Butwhatdoiknow, C. Trifle, Calmer Waters, Caltas, Cantiun, ChangChienFu, Changcho, CharlesHBennett, Chris Howard, ChristopherWillis, Chzz, Clay Juicer, Closedmouth, Cmeyer1969, Colonies Chris, Comrade009, ConfuciusOrnis, Count Iblis, Craig Stuntz, Csmiller, Curtmcd, Cybercobra, D-Notice, DESiegel, DJIndica, DMacks, DVdm, Danaudick, Daniel Vollmer, Danny Rathjens, Dauto, David R. Ingham, David edwards, Deagle AP, Delirium, Demaag, Demoni, Dionyziz, Discospinster, Djr32, DrFarhadn, DrKiernan, Droll, ESkog, Easchiff, Editwiki, Edreher, Edward Z. Yang, El C, Elungtrings, EncycloPetey, Erkenbrack, EryZ, Estrellador*, Everyking, FT2, Farrant, Fg2, Fones4jenke, Gandalf61, Gareth Griffith-Jones, Gareth Jones, Garuda0001, Gary King, Gdrg22, Geffb, Gene Nygaard, George100, GeorgeLouis, Georgelulu, Gifani, Giftlite, Gill110951, Gmalivuk, Goethean, Gogo Dodo, Gommert, Gonzonoir, Grika, Gunderburg, Harmfulman, Hbar12, Headbomb, Hqb, IPSOS, Ian Pitchford, Ignoranteconomist, Ilmari Karonen, Inwind, J04n, JWSchmidt, Jaganath, Jak86, Janechii, JocK, John Vandenberg, JoseREMY, Jpkole, KTC, Kasimant, Klonimus, KnowledgeOfSelf, Lambex, Larryisgood, Legare, Likebox, LilHelpa, Linas, Lisatwo, Lonappan, Loom91, Lorne ipsum, Lotje, Lue3378, Magog the Ogre, MarSch, Mario777Zelda, Masleyko, Materialscientist, Maurice Carbonaro, Mbell, Mejor Los Indios, Michael C Price, Michael Courtney, Michael Hardy, Midnight Madness, Mike sa, Mobius Clock, Modify, Mogudo, Morten Isaksen, Mr buick, Mteorman, Mukake, Mushin, Nageh, Nxtid, OpenScience, Orton 4 President, Owl, P.wormer, Paeosl, Papa November, Patrick0Moran, Paulc1001, Pedro Fonini, Pepper, Petersburg, Petri Krohn, PhySusie, Pjacobi, Promethean, Provider uk, Proxmi, Psyche825, Ptpare, Qbnmichael, QuasiAbstract, Qwertyus, Qwyrxian, R'n'B, RG2, RJSampson, Ramanpotential, Ranjithsutari, Rich Farmbrough, Richard001, Richwil, Riley Huntley, RobertG, Robth, RockMagnetist, Rocknrollanoah, Rose Garden, Rsocol, SCZenz, Samanthaclark11, Sarbruis, SarekOfVulcan, Sciamanna, ScienceApologist, Scolobb, Sean Whitton, Sergio.ballestrero, Sheliak, Shlomi Hillel, Sitar Physics, Skizzik, Sneakums, Sokari, Spoon!, Srleffler, Stephen Poppitt, Steve Quinn, Stevertigo, StradivariusTV, Svick, Swwright, T.M.M. Dowd, T2X, Tankred, Tcnuk, Tealish, Techdawg667, TheProject, Thekidmastergx, Tide rolls, Todayishere, Tone, Trackerseal, Twas Now, Tyler R, Ujm, UncleDouggie, Ventril2009, Viraj nadkarni, Voyajer, Vyznev Xnebara, WAS 4.250, Wafulz, Wakebrdkid, Watchayakan, Welsh, Weregerbil, Wernhervonbraun, WikHead, Wikaholic, Will Gladstone, Windowlicker, Woohookitty, Xt318, Yensin, Ytrewqt, Yumpis, Zadock9, Zarniwoot, Zundark, 440 anonymous edits Old Quantum Theory  Source: http://en.wikipedia.org/w/index.php?oldid=525806807  Contributors: AManWithNoPlan, Ace of Spades, AndreHahn, BD2412, Bakkedal, Bento00, Boardhead, Charles Matthews, ChrisGualtieri, Cogiati, Corrector of Spelling, Crowsnest, DVdm, Dogbert66, Fradeve11, Giftlite, GoingBatty, J.delanoy, JoeBruno, Julia W, Kient123, Kirsim, Likebox, Machine Elf 1735, Michael Hardy, Myasuda, Otto.fox, Owlbuster, ParticleStorm, Puzl bustr, Ragesoss, Rjwilmsi, RogierBrussee, Tesi1700, Thurth, Vsmith, WMdeMuynck, WTMitchell3, Wtmitchell, 36 anonymous edits Quantum Mechanics after 1925  Source: http://en.wikipedia.org/w/index.php?oldid=524590858  Contributors: 06twalke, 100110100, 11341134a, 11341134b, 128.100.60.xxx, 172.133.159.xxx, 1howardsr1, 21655, 28421u2232nfenfcenc, 3peasants, 4RM0, 5Q5, 64.180.242.xxx, 65.24.178.xxx, A. di M., APH, Aaron Einstein, Abce2, Academic Challenger, Acalamari, Achowat, Ackbeet, Acroterion, AdamJacobMuller, Addshore, AdevarTruth, Adhalanay, AdjustShift, Adolfman, Adrignola, Adventurer, Adwaele, AgadaUrbanit, Agge1000, Ahoerstemeier, Airplaneman, Aitias, Ajcheema, Akriasas, Akubra, Alai, Alamadte, Alansohn, Ale jrb, Alejo2083, AlexBG72, AlexiusHoratius, Alfio, Alison, Alvindclopez, Amareto2, Amarvc, Amatulic, AmiDaniel, Amit Moscovich, Ancheta Wis, Andonic, Andrej.westermann, Andris, AndriyK, Andy Dingley, Andy chase, Angelsages, AnnaFrance, AnonGuy, Anonwhymus, Antandrus, Antixt, Antonio Lopez, Anville, Ap, ApolloCreed, Apparition11, Aranea Mortem, Arespectablecitizen, Arjen Dijksman, Arjun01, Armegdon, Army1987, Arnero, Arthur chos, Asdf1990, Ashujo, Asmackey, AstroNomer, Astronautics, Asyndeton, AtticusX, AugustinMa, Austin Maxwell, Awolf002, AxelBoldt, Axlrosen, Azatos, BM, Bad2101, Badgettrg, Baker APS, Bakkedal, Barak Sh, Barbara Shack, Bart133, Batmanand, Bcasterline, Bci2, Bcorr, Bdesham, Beano, Beatnik8983, Beek man, Belsazar, Bender235, Bensaccount, Bento00, Benvirg89, Benvogel, Beta Orionis, Betacommand, Betterusername, Bevo, Bfiene, Bfong2828, Bigsean0300, Billcosbyislonelypart2, Billybobjow, Bkalafut, BlastOButter42, Blethering Scot, BlooddRose, Bmju, BobTheBuilder1997, Bobo192, Bongwarrior, Bookalign, Borki0, Bowlofknowledge, Braincricket, Brannan.brouse, Brazmyth, Brettwats, Brews ohare, BrightStarSky, Brina700, Brougham96, Brufnus, Bsharvy, BuffaloBill90, Bustamonkey2003, Byrgenwulf, C.c. hopper, CALR, CBMIBM, CIreland, CSTAR, CStar, CYD, Caiaffa, Cain47, Callum Inglis, Can't sleep, clown will eat me, CanadianLinuxUser, Canderson7, Capricorn42, Captain Quirk, Captain-tucker, Carborn1, CardinalDan, Caroline Thompson, Carowinds, Casomerville, Catgut, Centic, Chairman S., Chao129, Chaos, CharlotteWebb, Charvest, Chaujie328, Chenyu, Cheryledbernard, Chetvorno, ChooseAnother, Chris 73, Chris5858, Christian List, Christian75, ChristopherWillis, Chun-hian, Chzz, CiA10386, CieloEstrellado, Cigarettizer, Cipapuc, Ciro.santilli, Civil Engineer III, Clarince63, Clarkbhm, Cleric12121, Closedmouth, ClovisPt, Cmichael, Colinue, Cometstyles, Commander zander, CommonsDelinker, Complexica, Connormah, Conversion script, CosineKitty, Couki, Cp111, Cpcheung, Cpl Syx, Creativethought20, Crowsnest, CryptoDerk, Csdavis1, Cst17, Curlymeatball38, Cybercobra, Cyclonenim, Cyp, DJDunsie, DJIndica, DL5MDA, DMacks, DV8 2XL, DVdm, Dale Ritter, Dalma112211221122, Dan Gluck, Dan100, DangApricot, DanielGlazer, Dannideak, Darguz Parsilvan, DarkFalls, Darrellpenta, Darth Panda, Das Nerd, Dataphile, Dauto, Davejohnsan, David Gerard, David R. Ingham, David Woolley, Davidwhite18, Dbroesch, Dcooper, DeFoaBuSe, DeadEyeArrow, Deaddogwalking, Declan12321, Decumanus, Delldot, DenisDiderot, DerHexer, Derek Ross, Derval Sloan, Detah, DetlevSchm, Diafanakrina, Dickwet89, Dingoatscritch, Discospinster, Djr32, Dmblub, Dmr2, Doc Quintana, Dod1, Dominic, Donbert, Donvinzk, Dougofborg, Dpotter, DrKiernan, Dragon's Blood, Dreadstar, Dreftymac, Drmies, Dspradau, Dtgm, Dylan Lake, E Wing, EMC125, Ed Poor, Edsanville, Edward Z. Yang, El C, El Chupacabra, ElectronicsEnthusiast, Elenseel, Elitedarklord dragonslyer 3.14159, Ellmist, Elochai26, Eloquence, Enochlau, Enormousdude, Enviroboy, Epbr123, Eric-Wester, EricWesBrown, Eurobas, Evan Robidoux, Eveross, Ex nihil, Excirial, F=q(E+v^B), FF2010, FaTony, Factosis, FagChops, Falcon8765, Falcorian, Farosdaughter, FastLizard4, Fastfission, Favonian, Fearingfearitself, Feezo, Feinoha, Femto, Figma, Final00123, Finalnight, Firas@user, Firewheel, Firien, Fishnet37222, Fizzackerly, Floorsheim, FloridaSpaceCowboy, Foobar, Foober, Four Dog Night, Fpaiano, FrancoGG, Frankenpuppy, Freakofnurture, Fredkinfollower, Fredrik, Freemarket, FreplySpang, Frosted14, Fudoreaper, Fvw, Fæ, GLaDOS, GTubio, GaaraMsg, Gabbe, GabrielF, Gaius Cornelius, Gandalf61, Garuda0001, Gary King, Gcad92, Gdrg22, Geek84, Geometry guy, GeorgeLouis, GeorgeOrr, Gfoley4, Giants27, Gierens22, Giftlite, Givegains, Gjd001, Glacialfox, Glenn, Glennwells, Gnixon, Gogo Dodo, GoingBatty, Gpgarrettboast, Gr33k b0i, Grafen, Grafnita, Graham87, GrahamHardy, GreenReflections, Greg HWWOK Shaw, Gregbard, GregorB, Grey Shadow, Grim23, Grunt, Guitarspecs, Guy82914, Guybrush, Guythundar, Gwernol, Gyan, H.W. Clihor, H0riz0n, H1nomaru senshi, HCPotter, Hadal, HaeB, HamburgerRadio, Han Solar de Harmonics, Handsome Pete, HappyCamper, Happyboy2011, Harddk, Harp, Harriemkali, Harryboyles, Harrytipper, Hashem sfarim, Hatster301, Hdeasy, Headbomb, Hekoshi, Henry Delforn (old), Heptarchy of teh Anglo-Saxons, baby, Herakles01, Hermione1980, Hickorybark, Hidaspal, Hikui87, Hipsupful, Hobartimus, Holy Ganga, HounsGut, Hughgr, Huon, Husond, Hut 8.5, Hvitlys, II MusLiM HyBRiD II, IJMacD, IRP, IW.HG, Ieditpagesincorrectly, Ilyushka88, Imalad, Imperial Monarch, In11Chaudri, Info D, InverseHypercube, Iquseruniv, Iridescent, Islander, Isocliff, Itub, Ivan Štambuk, Ixfd64, Izodman2012, J-Wiki, J.delanoy, JDP90, JForget, JIP, JMK, JSpudeman, JSpung, JSquish, Jack O'Lantern, Jacksccsi, Jackthegrape, Jaganath, Jagbag2, Jagged 85, Jake Wartenberg, JakeVortex, Jaleho, JamesBWatson, Jamessweeen, Jan eissfeldt, Jaquesthehunter, Jason Quinn, Javafreakin, JayJasper, Jayunderscorezero, Jc-S0CO, Jebba, Jecar, JeffreyVest, JeremyA, Jesse V., Jfioeawfjdls453, JinJian, Jitse Niesen, Jj1236, Jmartinsson, Jni, Jnracv, JoJan, JocK, Joe hill, John Vandenberg, John of Reading, JohnCD, Johnleemk, Johnstone, Jon Cates, Jordan M, JorisvS, Joseph Solis in Australia, JoshC306, Joshuafilmer, Jossi, Jostylr, Jouster, Juliancolton, Jusdafax, JzG, K-UNIT, KCinDC, KHamsun, KTC, Kadski, Kaesle, Kan8eDie, Karol Langner, Katalaveno, Katzmik, Kbrose, Kding, Ke4roh, Keenan Pepper, Kenneth Dawson,

196

Article Sources and Contributors Kesaloma, Ketsuekigata, Kevin aylward, Kevmitch, KeyStroke, Khalid Mahmood, Khatharr, Kingdon, KnowledgeOfSelf, Knutux, Koeplinger, Krackenback, Krazywrath, Kri, Kroflin, Kross, Kyokpae, Kzollman, L Kensington, L-H, LWF, La hapalo, Lacrimosus, Lagalag, Latka, Laurascudder, Lavateraguy, Laye Mehta, Le fantome de l'opera, Lear's Fool, Leavage, Leesy1106, Lemony123, Leo3232, LeoNomis, Leoadec, Lerdthenerd, Lethe, Liarliar2009, Light current, Lightmouse, Ligulem, Liko81, LilHelpa, Linas, Lindsaiv, Linuxbeak, Lithium cyanide, Lithium412, Llightex, Lmp883, Logan, LokiClock, Lontax, Looie496, Loom91, Loopism, Looxix, Lotje, Lowellian, LucianLachance, Luk, Luna Santin, Lupo, MBlue2020, MC ShAdYzOnE, MER-C, MK8, MKoltnow, MPS, MR87, Machdeep, Machine Elf 1735, Mackafi92, Macven, Macy, Mafiaparty303, Magioladitis, Magister Mathematicae, Magnagr, Magnus, Magog the Ogre, Mahommed alpac, Manuel Trujillo Berges, Marek69, Mark Swiggle, Mark91, Marks87, MarphyBlack, Martarius, Martial75, Martin Hedegaard, Materialscientist, Matt McIrvin, Mattb112885, Matthew Fennell, Matthewsasse1, Matty j, Maurice Carbonaro, Mav, Maximillion Pegasus, Mbell, Mbenzdabest, McSly, Mcdutchie, Mckaysalisbury, Mebden, Megankerr, Mejor Los Indios, Melicans, Melmann, Menchi, Meno25, Metaldev, Metricopolus, Meznaric, Mgiganteus1, Michael C Price, Michael Courtney, Michael Hardy, Michael9422, MichaelMaggs, MichaelVernonDavis, Michele123, MicioGeremia, Midnight Madness, Midom, Mifter, Mikaey, Mike Peel, Mike6271, Mikeiysnake, Mikeo, Minimac's Clone, Mitchellsims08, MithrandirAgain, Mjamja, Mjb, Mk2rhino, MoForce, Mod.torrentrealm, Modify, Modusoperandi, Moeron, Moink, Mojska, Mollymop, Moncrief, Monedula, Monty845, Moogoo, Moskvax, Mouselb, Mpatel, MrJones, MrOllie, Mrsastrochicken, Mrvanner, MtB, Muijz, Mustbe, Mxn, Myasuda, Mygerardromance, N1RK4UDSK714, N5iln, Nabla, Nagy, Nakon, Nanobug, Nathan Hamblen, Naturelles, NawlinWiki, NellieBly, Netalarm, Netrapt, Neverquick, NewEnglandYankee, Nibbleboob, Nick Mks, Nick Number, Nick125, NickBush24, NickJRocks95, Nielad, NightFalcon90909, NijaMunki, Nikai, Nin0rz4u 2nv, Niout, Niz, NoEdward, Noisy, Norm, North Shoreman, Northamerica1000, NotAnonymous0, Notolder, Nsaa, Nturton, NuclearWarfare, Numbo3, Nunquam Dormio, Nv8200p, Nxtid, Octavianvs, Odenluna, Olaniyob, Oleg Alexandrov, Olivier, Ondon, Onesmoothlefty, Onorem, Oo64eva, Optim, OrgasGirl, Otolemur crassicaudatus, Ott, OwenX, Owl, Oxymoron83, P99am, PAR, PGWG, Palica, PandaSaver, Papadopc, Pascal.Tesson, Pasky, Pat walls1, Pathoschild, Patrick0Moran, Paul August, Paul Drye, Paul E T, Paul bunion, Paulc1001, Paulraine, Pcarbonn, Pchov, Peace love and feminism, Penguinstorm300, Pepper, Peter, Peterdjones, Petgraveyard, Petri Krohn, Pfalstad, Pgb23, Pgk, Phil179, Philip Trueman, Philippe, Philmurray, Phoenix2, PhySusie, Phyguy03, Phys, Physics1717171, PhysicsExplorer, Physika, Physprof, Piano non troppo, Pilotguy, Pinethicket, PinkShinyRose, Pippu d'Angelo, Pj.de.bruin, Pjacobi, Pkj61, Pleasantville, Plotfeat, Pokemon274, Polgo, PorkHeart, Positivetruthintent, Possum, Prashanthns, Pratik.mallya, Professormeowington, Psaup09, Pxfbird, Qmtead, Quackbumper, Quantummotion, Quercus basaseachicensis, Quuxplusone, Qwertyuio 132, Qxz, Qzd800, RG2, RJaguar3, Raaziq, Radzewicz, Raggot, Rakniz, RandomXYZb, Randy Kryn, Raymondwinn, Razataza, RazorICE, Rdsmith4, Reaper Eternal, Reconsider the static, Rednblu, Reinis, Rettetast, Rex the first, Rgamble, Rich Farmbrough, Richard L. Peterson, Richard.decal, Richard001, Ridge Runner, Rightly, Rjwilmsi, Rnb, Rnpg1014, Roadrunner, Robb37, Robbjedi, RobertG, Robertd514, Roberts Ken, Robin klein, RobinK, RockMagnetist, Rodier, Rodri316, Rogper, Romaioi, Ronhjones, Roopydoop55, Rory096, Roxbreak, RoyBoy, Royalbroil, Royalguard11, Rursus, Russel Mcpigmin, RxS, Ryan ley, RyanB88, SCZenz, SCriBu, SFC9394, SHCarter, SJP, SWAdair, Sabri Al-Safi, Sadi Carnot, Saibod, Sajendra, Salamurai, Samlyn.josfyn, SanfordEsq, Sango123, Sankalpdravid, Saperaud, Saurabhagat, Savant1984, Sbandrews, Sbyrnes321, SchreiberBike, Schumi555, ScienceApologist, Scott3, Scottstensland, ScudLee, Sean D Martin, Search4Lancer, Seawolf1111, Seba5618, SeriousGrinz, SeventyThree, Sfngan, Shadowjams, Shanemac, Shanes, Sheliak, Shipunits, Shoyer, Signalhead, Silence, Silly rabbit, Sionus, Sir Vicious, SirFozzie, Sjoerd visscher, Skewwhiffy, Skizzik, SkyWalker, Slaniel, Smallpond, Smear, Smyth, SnappingTurtle, Snowolf, Snoyes, Sodicadl, Sokane, Sokratesinabasket, Soliloquial, Solipsist, Solo Zone, Solomon7968, Someguy1221, Soporaeternus, SpaceMonkey, Spakwee, Spearhead, Specs112, Spellcast, Spiffy sperry, Spinningspark, Sriram sh, Srleffler, St33lbird, Stanford96, StaticGull, StefanPernar, Stentor7, Stephan Leclercq, Stephen Poppitt, Stephen e nelson, Steve Quinn, Stevenganzburg, Stevenj, Stevenwagner, Stevertigo, Stiangk, Stone, StonedChipmunk, StopTheCrackpots, Storm Rider, Stormie, StradivariusTV, Sukael, Sukaj, SunDragon34, Superior IQ Genius, Superlions123, Superstring, Supppersmart, Surfcuba, Swegei, Swwright, Syrthiss, TTGL, Ta bu shi da yu, TakuyaMurata, Tangotango, Tantalate, Taral, Tarotcards, Taylo9487, Tayste, Tbhotch, Tcncv, Teardrop onthefire, Teles, Template namespace initialisation script, Terra Xin, Texture, Tgeairn, The Anome, The Captain, The Firewall, The High Fin Sperm Whale, The Mad Genius, The Mark of the Beast, The Red, The Spam-a-nata, The Thing That Should Not Be, The beings, The ed17, The wub, TheKoG, Thepalerider2012, Theresa knott, Thingg, ThisLaughingGuyRightHere, Thunderhead, Tide rolls, Tim Shuba, Tim Starling, TimVickers, Timwi, Titoxd, Tkirkman, Tobby72, Tobias Bergemann, Tohd8BohaithuGh1, Tom.Reding, Tommy2010, Toofy mcjack34, Tpbradbury, Train2104, TripLikeIDo, Tripodics, Tromer, Trusilver, Truthnlove, Ttasterul, Tycho, TylerM37, Tylerni7, UberCryxic, Ujjwol, Ujm, Ultratomio, Uncle Dick, UncleDouggie, Updatehelper, Uploadvirus, Urbansky, User10 5, Utcursch, V81, Vagigi, Van helsing, VandalCruncher, Vb, Velvetron, Venu62, Vera.tetrix, Versus22, Viduoke, Vitalikk, Vlatkovedral, Voyajer, Vsmith, Vudujava, WAS 4.250, WMdeMuynck, Wafulz, Waggers, WarEagleTH, Warbirdadmiral, Wareagles18, Wavelength, Waxigloo, Wayne Slam, Wayward, Weiguxp, Welsh-girl-Lowri, Werdna, WereSpielChequers, Weregerbil, Wfaze, Wham Bam Rock II, White Trillium, Wik, WikHead, Wikiklrsc, Wikipelli, Wikishotaro, William Avery, William.winkworth, Williamxu26, Willieru18, WilliunWeales, Willking1979, Willtron, Wimt, Winston Trechane, Winston365, Wk muriithi, Wknight94, Woohookitty, Word2need, Wquester, Wrwrwr, Wwoods, X41, XJamRastafire, XTRENCHARD29x, Xana's Servant, Xanzzibar, Xenfreak, XeniaKon, Xezlec, Xihr, Yamaguchi先 生, Yath, Yeahyeahkickball, Yellowdesk, Yokabozeez, Yowhatsupdude, Zakuragi, Zarniwoot, Zenbullets, Zero over zero, Zhinz, Zigger, ZiggyZig, Zolot, Zondor, Zooloo, Zslevi, Александър, 2148 anonymous edits Interpretations of Quantum Mechanics  Source: http://en.wikipedia.org/w/index.php?oldid=525750279  Contributors: 23haveblue, A. di M., Aarghdvaark, Adwaele, Afshar, Agondie, Airplaneman, Alessandro70, Alienus, Alison, Alkivar, Amourfou, Arichnad, Arjen Dijksman, Ark, Army1987, Arthena, Bardon Dornal, Bduke, Beetstra, Belsazar, Berteun, Bfiene, Billinghurst, Blainster, Bm gub, Bmju, Brazmyth, Brews ohare, Burkhard.Plache, Burn, Byelf2007, Byrgenwulf, C S, C h fleming, CSTAR, Camcolit, Carbon-16, Carlorovelli, Caroline Thompson, CesarB, Charles Matthews, Cholling, Chris Howard, Cic, Citynoise, Crust, Cryptonaut, CubsMan91, Dan Gluck, Dan Pelleg, Daniel Weissman, Danthewhale, David R. Ingham, Decumanus, Deniskrasnov, Dieksje, Djr32, Dragon's Blood, Dreamer08, Duff man2007, Dvtausk, EdJohnston, ElBenevolente, Elimisteve, Enormousdude, ErkDemon, Frau Holle, Freakofnurture, Gabbe, Gaius Cornelius, GangofOne, Gauge, Geremia, Giftlite, GoingBatty, Golbez, Graham87, Gramschmidt, Gregbard, Guevara's Revenge, GuidoGer, Gurch, H0riz0n, H2g2bob, Hairy Dude, Happyboy2011, Harizotoh9, Headbomb, IIBewegung, INic, Ilmari Karonen, InXistant, Iridescent, JEN9841, JIP, Jackol, Jefffire, John Broughton, John of Reading, JohnBlackburne, JohnMcFerlane, JohnSankey, Jordgette, Jostylr, Kane5187, Karol Langner, KasugaHuang, Kbh3rd, Keithbowden, Kevin aylward, Kishanvbhatt, Kmarinas86, KnightMove, KnightRider, Kripkenstein, L33tminion, LC, Lacatosias, Laurascudder, Lethe, Light current, Likebox, Linas, Luckstev, Lumidek, Machine Elf 1735, Materialscientist, Maurice Carbonaro, Mbell, Mheaney, Michael C Price, Michael Hardy, Mike4ty4, Modocc, Mogudo, Mystic.Ventus, Naasking, Nordisk varg, Normxxx, Oleg Alexandrov, Omeganumber, Owl, Pablo X, PaddyLeahy, Pak21, Palindromatic, Paolo.dL, Paul August, Paul D. Anderson, Pekka.virta, Peterdjones, Pfalstad, Physics108, Pjacobi, Pkeastman, Pleasantville, Plumbago, Pmcmahon, Portillo, Protohiro, RK, RQG, Rafaelgr, Rami radwan, Randallbsmith, Randommuser, Rasmus Faber, Reddi, Redsolarearth, Reedy, Resuna, Rjwilmsi, Roadrunner, Rtc, Ryan.vilbig, SPrasanth1991, SQL, Samboy, Samlyn.josfyn, Sbyrnes321, Schlafly, Sdornan, Sebastianlutz, Sheliak, Sigmundur, Simetrical, Sokane, Spireguy, Stephen Poppitt, Stevertigo, Tabletop, Tanath, Tarotcards, Tbleher, TedPavlic, Tercer, Tesseract2, Texture, Tgr, Tito-, TonyW, Tparameter, Truthnlove, Ujm, Vashti, Vroman, WMdeMuynck, Weregerbil, William M. Connolley, Wooster, Wragge, WuTheFWasThat, Xanzzibar, Yensin, Yetisyny, Yuttadhammo, Zarniwoot, Zujua, 285 anonymous edits The Copenhagen Interpretation  Source: http://en.wikipedia.org/w/index.php?oldid=522242848  Contributors: 24.93.53.xxx, Afshar, AgadaUrbanit, Agge1000, Agger, Akriasas, Alienus, Alma Pater, Amakuha, Amble, Amdayton, Andybiddulph, Ark, Atomota, AxelBoldt, Barbara Shack, Barraki, Batmanand, Bazzargh, Bbbl67, Belsazar, Ben Thuronyi, Berkay0652, Bigmantonyd, Blaine Steinert, Bm gub, Bmatthewshea, Boatteeth, Bonaovox, Bookalign, Boozerker, Brandonsmells, Bryan Derksen, CLW, CYD, Carey Evans, Cfp, Charles Matthews, Chas zzz brown, Chjoaygame, Complexica, ConradPino, Conversion script, Cortonin, Cpiral, Crochet, Cybercobra, DAGwyn, DJIndica, Dan Gluck, David R. Ingham, David R. Ingharn, DavidRideout, Dicklyon, Dirac66, Djr32, Dominic, Don4of4, Dragon's Blood, Drunken Pirate, Duendeverde, EdJohnston, Edward, Eequor, El C, Ewlyahoocom, Fatka, Fazalmajid, Fernbom2, Fph, Freakofnurture, Gcbirzan, Geremia, Giftlite, Glenn, Graham87, Gregbard, Gretyl, Guoguo12, Haham hanuka, Hairouna, Harry Potter, Hbayat, Headbomb, Heron, Igni, J-Wiki, Jamshearer, Jeltz, Joel7687, JohnArmagh, Joriki, K-UNIT, KHamsun, Kahn, Kate, Kevin aylward, KillerChihuahua, Kriegman, Kwertii, LC, Light current, Likebox, Linas, Loe988, Looie496, Looxix, Lotu, Lumidek, Mac Dreamstate, Machine Elf 1735, Manicsleeper, Marcus Qwertyus, Matttoothman, Maurice Carbonaro, Mav, Mbell, McGinnis, Mdwh, Michael Hardy, Mm1671, Moocha, MorphismOfDoom, Murf42, NYKevin, Nabla, Nbarth, Nercromancy?, Nick Number, OlEnglish, ParticleMan, Patrick0Moran, Pcbene, Peterdjones, Phys, Pjacobi, PointedEars, Prickus, Quadell, RCSZach, RFWerner, Rbedford, Realneyc, Rich Farmbrough, Richard001, Rickard Vogelberg, Rjwilmsi, Roadrunner, Rorro, Ross Fraser, Runningonbrains, Rursus, SalineBrain, Samboy, SaveTheWhales, Sbiki, Scapermoya, Schaefer, Schneelocke, Schuermann, Shambolic Entity, Sheliak, Sokane, Sperling, Stanford96, Stephen Poppitt, Stevenwagner, Stuidge, Superpaul3000, TYelliot, Tabish q, Tanuki Z, Tesseract2, The Anome, TheMathinator, Thecheesykid, TheloniousBonk, TimidGuy, Timo Honkasalo, Timwi, Tparameter, Trilobitealive, Trugster, Ujm, Unyoyega, Uppland, User2004, Venny85, Victor Gijsbers, Vssun, Vyznev Xnebara, WMdeMuynck, William.winkworth, Wjmallard, Xezbeth, Yonatan, 229 anonymous edits Principle of Locality  Source: http://en.wikipedia.org/w/index.php?oldid=526695842  Contributors: Agge1000, Avirab, Bility, Boson, Burn, Caroline Thompson, Charles Matthews, Clicketyclack, ClockworkSoul, Contact '97, Cybercobra, Dave Kielpinski, David Eppstein, Deathphoenix, Doquynh2k7, DÅ‚ugosz, Ehrenkater, Ettrig, Evercat, Gill110951, Grutter, Hairy Dude, Jmnbatista, Joriki, JorisvS, LarsPensjo, LedgendGamer, Len Raymond, Leobold1, Linas, Lumidek, Machine Elf 1735, Maurice Carbonaro, Maxw41, Michael C Price, Mpatel, NOrbeck, Naasking, Nwbeeson, Onebyone, Paine Ellsworth, Peterdjones, Petri Krohn, Phys, Populus, Ross Fraser, Ruud Koot, Ryan Roos, Stephen Poppitt, Stirling Newberry, Traitor, Tweisbach, Vyznev Xnebara, Wiggles007, Zvis, 64 anonymous edits EPR Paradox  Source: http://en.wikipedia.org/w/index.php?oldid=525672349  Contributors: -- April, .:Ajvol:., 207.171.93.xxx, 2over0, ASCWiki, Agge1000, Agger, Alan McBeth, Alkivar, Amakuha, AmarChandra, Andejons, Apoc2400, Ark, Arpingstone, Astrofan7, AxelBoldt, B4hand, Ballhausflip, Bevo, Bigbluefish, Binksternet, Blauhart, Bob K31416, Brews ohare, Brianga, Brrk.3001, Bryan Derksen, CSTAR, CYD, Camrn86, Canadian-Bacon, Carbuncle, Cardmagic, Caroldermoid, Caroline Thompson, Chas zzz brown, Chjoaygame, Chris 73, Chris Howard, Clarityfiend, CobbSalad, Complexica, Cortonin, CosmiCarl, Cpiral, Craig Pemberton, Crowsnest, Cspan64, D0nj03, DAGwyn, DBooth, Dalit Llama, Darktaco, DaveBeal, David R. Ingham, David spector, Declare, Dogcow, Dpbsmith, Dr Smith, DrBob, DrChinese, DragonflySixtyseven, Dratman, Dryke, Dzhim, Długosz, EdH, Edward, Eequor, Egmontaz, Ehn, Eiffel, Ejrh, Elassint, Emvan, Etabackman, Ettrig, Falcorian, Finemann, Finlay McWalter, Fredkinfollower, Frish, Fulldecent, Fwappler, GeorgeMoney, Giftlite, Goldfritter, Graham87, GregorB, Gretyl, HCPotter, Hackwrench, Harald88, Headbomb, Hephaestos, Hhhippo, Hirak 99, Houftermann, Hugo Dufort, Hydnjo, IAdem, Iamthedeus, J-Star, JaGa, JamesMLane, Jan-Åke Larsson, Jcajacob, Jinxman1, Jnc, JocK, JohnBlackburne, Jpittelo, Jpowell, Jwrosenzweig, KHamsun, Kapalama, Karada, Karol Langner, KasugaHuang, Keenan Pepper, Kuratowski's Ghost, Larsobrien, Lethe, Lf89, LiDaobing, Linas, Linus M., Looxix, Lumidek, Maher27777, Marek.zukowski, Marie Poise, Mark Arsten, Mark J, Masudr, Maurice Carbonaro, Mav, Meiskam, Metron4, Michael C Price, Michael Hardy, Moink, MorphismOfDoom, MrJones, Msridhar, Myrvin, Naddy, Natevw, NathanHurst, Nickyus, Noca2plus, ObsidianOrder, Orange Suede Sofa, Owen, Pace212, Paine Ellsworth, Paranoid, Pateblen, Patrick0Moran, Peashy, Pekka.virta, PenguiN42, Peter Erwin, PeterBFZ, Peterdjones, PetrGlad, Phancy Physicist, PierreAbbat, Prezbo, Publicly Visible, Pwjb, Pérez, Quondum, RTreacy, Radiofriendlyunitshifta, Rama, Razimantv, Rentzepopoulos, Rich Farmbrough, Rjwilmsi, Roadrunner, Robert K S, Robertd, Robertefields, Ronjoseph, Rracecarr, Ryft, SGBailey, Sanders muc, Schneelocke, ScienceApologist, Seb, Shadanan, Shakyshake, Shalom Yechiel, Sidasta, Sigmundur, Skierpage, Snoyes, Sonicsuicide, Srleffler, Stain, Stephen Poppitt, Steve Quinn, Suisui, Sundar, Syko, Tarotcards, Tempshill, Tercer, Texture, ThomasK, Thrain2, Timwi, Tlabshier, Trilobitealive, Tsop, Ty8inf, TylerDurden8823, Vasiľ, Victor Gijsbers,

197

Article Sources and Contributors Vodex, Voyajer, WMdeMuynck, Waleswatcher, Weekwhom, Wik, Wile E. Heresiarch, William M. Connolley, WillowW, XJamRastafire, Xgrrr, Xnquist, YUL89YYZ, Yill577, Zeycus, Zootm, Александър, 322 anonymous edits Bell's Theorem  Source: http://en.wikipedia.org/w/index.php?oldid=526872304  Contributors: &Delta, A. di M., AC+79 3888, AdamSiska, AdamSolomon, Agge1000, Aldux, AmarChandra, Amareshdatta, Anakin101, Andrewthomas10, Android Mouse, Andyjsmith, Antonielly, Aranel, ArnoldReinhold, Arthur Rubin, Ashmoo, Avb, B9 hummingbird hovering, Ballhausflip, Barticus88, Baxxterr, BeatePaland, Bender235, BeteNoir, BiT, Bo Jacoby, BobKawanaka, Bobo192, Bongwarrior, Brad7777, Brazmyth, BrianWren, Bryan Derksen, Byrgenwulf, CRGreathouse, CSTAR, CYD, Cacycle, Calvero JP, Caroline Thompson, Catchanil, Cgingold, Chalst, Charles Matthews, Chkno, Chris Howard, Chris the speller, Christopher Cooper, Complexica, Count Iblis, Cremepuff222, Crocodealer, Curious1i, DarwinPeacock, Dauto, Dave Runger, Deathphoenix, Deniz195, Dfrg.msc, Dirac66, Dmr2, DomenicDenicola, Don warner saklad, DrChinese, Drilnoth, Drmies, Dtgriscom, EcoMan, EdC, Eequor, Egg, Ekabhishek, Endlessmike 888, F=q(E+v^B), Fastily, FormerNukeSubmariner, Franl, Frish, Fulldecent, Fwappler, GangofOne, Gareth Griffith-Jones, Gene Nygaard, Geremia, Giftlite, Gill110951, Giraffedata, Gmusser, Gregbard, GregorB, Grok42, Groyolo, Gsjaeger, Guy Harris, Gzornenplatz, HaeB, Hairy Dude, Harald88, Headbomb, Henry David, Hmonroe, Hqb, IRWolfie-, IWillNeverLearn, Iconofiler, Incnis Mrsi, Informatorium, Interintel, Isaacdealey, Isocliff, J-Wiki, Jack who built the house, Jcobb, Jim E. Black, Jim.belk, Jncraton, Jpittelo, Jules.lt, Jwpitts, KHamsun, Katherine Pendleton, Keenan Pepper, Keithbowden, Kingmundi, Kmmertes, KnowledgeOfSelf, Konstable, Ksolway, Kyrisch, LC, Laudaka, Leobold1, Lethe, Likebox, Linas, Loggerjack, Lumidek, Lycurgus, Malcohol, Mandolinface, Maniatis, Marek.zukowski, Mastertek, MathKnight, Maxw41, Mbweissman, Michael C Price, Michael Hardy, Mike mian, Mike2vil, Mister fu-ck you, MisterSheik, Mmyotis, Mogism, Monslucis, MorphismOfDoom, Mujokan, Myrvin, NOrbeck, Nihil, Nilock, Nowakpl, NuclearWarfare, Oakwood, Oleg Alexandrov, Olek Bijok, Orangedolphin, Ott2, Paddu, Patrick0Moran, Patsobest, Paul August, Paulginz, Petri Krohn, Phoenix-antiscammer, Physicskev, Physis, Pjacobi, Plrk, Pokipsy76, Quibik, RJFJR, RK, Rbodgers, RekishiEJ, Rich Farmbrough, Richard75, Richwil, Rickard Vogelberg, Rl, Roadrunner, Robert1947, Rodasmith, Roke, Rror, Ryanmcdaniel, SJLPP, Sabri Al-Safi, Salsb, SchreiberBike, Seth Bresnett, Simetrical, Skeppy, Slawekb, SlimVirgin, Stdjmax, Steady unit, Stephen Poppitt, StevenJohnston, Stevertigo, Stirling Newberry, Suffusion of Yellow, Suslindisambiguator, Susvolans, Syncategoremata, Tethys sea, Thorseth, Tide rolls, Tothebarricades.tk, Tox, TriTertButoxy, Trusilver, Tsirel, Ueit, Ulrich Utiger, Vessels42, WMdeMuynck, Waleswatcher, Wfku, Wikikam, WookieInHeat, Wwoods, Ybband, Yecril, Ymblanter, Yuttadhammo, Zell2929, Zeycus, Zvis, 老 陳, 280 anonymous edits Schrödinger's Cat  Source: http://en.wikipedia.org/w/index.php?oldid=526614467  Contributors: -jmac-, .:Ajvol:., 100110100, 198, 24.93.53.xxx, 64.12.104.xxx, A. di M., A2326xyz, ABCD, AEdwards, AVJP619, Abtract, Access Denied, Acroterion, Adashiel, Adpete, Af648, AgadaUrbanit, Agamemnon2, Agge1000, Ajfweb, Albanharrison, Aleenf1, AlexPenson, Alfredr, Amakuru, Amcbride, Amorymeltzer, AnOddName, Andrewjlockley, Andrewpmk, Angr, Angry Lawyer, Anonymous Dissident, Antandrus, Anupamsr, Anville, Aperisic, Asbestos, Ashley Pomeroy, Ashmodai, AstroNomer, AussieLegend, Avb, Bakkster Man, Beeblebrox, Benhocking, Bensmith53, Betterusername, Bhoch14, BigFatBuddha, Bill Spencer5, Bjdehut, Bmicomp, Bob K31416, Bob toben, BobKawanaka, Bobby D. Bryant, Bobbydowns, Bobo192, BookgirlST, Borisblue, Bovineone, Brews ohare, Brion VIBBER, Brookston, Broski, Brusegadi, Bryan Derksen, Bth, Bubba hotep, Bucephalus, CShippee, CYD, Calvinkrishy, Camembert, Canadian Paul, CanadianLinuxUser, Canterbury Tail, Capricorn42, Carbuncle, Catbag, Catgut, Ccscott, Centrx, Cgay88, Chaos, Charlesrkiss, Chase2032, Chemguy2, Chetvorno, Chitomcgee, Chris Longley, ChrisHodgesUK, Christofurio, Ciphergoth, ClockwrokSoul, Closetsingle, Cmsreview, Cohesion, Colincbn, Commonbrick, Complexica, Comrade009, Conversion script, Corporal Tunnel, Cpl Syx, Craw's Revenge, Crazynas, Crosbie Fitch, Crystallina, DAGwyn, DAID, DJ Clayworth, DJIndica, DV8 2XL, Danlibbo, Darkwhistle, Darth Panda, David R. Ingham, David R. Ingharn, David.Monniaux, Davidizer13, Davidpdx, Dc987, Deanlaw, Den fjättrade ankan, Dennisthe2, Derwyk, Dfhussey, Dhatfield, Dicklyon, Discospinster, Djpadz, Dmoulton, Doorag1, Dr.Gonzo, Dr.K., Drmies, Drunkenmonkey, Dspradau, Eaglizard, Ebehn, Eblingdp, Ed Cormany, Eeekster, El C, Elation, Ennerk, Epbr123, Eric Shalov, Escape Orbit, Evil Monkey, Falcon8765, Famousdog, Faseidman, Faseidrnan, Feldspaar, Fenway4201912, Fetchcomms, Fibonacci, Fieldday-sunday, FlamingSkullComics, Floria L, FlorianMarquardt, Foobar, Forai, Francis Ocoma, Free Bear, FriendlyRiverOtter, Gabbe, Gagueci, Galaxiaad, Gavin.perch, Gawaxay, Gerhardvalentin, Giftlite, Giggy, Gilly3, Glacialfox, Gore Physicist, Graham87, Greenleaf, Gregbard, Greudin, Grunt, Gruntree, Gurch, Gz33, HDarke, HPRappaport, Haakon, Hadal, HaeB, Hamiltondaniel, Harkleroad, Harry, Hasek is the best, Headbomb, Henning Makholm, Heracles31, Heron, Hhhippo, Hillbrand, Hires an editor, Hmhur, Holylampposts, Hondalube32, HorsePunchKid, Hovden, Howellpm, I am a Wikipedian, IAmAndrewSoul, IRP, Identity99uk, Ikarus14, Intelligenttheorist, Ixfd64, J.delanoy, JE, JIP, JPX7, Jack Mickkelson, Jaganath, Jayden54, Jblatt, Jcfolsom, Jeltz, Jim Birkenshaw, Jim McKeeth, Jim1138, Jivecat, Jjamison, Jncraton, Joao, John, JohnArmagh, JohnSankey, Johnvdenley, Jordgette, Josh Grosse, Jtmichcock, JustAddPeter, K, KagakuKyouju, Kanthoney, Karl, Kevin aylward, Khalid Mahmood, Khazar, KittySilvermoon, Klonimus, Koyaanis Qatsi, Kukec, Lambiam, Lancereq, LeadSongDog, Lear's Fool, LearningIsLiving, Lectonar, Lefty, Light current, Lignomontanus, Likebox, Linas, Lkstrand, Loodog, LookingGlass, Lord Anubis, Luckyherb, Lumidek, MER-C, Maelin, MafiaCapo, MagneticFlux, Mako098765, Mandarax, Marc711, MartinHarper, MathKnight, Maurice Carbonaro, Max Terry, Maziotis, Me.play, MercuryBlue, Mesoderm, Michael C Price, Mieszko the first, Mike Rosoft, MikeBartlett, Mikoangelo, Mild Bill Hiccup, Mindmatrix, Minimac, Mion, Mipadi, Mjodonnell, Mono mano, Monsieur Prolong, Mrawesomemanforeverwoo, Mrdude, Ms2ger, Mungukwachupi, Muu-karhu, N5iln, NTK, Nabla, Nabokov, Nakon, Nard the Bard, NawlinWiki, NellieBly, Nessus, Nick Mks, Nile, No Guru, NormanEinstein, Northside777, Nullx42, Nvmandi, O18, Ofekalef, Old Moonraker, Opelio, Optichan, OranL, OrangeAipom, Oski97, Otis182, Owen, Paste, Paul August, Pernicat, Personman, Peter Meyrathing, Peter Tomblin, Peterdjones, Pfalstad, Pgan002, Pharaoh of the Wizards, Pharos, Philg88, Philip Trueman, Philosophus, Phoenixrod, Picaroon, PierreAbbat, Pinethicket, Piquan, Pitamus, Pitr C, Pladask, Polyparadigm, PonyToast, Prezbo, Profugum, Protohiro, Prunesqualer, PseudoSudo, Psternenberg, Pureaddiction, Quaestor23, Quandon, RG2, RK, RMFan1, Racerx11, Raisesdead, RajZsingh, RandMC, RandomCritic, Reach Out to the Truth, Rearete, RedWolf, Redirect fixer, Reportica, Rich Farmbrough, Richard Weil, RickDeNatale, Riconaps, Riki, Roadrunner, RobertG, Ronaldomundo, Ronhjones, Rwjensen78, Ryan4314, RyanCross, RyoGTO, SMP, SabineLaGrande, Sabrebd, Saebjorn, Saigon punkid, Sam Spade, Samaritan, Sampi, Sanders muc, Sango123, Saperaud, SavinSav, SchnitzelMannGreek, ScienceApologist, Scimitar, Scoutersig, Sean D Martin, Sean.hoyland, Seaphoto, Seirscius, Serendipodous, Serpens, Sewblon, Shadowjams, Shanes, Sheliak, Shervinafshar, Shizhao, Sidesteal, Silly rabbit, Sillybilly, SimonD, Sjorford, Skinnyweed, Sky Harbor, Skychrono, Slimshady72397, Smartneddy, Smartyhall, Smitz, Smyth, Snillet, Snoyes, Sophitus, Spade285, Spearhead, Splash, Spring1, Sprowl, Srnitz, Sten, Stepa, Steven J. Anderson, Stevertigo, Sumsum2010, Sweetser, TUF-KAT, Tabster0000, Taejo, TaintedMustard, Tannkrem, Teflon Don, Teutonic Tamer, The Cunctator, The Monster, The Rambling Man, The Thing That Should Not Be, The wub, TheSun, Theendofgravity, Thorncrag, Tide rolls, Timotheus Canens, Timwi, Tobias Bergemann, Tom Morris, Tom Peters, Tommy2010, Toytoy, Tregoweth, Trevor MacInnis, Trilobitealive, Trusilver, Tufflaw, Twas Now, Typobox, Ujjwol, Ukcreation, Ukexpat, Uogl, Useight, Vacation9, Vadim Makarov, Val42, Vanischenu, Varuna, Veghead, Versehalf, VincentCalci, Violetriga, Virogtheconq, VolatileChemical, WCFrancis, WMdeMuynck, Wandall Schwartz, Wapcaplet, Waterfall117, Wenli, Weregerbil, Wereon, Wetllama, Whitehorse1, Whoop whoop pull up, Whosyourjudas, Wik, Wikiliki, Wikimancer, Wikipelli, Will2k, Wind-up Spirit, Wisq, Worthawholebean, Wwheaton, XJamRastafire, Yamamoto Ichiro, Yar Kramer, Yeezil, Yngvarr, Ytrewq05, Yvwv, Zakhebeone, Zeerak88, Zenosparadox, ZeroOne, Zginder, ^demon, Мурад 97, 857 ,‫ דוד‬anonymous edits The Measurement Problem  Source: http://en.wikipedia.org/w/index.php?oldid=517381855  Contributors: Aarghdvaark, Angelpbj, Apoc2400, ArglebargleIV, Bm gub, Brews ohare, Charles Matthews, Chetvorno, David R. Ingham, Dchristle, Doradus, Dragon's Blood, EMPE, Edward, El C, Erik Carson, Freakofnurture, Impossiblepolis, Jdforrester, Jean-Christophe BENOIST, Jeffq, JocK, Jockocampbell, JohnBlackburne, Jordgette, Jostylr, Karol Langner, Lendtuffz, Likebox, Linas, Lumidek, Machine Elf 1735, Maury Markowitz, Michael C Price, Michael Hardy, Mjszzz, Nihilo 01, Patrick0Moran, Pekka.virta, Peterdjones, Pfa alkemade, Pfalstad, Phil Boswell, Rjwilmsi, Roberts Ken, RoyBoy, Sam Hocevar, Sigmundur, Solid Contributor, Stevertigo, Tesseract2, Tgeairn, Ujm, Vyznev Xnebara, Woz2, Xihr, Zoegernitz, Zwikki, 57 anonymous edits Measurement in Quantum Mechanics  Source: http://en.wikipedia.org/w/index.php?oldid=522251869  Contributors: Alba, Algebraist, Ancheta Wis, Apoc2400, Aqerto, Arpingstone, Auspex1729, Baltorn, Bdesham, Bm gub, Bobo192, Brenont, Brews ohare, Bronix, Bth, Cohesion, Commander Keane, CosmiCarl, DJIndica, Danlaycock, Dave Kielpinski, Dicklyon, Fredrik, Fuhghettaboutit, Fwappler, GangofOne, Grayshi, H2g2bob, Headbomb, Henry Delforn (old), Hudon, Incnis Mrsi, J S Lundeen, JFlav, Jake Wartenberg, Jantov, Jeepday, Jetherrie, JocK, JonathanD, Joriki, Judgesurreal777, Karol Langner, KasugaHuang, Korepin, Kzollman, Laurascudder, Li3939108, Likebox, Linas, Lofor, M0532062613, Macmelvino, Makiteo, MathKnight, MathMartin, Matthew Yeager, Maurice Carbonaro, Mct mht, Michael C Price, Michael Hardy, MountainSplash, NJWard, Peterdjones, Philip Trueman, Phys, RCSB, Rjwilmsi, RobHar, Robert L, STHayden, SWAdair, Sam Hocevar, Sbyrnes321, SeventyThree, Sheliak, Smack, Spoon!, StaticGull, Stevertigo, Stongli, TStein, TakuyaMurata, Tedickey, Tercer, Ujm, WMdeMuynck, Who, William M. Connolley, Yaoyun.shi, Zarniwoot, 81 anonymous edits Quantum Number  Source: http://en.wikipedia.org/w/index.php?oldid=525553913  Contributors: Acalamari, Akshanshshrivastava, Albrozdude, Aliekens, Allmightyduck, Arthena, Arthur Rubin, Azure8472, BD2412, Bambaiah, Barticus88, Br77rino, Brownsteve, BryanD, Bubbachuck, Canopus49, Chaos, Charles Baynham, Choihei, Christopherlin, Cobaro, Complexica, Craig Pemberton, Cubbi, DARTH SIDIOUS 2, DJIndica, Dan Gluck, DaveTheRed, Davemcarlson, Decumanus, Demeza13, Dmmaus, Donarreiskoffer, Drova, EarthCom1000, EddEdmondson, Edsanville, El.vegaro, F=q(E+v^B), Fred Bradstadt, Fresheneesz, Gfoley4, Giftlite, Go2slash, Hadal, Hairy Dude, HappyCamper, Hidaspal, Im.a.lumberjack, Imnotoneofyou, Itub, Ivy martin08, J.delanoy, JWB, JabberWok, Jasper Deng, Jengelh, John, Joyonicity, Just granpa, Kanags, Kareeser, Katoa, Keraunoscopia, Kushal one, L0ngpar1sh, Larryisgood, Law, Maatghandi, ManoaChild, Matrix61312, MaximH, Mets501, Modify, Mpatel, Mrfair, Mushin, Musicality213, Mxn, Myasuda, NewEnglandYankee, Nsaa, PV=nRT, Phantombantam, Pol098, Qmonkey, Ravilovefriends, Redrose64, RetiredUser2, RobinK, Sadeq, Sealbock, SeventyThree, Shivsagardharam, SkyLined, Somdebg, Sp4cetiger, SpK, SpeedyGonsales, Spoon!, Stevertigo, Suspekt, Sverdrup, Syntax, Teply, That Guy, From That Show!, The Anome, Tim Starling, Tommy2010, Trapezoidal, Trevor MacInnis, Troller Hi, Tsemii, Venny85, Versus22, Voyajer, Vsmith, Wake chaser, Wavelength, Waxigloo, Waza, WikiUserPedia, Wknight94, Xavic69, Yevgeny Kats, Yoshigev, Zahd, 238 anonymous edits Quantum Information  Source: http://en.wikipedia.org/w/index.php?oldid=524464422  Contributors: 3malchio, Abdelaty, Alinihatekenwiki, Ammarsakaji, Antandrus, Ballhausflip, Bearcat, CSTAR, Charles Matthews, Chemako0606, Cipapuc, Ckwongb, Cnilep, Conversion script, Creidieki, DFRussia, DGG, Dave Kielpinski, Dcoetzee, Dgrant, Dojarca, Eliium, Gene.arboit, Guillaume2303, Isnow, Itchy, J S Lundeen, Jamelan, KathrynLybarger, Kenneth M Burke, Kilor, Kipton, Knotwork, Korepin, Linas, Looie496, Mandarax, Mastertek, Materialscientist, Maurice Carbonaro, Mct mht, Michael Hardy, Mwilde, Naddy, OrgasGirl, Pengo, Phys, Pieterkonings, Robertvan1, Sanders muc, Sbalian, Seb, Shepelyansky, Shishir0610, Skatche, Skippydo, Sole Soul, Stevertigo, Template namespace initialisation script, The JPS, Todrobbins, Tracy Hall, Uukgoblin, Voorlandt, 52 anonymous edits Quantum Statistical Mechanics  Source: http://en.wikipedia.org/w/index.php?oldid=525860783  Contributors: B7582, CSTAR, Cuzkatzimhut, Dani setiawan, Dyunvort, JSquish, JamesTeterenko, Jheald, Linas, Mct mht, Mets501, Nsk92, PAR, Passw0rd, Patrick, Phys, Sam Hocevar, Sietse Snel, Zak.estrada, 11 anonymous edits Quantum Field Theory  Source: http://en.wikipedia.org/w/index.php?oldid=523977367  Contributors: 130.64.83.xxx, 171.64.58.xxx, 2001:610:308:693:16E:465E:CAE1:DD39, 9.86, APH, Acalamari, Ahoerstemeier, Aj7s6, Albertod4, Alfred Centauri, Alison, AmarChandra, Amareto2, Amarvc, Ancheta Wis, Andrea Allais, Archelon, Arnero, AstroPig7, AxelBoldt, Bambaiah, Bananan, Banus, Bci2, BenRG, Bevo, Bkalafut, Bktennis2006, Blckavnger, Bob108, Brews ohare, Bt414, CALR, CYD, CambridgeBayWeather, Caracolillo, CarrieVS, Charles Matthews, Chris

198

Article Sources and Contributors the speller, Ciphers, Cogiati, Complexica, ConCompS, ConradPino, Conversion script, Count Iblis, Custos0, Cybercobra, Cyp, D6, DJIndica, Dan Gluck, Dauto, Davidaedwards, DefLog, Dj thegreat, Dmhowarth26, Docboat, Dratman, Drschawrz, Dylan1946, Earin, Egg, El C, EoGuy, Erik J, Etale, Ettrig, Evanescent7, Fortesque666, Fortune432, Four Dog Night, Gandalf61, Garion96, Garuda0001, Gcbirzan, GeorgeLouis, Giftlite, Gilderien, Gjsreejith, Glenn, GoingBatty, Graham87, GrahamHardy, HEL, Hairy Dude, Hanish.polavarapu, Headbomb, Helix84, Henry Delforn (old), HexaChord, Heyheyheyhohoho, Hickorybark, HowardFrampton, Ht686rg90, Hu12, Hyqeom, IRP, IZAK, Ibapahombreperomequedeenmono, Igodard, Igorivanov, Iloveandrea, Iridescent, Itinerant1, J.delanoy, JEH, Jamie Lokier, Jecht (Final Fantasy X), Jeepday, Jim.belk, Jjspinorfield1, Jmnbatista, Joseph Solis in Australia, Juliancolton, KarlHallowell, Karol Langner, Kasimann, Kenneth Dawson, Khazar2, Klilidiplomus, Knowandgive, Kromatol, LLHolm, Lambiam, Leapold, Lerman, Lethe, Likebox, LilHelpa, Lom Konkreta, Looxix, Lotje, MIRROR, Maliz, MarSch, Marie Poise, Markisgreen, Marksr, Martin Kostner, Martin.uecker, Masudr, Materialscientist, Mav, Mbell, Mboverload, Mcld, Melchoir, Mendicus, Mentifisto, Mgiganteus1, Michael C Price, Michael Hardy, Minimac, Moltrix, Momo1381, Moose-32, Mpatel, Msebast, MuDavid, N shaji, N4tur4le, Neparis, Newt Winkler, Nick Number, Nikolas Karalis, Niout, Northgrove, Northryde, Nvj, Odddmonster, Opie, PSimeon, PWilkinson, Palpher, Paolo.dL, Pcarbonn, Pfeiferwalter, Phys, Physicistjedi, Pinkgoanna, Pjacobi, Plasmon1248, Policron, Professor J Lawrence, Pt, Puksik, QFT, R.ductor, R.e.b., RE, RG2, Raphtee, Reaper Eternal, Rebooted, Rjwilmsi, Roadrunner, Robert L, RogueNinja, Rotem Dan, Rudminjd, Rursus, RuudVisser, SCZenz, SebastianHelm, Semperf, Shanes, Sheliak, Shvav, Sietse Snel, SirFozzie, Skou, Spellage, Srleffler, St3vo, Stephan Schneider, Stevertigo, StewartMH, Stupidmoron, Sue Rangell, Tamtamar, Telecomtom, Template namespace initialisation script, Tetracube, The 80s chick, The Anome, The Original Wildbear, The Wild West guy, The ubik, Thecheesykid, Thepalerider2012, Thialfi, Thingg, Threepounds, Tim Starling, TimNelson, Timwi, Tlabshier, Tobias Bergemann, Tom Lougheed, Truthnlove, Tugjob, UkPaolo, Urvabara, Van helsing, Vanderdecken, Varuna, Victor Eremita, Vsmith, W.D., Walterpfeifer, Wavelength, Wik, Witten Is God, Wwheaton, XJamRastafire, Yaush, Ykentluo, Yndurain, Zak.estrada, Zarniwoot, Zueignung, संजीव कुमार, 358 anonymous edits String Theory  Source: http://en.wikipedia.org/w/index.php?oldid=526349924  Contributors: --=The Doctor=--, 100110100, 137.111.13.xxx, 195.5.70.xxx, 1exec1, 2001:db8, 2602:306:CDFE:6520:5DB0:7409:4584:294F, 4lex, 4v4l0n42, 4wajzkd02, 62.104.216.xxx, 69gangsta420, 8digits, 90 Auto, A. di M., A23649, ABigGreenHippo, Aaron35510, Ab1, Academic Challenger, Adamstevenson, Addshore, Aero66, Aero77, Afteread, Ahoerstemeier, Aidan Croft, Aigisthos, Aktsu, Alaithiran, Alansohn, Aldrich2122, Alex Modzz, Alex.muller, AlexGWU, Alexcalamaro, Alexdamaino9, Alexdeburca18, Algebraist, Algri, Alien life form, AlisonW, Alman1234321, AlphaEta, AlphaNumeric, AlysTarr, Amorim Parga, Anaraug, Andrej.westermann, AndrewWTaylor, Andrewlp1991, Andrius.v, Angela, Angela26, AngryBacon, Angus Lepper, Anna Frodesiak, Anonymous Dissident, Anpecota, Anruy, Antandrus, Antimatter15, Anville, Arabani, Arbnos, Archanamiya, ArchonMagnus, Areldyb, ArielGold, Arjun01, Artaxiad, Arthurcprado, Aruben537, Asaa82, Asafavi, Astiburg, AstroGod, AstroHurricane001, Athelwulf, Atomota, Audree, Auspex1729, Avihu, Awolf002, AxelBoldt, Axl, Ayleuss, Az29, Azeraphale, Aziz1005, Aznhero3793, B9 hummingbird hovering, BQND, BVBede, Babawhitemoose, Babemachine, Babemonkey, BahTab, Bandit5005, Baronnet, Bart133, Bassbonerocks, Batong, Bbatsell, Bdiah, Before My Ken, Beland, Belovedfreak, Bemoeial, Ben Tibbetts, Benhenchdickthomas, Benjamin1414141414141414, Benzband, Bethovenn, Bevo, Bewildebeast, Bhny, Bigdaddy4x4, BillC, Bkazaz, Blackhall616, Bobo192, Bobrayner, Bongoo, Bongwarrior, Borgx, Borisblue, Born Gay, BossEditors, Bovineone, Brahmajnani, Branman515, Brechbill123, Brews ohare, Brian0918, Brianlucas, Brina700, Bryan Derksen, Btphelps, Byelf2007, CBM, CHIAGEHYANG, COGDEN, CRGreathouse, Caidh, Cal 1234, Calwiki, Can't sleep, clown will eat me, CanadianLinuxUser, Canthusus, Capricorn24, Capricorn42, Captain-tucker, Capudo, Carandol, CardinalDan, CaseyPenk, CatFiggy, Catgut, Catsloveit07, Causa sui, Caz34, Cdarrai1, Cenarium, CesarB, Ceyockey, Cg-realms, Charibdis, Charles Matthews, Charlesvi, ChazBeckett, Cheesus, Chowbok, Chris 73, Chris Brennan, ChrisGualtieri, Chrispaps2413, Christopher Parham, Christopher Thomas, Chzz, Ckatz, Cleared as filed, Cleeseheb, Closedmouth, CloudNine, Cogiati, Coltonhs, CommanderMoka, Complexica, Computerjoe, Concertmusic, Conorific, Conversion script, Cool3, Couldbenoway66, Courcelles, Cpl Syx, Craig Stuntz, Crazycomputers, Crazynas, Crazytales, Crohnie, Csigabi, Cst17, Cureden, Curps, Cybercobra, Cyferx, Cyrus Andiron, D3gtrd, D6, DAGwyn, DHN, DJBullfish, DKqwerty, DMacks, DV8 2XL, DVD R W, DaFalk, Dan Gluck, Danga1988, Daniel Olsen, Dante Alighieri, Dark dude, Dauto, David136a, David424, Davidmedlar, Daynightrader, Db63376, Dbelange, Dc987, Decoy, DeletionUK, DemonThing, Den fjättrade ankan, Denali134, Devourer09, Dewayne76, Dhatfield, Dirac66, Directorstratton, Discospinster, DoItAgain, Doc Tropics, Don4of4, Dotdotdotdash, DoubleBlue, Dougweller, Dpotter, Dr. Aakash Patel, Dream of Nyx, Dreamafter, Drphilharmonic, Drseudo, Drweetmola, Dsabo74, Duchesserin, Duduong, Duncan McB, Dunkaroo207, Dwielark, Dylan Lake, Dylanlatham, Dysepsion, Earthlyreason, Ed270791, EdJohnston, Editmyhandman, Edward, Eean, Eeekster, Egemont, Egmontaz, Ekwos, El C, ElBenevolente, ElTyrant, Eleassar, Elemented9, Elvey, Emarv, Enauspeaker, Encyclopadia, Enirac Sum, Ensign beedrill, Enviroboy, Epastore, Epbr123, Ergative rlt, Eric Drexler, Erik4, ErikStewart, Erth64net, Escher26, Etov, EvelinaB, Evil saltine, Exert, Exolius, Eyelidlessness, FF2010, Fadereu, Faigl.ladislav, Falcorian, Fanghong, Felix Wan, Fences and windows, Ferkelparade, FireMouseHQ, Fizzycyst, FlieGerFaUstMe262, Flowerhat15, Fortdj33, Fotoni, Frankenpuppy, FredrickS, Fredrik, Fredsie, Froid, Fropuff, Fuhghettaboutit, Furrykef, Fuzzy Bob Saget, GSlicer, Gabbe, Gadfium, GaeusOctavius, Gaius Cornelius, Galwhaa, Gary King, Gautam10, Gebrah, Geekdom04, Gene Nygaard, Gensanders, Geoffrey.landis, GeorgeOrr, Georgepowell2008, Geschichte, Ggb667, Giftlite, Giko, Gil Gamesh, Gil987, Gitman4, GlassCobra, Gmusser, Gogo Dodo, Goh ryangoh, Goldenglove, Goodbye Galaxy, Graham87, Green Lane, Greenrd, GregorB, Gtstricky, Gurch, Guy392, HEL, HFarmer, HJ Mitchell, Hadal, HaeB, Haemo, Hairy Dude, HamatoKameko, Harold f, Harp, Harryboyles, Headbomb, Heidisql, Helicoptor, Hempfel, Hephaestos, Heqs, HerpesVirus, Hilander316, Hillbrand, Hillman, Hindol, Hitchhiker89, Hoaxinator, Housegeek224, HowardFrampton, Hqb, Humanino, Huntscorpio, Huperphuff, Hvitlys, Hydrogen Iodide, I dream of horses, IRP, IVAN3MAN, Ianreisterariola, Icairns, Ihatetoregister, Ikiroid, Illusion96, Ilmari Karonen, Immunize, ImperialismGo, Impunv, Insineratehymn, Introductory adverb clause, InvisibleK, Invisifan, Ionfield, Iridescent, Irish Souffle, IronGargoyle, Isaac, Isocliff, It Is Me Here, Izno, J.A.Ireland, BA (IHPST), J.delanoy, JForget, JGallardo2600, JIP, JJ Harrison, JYolkowski, Jacobolus, Jagbag2, Jake Wartenberg, Jakemarz197, James McNally, JamesBWatson, Jamesofur, Janus Shadowsong, JavierMC, Jay Litman, Jdforrester, Jdlambert, Jeff Song, Jespinos, Jibal, Jim1138, Jimbo Mahoney, Jimp, Jitse Niesen, Jkl, Jmcc150, JoJan, John Vandenberg, JohnBlackburne, Joke137, Jomoal99, JonezyKiDx, Jonnyk aus, Joriki, Joshua Davis, Jrcla2, Juan Marquez, Julesd, Julia W, Jusdafax, Justanother, JustinHagstrom, K, Kaenneth, Kahananite, Kartano, Keegan, Keegscee, Kelvinc, KerathFreeman, Kevinmon, Khaosworks, Khlo, Kim Bruning, Kirbytime, Kjoonlee, Kjramesh, KnightLago, Knome335, Knotwork, KnowledgeOfSelf, KoalamaN2, Kocio, Kongr43gpen, Konstable, Kranix, Kriak, KrishSundaresan, Kronix35, Kurochka, Kusma, Kvdveer, Kwiki, Kylemew, Kzhang1025, Kzzl, L Kensington, Lacrimosus, Laurascudder, Lbr123, LeaveSleaves, LedgendGamer, Leflyman, Lemonflash, Leujohn, Lightdarkness, Likeaboss189, Likebox, Linas, LindsayH, Linnell, LinusE8, Linuxlad, Lionelbrits, Little Mountain 5, Livajo, Looxix, Lostart, Lottamiata, Lozeldafan, Lugia2453, Luke Walkerson, Lulzprotuns, Lumidek, Luna Santin, Lunch, M C Y 1008, M00npirate, M1k3 101, MER-C, MM4EVAH, MONGO, MSTCrow, MacsBug, Madagaskar07, Magister Mathematicae, MagnaMopus, MahRanch, Makeitnasty, Malcolm Farmer, Manheat84, Mani1, Manzeet, MaooaM, MarSch, Marcdean123, Marek69, Martin451, Martti Muukkonen, Mashford, Materialscientist, MathStuf, Mathgenius3141592, Mattfat8, Matthew Sanders, Mattninja, Maurice Carbonaro, Mav, Maxim, Maximilian77, Mbell, Mboverload, Mcarling, Mdl53711, MeekMelange, Mel Etitis, Melchoir, Mentifisto, Mephistophelian, Merenta, Merope, Mgiganteus1, Mhking, Michael Anon, Michael C Price, Michael Hardy, MichaelBillington, MichaelMaggs, Mick le pick, Mike Doughney, Mike18xx, Mikebernstein, Mikeshelton1, Mikez, Milkocookie, Minimac, Mintrick, Mirv, MissMJ, Mkroh, Mmortal03, Moemajdi, MontanNito, Moronoman, Morris729, Moshe Constantine Hassan Al-Silverburg, Moyogo, Mpatel, Mporter, Mrshabam, My76Strat, Myasuda, Myfriendganesha, NPguy, Nakon, Name Omitted, Narayanese, Nasulikid, Naturespace, NawlinWiki, Ned Scott, Neilanderson, Nergaal, Nestamachine, Netdragon, NewEnglandYankee, Newbyguesses, Nickslspride34, Nishch, NithinBekal, No Guru, NoPetrol, Nobelprizewinner, Nomad, Nonsuch, Norm mit, Northfox, Notpayingthepsychiatrist, NuclearFusion, NuclearWarfare, Nufy8, Nunquam Dormio, O, OlEnglish, Olaf Davis, Oli Filth, Olivier, Ollainen, Omegatron, Omgwaffels, Omnipaedista, OnePt618, Onorem, Opelio, Ophion, Orange Suede Sofa, Orenburg1, Ornamentalone, Oswaldo Zapata, Oxymoron83, Paine Ellsworth, Pallab1234, Panser Born, Panu, Paul August, Pavel Vozenilek, Pentasyllabic, Pepsidrinka, Perlman10s, Perspectival, Peteryoung144, Philip Trueman, Phillip.phillipson, Phils, Phoenix2, Phys, PhysPhD, Physicistjedi, Phyte, Pigman, Pimpin101, Pinethicket, Pip2andahalf, Pippo2001, Pjacobi, Pluma, Plumbago, PoisonGM, Poli-Psy, Policron, Polyamorph, Poshzombie, Pra1998, Pred, Pretty Green, ProGeek314, Prodego, Pwqn, Pym98, Quadell, Quajafrie, Quaker phil, Quaqa, Qutt, Qweeveen, Qxz, R Calvete, R.e.b., R3m0t, RA0808, Rabinzkaman, Rachel weld, Ratanmaitra, Rawling, RayAYang, Raymond Hill, Razorflame, Rd232, Reaper Eternal, Reaperfromhell, Reconsider the static, Red Act, Rediahs, Remingtonhill1, Res2216firestar, RexNL, Reyk, Rhetoric Of A Sophist, Rhobite, Rich Farmbrough, Rickstauduhar, Ringbang, Rjwilmsi, Rnricklefs, Robdunst, RobertM525, Robin S, Ronhjones, Rory096, Roy Fultun, RoyBoy, Rpcappello, Rustyjamsen, Rwmnau, RxS, Ryanjunk, Ryanmcdaniel, S3000, SCZenz, SColombo, SDJ, SRoughsedge, SSJ 5, SWAdair, Saalstin, Saeed.Veradi, Salsa Shark, Sardinita, Sarvagnya, Sashhere, Savant13, Sbharris, SchfiftyThree, Schneelocke, SchnitzelMannGreek, ScienceApologist, Sciurinæ, Scoobey, Scott1329m, Scourge of God, Sct72, Scwlong, Seahorseruler, Sean271293, Seaphoto, Seraphim, Several Pending, Sewblon, Sh4wz0r, Shamvil, Shanel, Shanes, Shawn in Montreal, Sheliak, Shidobu, Shimwell, Shirik, Shlomi Hillel, Shreyakstring, Sideswiper, Silence, Silly rabbit, Simon_J_Kissane, Sjakkalle, SkepticVK, Skrewtape, SlashDot, Slightsmile, Smamaret, Smokizzy, Sodium, Solancel, Solipsist, Some jerk on the Internet, Somebody2292, SpacePyjamas, Specs112, SpiderJon, Spiff, Spliffy, Spongbob456789, SpuriousQ, Staffwaterboy, Stephen shenker, Stephenb, Steuard, Steven.w.kowalski, Steven66s, Steveo27five, Stevertigo, Storm.sarup, StormbringerUK, Street Scholar, String4d, StringLove, StringyGuy, Stu181, Suisui, Sunoco, SuperCalzer, Superior IQ Genius, Sverdrup, Sweaty maori sphincter, Swisstingle, Syebo, Synchronism, T-dot, T00g00d96, TERBAFAN, TKD, Taco Manipulator, Tallboyhoops1991, TallulahBelle, Tamaratrouts, Tarquin, Tassedethe, Taw, Tayste, Tcturner2002, TeaDrinker, Teardrop onthefire, Teleprinter Sleuth, Terrifictriffid, Terrisknickers, TestMaster, Tgeairn, Tgeorgescu, That Guy, From That Show!, The Anome, The Thing That Should Not Be, The tooth, TheMonkeyboy524, TheRingess, TheTito, Theresa knott, Theshadow444, Thesis4Eva, Thincat, Thingg, Thor cherubim, Three887, Thubsch, Thumperward, Tide rolls, Timeitsways, Timneu22, Timothy Clemans, TimothyRias, Timwi, Titus III, Tobias Bergemann, Tom Lougheed, Tom harrison, Tommy2010, Tomruen, Tong, Tony Fox, Torrazzo, Tothebarricades.tk, Trent, Trevor MacInnis, Trevorkid45, Truthnlove, Tschach, Turgan, Twri, Tycho, UU, UberScienceNerd, Ucanlookitup, Ugha, Unique and proud of it, Uruk2008, Urvabara, Vald, Van helsing, Vanka5, Vapour, Varrey280303, Vasiliy Faronov, Vastly, Versus22, Verum, Vhann, Vicarious, Visium, Visor, VolatileChemical, Vollrath2323, Vrkaul, WJBscribe, Waffleboy36, Wakabaloola, Waleswatcher, Watsup1313, Wavelength, Wayne Slam, Weaselstomp, Where, Widr, Wiggl3sLimited, Wiki incorp, WikiFew, WikiPuppies, WikiZorro, Wikipelli, Willy Weazley, Winner 42, Witten Is God, Wmahan, Woland37, Wolfmankurd, Wooba doob, Wpegden, XAXISx, Xanchester, Xgamer4, Xhaoz, Xiahou, Y.t., Yandman, Ybungalobill, Yevgeny Kats, You? Me? Us?, Ytomem, Yukiseaside, Zalgo, Zazaban, Zegoma beach, Zelos, Zidonuke, Zifnabxar, Zro, Zundark, Zunz, Zymurgy, と あ る 白 い 猫, 陽, 1906 anonymous edits Quantum Gravity  Source: http://en.wikipedia.org/w/index.php?oldid=525524384  Contributors: 194.117.133.xxx, 200.191.188.xxx, 2over0, 336, Acalamari, Allowgolf, Alvatros, Amareto2, Anarchimede, Ancheta Wis, Andwor9, Apelikedawg, Apratim07, Apyule, Arbnos, Army1987, AstroHurricane001, AstroNomer, Avono, B, Bardsley Rides a Segway, Bdalevin, Ben.c.roberts, BenRG, Bender235, Bevo, Bobby D. Bryant, Brainssturm, Brazmyth, Brews ohare, Caco de vidro, Cal 1234, CalebNoble, CanadianLinuxUser, CardinalDan, CaseyPenk, Casimir9999, Charles Matthews, Chris the speller, Christopher Thomas, Cjthellama, Ck lostsword, Clement Cherlin, Complexica, Conversion script, Coren, Count Iblis, Craig Pemberton, Cthuljew, DAGwyn, DJIndica, Dabbler, Dac04, Dan6hell66, DancingPenguin, Daniel Arteaga, DanielBurnstein, David Schaich, Davidclifford, Delaszk, Dicklyon, Dllahr, DonJStevens, DrArthurRubinPHD, Dude1818, Duduong, Eeekster, Ekwos, El C, Ems57fcva, Endlessnameless, Epbr123, Eric Drexler, ErkDemon, FT2, FayssalF, Finn-Zoltan, FiveColourMap, Follyland, Fotoni, Francophile124, Frazzydee, Fropuff, Frosty726, Fullmetal2887, Fæ, GarbagEcol, Garfield Salazar, Garuda0001, Giftlite, Googledin!, Graham87, Gravitophoton, Grayfell, GregorB, Harold f, Hbackman, Headbomb, Hep thinker, Herbee, Hhhippo, Hillman, Hirvenkürpa, Hmonroe, IRP, Igny, Ilya (usurped), IronGargoyle, Isocliff, Itinerant1, JHMM13, JSquish, Jaganath, Jason Quinn, Jeffq, Jhmmok, JimJast, JocK, Joke137, JonathanD, JorisvS, Joyous!, Jpod2, Jrasowsky, KasugaHuang, Keenan Pepper, Keraunos, Klasovsky, Knowandgive, Koeplinger, Korepin, KrisBogdanov, Kroggz, Kryomaxim, Kurtan, Kusername, L Kensington, Lagelspeil, Lambiam, Lambtron, LilHelpa, Lmatt, LorenzoB, Lumidek, MER-C, Malyctenar, Marcus2, Marek69, Markus Poessel,

199

Article Sources and Contributors Maschen, Masudr, Materialscientist, Matusz, Mcarling, Mhsb, MichaelMaggs, Miguel, Modify, Mortense, Mpatel, Mr.viktor.stepanov, Mstuomel, Musictime4me, N0814444, NewEnglandYankee, Notburnt, NuclearWinner, Nucleophilic, Nunc aut numquam, Onebravemonkey, ParadiZio, Patrick O'Leary, Paulmlieberman, Perlygatekeeper, PhilHibbs, Phys, Piccor, Pie4all88, Pigetrational, Pjacobi, Pleasantville, Pmokeefe, Polyamorph, Preon, QFT, Quantity, RG2, RJFJR, Rabsmith, Raidr, Raul654, Rdekleer, Reddi, ReluctantPhilosopher, Resolute, Rettetast, Rholton, Rjwilmsi, Roadrunner, Roiwallace, Rolfguthmann, Roy Brumback, Rror, Ryanr666, SCZenz, SHCarter, ST47, Saehry, Saibod, Sam Staton, Samdutton, Sanathdevalapurkar, Sanathlab, Scarykitty, Science writer, Sdedeo, Seattle Skier, Seth Ilys, Sheliak, Shushruth, Silly rabbit, SirFozzie, Slightsmile, Smithfarm, Soosed, Spencerfjase, SpikeTorontoRCP, StaticG, StealthCopyEditor, StevenJohnston, Stevertigo, Susurrus, TAz69x, TVC 15, Taketa, Tamaratrouts, Tassedethe, Terra Novus, The Thing That Should Not Be, Theanphibian, Theonlydavewilliams, ThomasWinwood, Throwmeaway, Tide rolls, Tim Shuba, Timwi, Titan1129, Tms9, ToddFincannon, TonyMath, Tpbradbury, Treyp, Trifonov, TrueTeargem, Truthnlove, Ttimespan, Twunchy, Tycho, Udifuchs, UncleBubba, VBGFscJUn3, Vacuunaut, Vald, Valeriy Pischenko, Vampus, Van Speijk, Victor Blacus, Vincenzo.romano, Vitaleyes, Wereon, Wiki Roxor, WikiDan61, Wikiwikimoore, Wireader, Woohookitty, YapaTi, Yevgeny Kats, Yurik, Zunaid, 350 anonymous edits Quantum  Source: http://en.wikipedia.org/w/index.php?oldid=526163092  Contributors: 4C, Academic Challenger, Acalamari, Achowat, AdjustShift, Ahoerstemeier, Ajmah 200, AlexiusHoratius, Alison, Andreas Kaufmann, Andrewpmk, Andypandy.UK, Anthony Appleyard, Ashmoo, Atlant, Bbq332, Bensaccount, Bicala, Bilbo1507, Bjankuloski06en, Bongwarrior, Booknotes, BrightStarSky, BryanD, Bupper, Burakburak, C. Trifle, Calabe1992, Capricorn42, Ceyockey, Charles Matthews, Chymicus, Complexica, Crunchy Numbers, DaGizza, Dan Granahan, Dannyeder, Darsie42, DavidBrahm, Davidhorman, Db099221, Deasmumhain, Deathphoenix, Dennis Estenson II, Djkrajnik, Dr Miles Long, Dragon's Blood, DragonHawk, Drakonicon, Dungodung, El C, Epbr123, EugeneZelenko, Falcon8765, Fdmt, Filelakeshoe, Finn-Zoltan, Fjjf, FlamingSilmaril, Flightx52, Francvs, Freakofnurture, Fresheneesz, Garion96, Geoffrey.landis, Gexmeansgecko, Graeme Bartlett, Gragox, Grapeguy7, Gratedparmesan, Greenbreen, Guruspiritual, Harishng, HiDrNick, Hosterweis, Hujaza, Iisthphir, JHolman, Jaan513, Jag123, Jalexsmith1991, Jeff02, JerrySteal, Joefromrandb, John Vandenberg, Jsjunkie, Julia W, Juventas, K.zaman1710, Kilmer-san, Kinston eagle, KnowledgeBased, Kurt hueston, Laplace's Demon, Laurascudder, Leebo, Lethe, Loggin12354, Lova Falk, LoveEncounterFlow, LtBert44, Marco Polo, Mariodivece, Masudr, Materialscientist, Maurice Carbonaro, Mebden, Melchoir, Mental Blank, Mhocker, Mikeblas, NCDane, Nacefe, NawlinWiki, NerdyNSK, Ngexpert5, Nick Number, Nihiltres, Nsaa, Nutfortuna, Omoo, Ontarioboy, Openforbusiness, Oreo Priest, Orphan Wiki, Orthografer, Oxymoron83, Parkyere, Pdcook, Pengo, Pengyanan, Platypus222, Programming gecko, Pundit, Quantumpundit, Qwertzy2, RG2, Raidon Kane, Raylu, Ricky81682, RodC, RoyBoy, Sadi Carnot, Salvatore Ingala, Sam Korn, Sampi, Scorpionman, Scwlong, Seanruiz, Smack, Smartcowboy, Spangineer, Stevenmitchell, Stevertigo, Stewartrfc, Sverdrup, TastyPoutine, The Anome, Tide rolls, Tiptoety, Too Old, Topbanana, Truthflux, Truthnlove, Tsemii, Vald, Viriditas, Vishakh24, Vogone, Vortico, Vrenator, Wafulz, War sharks, Whodunit, WikiWikiPhil, Wimt, Yougotavirus, Yungjui, Zamb, අඛිල, 242 anonymous edits Quantum state  Source: http://en.wikipedia.org/w/index.php?oldid=525606565  Contributors: ARTE, Agent Foxtrot, Alksentrs, Andres, B. Wolterding, Bambaiah, BasvanPelt, BenRG, Bevo, Bgwhite, Bizzon, Bkalafut, Bob K31416, CSTAR, CapitalR, Chetan666, Chris Howard, Colin Watson, Colincmr, Cortonin, Dan East, DrBugKiller, Dragon's Blood, Dragonfi, Dzordzm, Eagleclaw6, EoGuy, Erwin, F=q(E+v^B), Freakofnurture, Freddy78, Fresheneesz, Ganitvidya, Gaurav biraris, GeorgeLouis, Geschichte, Giftlite, Hans Dunkelberg, Harddk, Headbomb, Hidaspal, Hulten, IRWolfie-, J04n, JTXSeldeen, Julesd, Jutta234, Kbrose, Kinneytj, Larsobrien, Laussy, LokiClock, Lotje, Machine Elf 1735, MathKnight, Maurice Carbonaro, Mct mht, MelbourneStar, Michael Hardy, MichaelHaeckel, Mpatel, Mschlindwein, Nathanielvirgo, Ott, Oxonienses, Papadopc, Patrick0Moran, Phatom87, Physis, Pierluigi.taddei, PoorLeno, Ptpare, Pvkeller, RTC, RealityDysfunction, RobinK, Rockfang, Rorro, Sbyrnes321, SchreiberBike, Sheliak, Slipstream, Stephen Poppitt, Steve Quinn, StewartMH, Tercer, The-tenth-zdog, Thurth, V81, Waxigloo, WaysToEscape, Woohookitty, Xronon, 老 陳, 92 anonymous edits

200

Image Sources, Licenses and Contributors

Image Sources, Licenses and Contributors File:10 Quantum Mechanics Masters.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:10_Quantum_Mechanics_Masters.jpg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: derivative work: Patrick Edwin Moran (talk) Einstein1921_by_F_Schmutzer_2.jpg: Ferdinand Schmutzer (1870-1928) Niels_Bohr_Date_Unverified_LOC.jpg: unknown Broglie_Big.jpg: unknown Max_Born.jpg: unknown Dirac_4.jpg: unknown Werner_Heisenberg_at_1927_Solvay_Conference.JPG: Photograph Institut International de Physique Solvay, Brussels, Belgium Wolfgang_Pauli_young.jpg: unknown Richard_Feynman_ID_badge.png: unknown File:Boltzmanns-molecule.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Boltzmanns-molecule.jpg  License: Public Domain  Contributors: Original uploader was Sadi Carnot at en.wikipedia File:Photoelectric effect.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Photoelectric_effect.svg  License: GNU Free Documentation License  Contributors: Wolfmankurd File:Black body.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Black_body.svg  License: Public Domain  Contributors: Darth Kule File:Bohr model 3.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Bohr_model_3.jpg  License: GNU Free Documentation License  Contributors: User:Sadi Carnot File:Feynmann Diagram Gluon Radiation.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Feynmann_Diagram_Gluon_Radiation.svg  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors: Joel Holdsworth (Joelholdsworth) File:Hot metalwork.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Hot_metalwork.jpg  License: unknown  Contributors: Contributor, Fir0002, Jahobr, Wst, 1 anonymous edits File:RWP-comparison.svg  Source: http://en.wikipedia.org/w/index.php?title=File:RWP-comparison.svg  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors: sfu File:Einstein.Painting.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Einstein.Painting.jpg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: GeorgeLouis File:Emission spectrum-H.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Emission_spectrum-H.svg  License: Creative Commons Zero  Contributors: Merikanto, Adrignola File:Bohr atom model English.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Bohr_atom_model_English.svg  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors: Brighterorange File:Young+Fringes.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Young+Fringes.gif  License: GNU Free Documentation License  Contributors: Young.gif: Mpfiz derivative work: Patrick Edwin Moran (talk) File:Single & double slit experiment.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Single_&_double_slit_experiment.jpg  License: GNU Free Documentation License  Contributors: Patrick Edwin Moran File:Erwin Schroedinger.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Erwin_Schroedinger.jpg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: User:GeorgeLouis File:Heisenberg 10.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Heisenberg_10.jpg  License: Public Domain  Contributors: JdH, Palamède, Quiris File:neon orbitals.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Neon_orbitals.JPG  License: Public Domain  Contributors: Benjah-bmm27, Mortadelo2005, 1 anonymous edits File:Dirac 3.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Dirac_3.jpg  License: anonymous-EU  Contributors: Cambridge University, Cavendish Laboratory File:Superposition.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Superposition.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Patrick Edwin Moran File:Paul.Dirac.monument.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Paul.Dirac.monument.jpg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: GeorgeLouis, Ww2censor File:Max Planck (1858-1947).jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Max_Planck_(1858-1947).jpg  License: anonymous-EU  Contributors: Infrogmation, Lobo, QWerk, TwoWings, 1 anonymous edits File:Solvay conference 1927.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Solvay_conference_1927.jpg  License: Public Domain  Contributors: Benjamin Couprie, Institut International de Physique de Solvay Image:HAtomOrbitals.png  Source: http://en.wikipedia.org/w/index.php?title=File:HAtomOrbitals.png  License: GNU Free Documentation License  Contributors: Admrboltz, Benjah-bmm27, Dbc334, Dbenbenn, Ejdzej, Falcorian, Hongsy, Kborland, MichaelDiederich, Mion, Saperaud, 6 anonymous edits File:Rtd seq v3.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Rtd_seq_v3.gif  License: Creative Commons Attribution 3.0  Contributors: Saumitra R Mehrotra & Gerhard Klimeck File:QuantumDot wf.gif  Source: http://en.wikipedia.org/w/index.php?title=File:QuantumDot_wf.gif  License: Creative Commons Attribution 3.0  Contributors: Saumitra R Mehrotra & Gerhard Klimeck File:Qm step pot temp.png  Source: http://en.wikipedia.org/w/index.php?title=File:Qm_step_pot_temp.png  License: Public Domain  Contributors: User:F=q(E+v^B) File:Infinite potential well.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Infinite_potential_well.svg  License: unknown  Contributors: User:Bdesham File:QuantumHarmonicOscillatorAnimation.gif  Source: http://en.wikipedia.org/w/index.php?title=File:QuantumHarmonicOscillatorAnimation.gif  License: Creative Commons Zero  Contributors: Sbyrnes321 Image:EPR-paradox-illus.png  Source: http://en.wikipedia.org/w/index.php?title=File:EPR-paradox-illus.png  License: GNU Free Documentation License  Contributors: Original uploader was CSTAR at en.wikipedia Image:Bell's theorem.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Bell's_theorem.svg  License: Creative Commons Attribution 3.0  Contributors: Manwesulimo2004 Image:StraightLines.svg  Source: http://en.wikipedia.org/w/index.php?title=File:StraightLines.svg  License: Public Domain  Contributors: Caroline Thompson Image:Bells-thm.png  Source: http://en.wikipedia.org/w/index.php?title=File:Bells-thm.png  License: GNU Free Documentation License  Contributors: Bdesham, Common Good, It Is Me Here, Joshbaumgartner, Karelj, Maksim, Mdd, Pieter Kuiper, Tano4595 Image:Bell-test-photon-analyer.png  Source: http://en.wikipedia.org/w/index.php?title=File:Bell-test-photon-analyer.png  License: GNU Free Documentation License  Contributors: Chetvorno, Glenn, Joshbaumgartner, Karelj, Maksim, Mdd File:Schrodingers cat.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Schrodingers_cat.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Dhatfield File:Schroedingers_cat_film.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Schroedingers_cat_film.svg  License: Creative Commons Zero  Contributors: Christian Schirm File:String theory.svg  Source: http://en.wikipedia.org/w/index.php?title=File:String_theory.svg  License: Creative Commons Attribution 3.0  Contributors: MissMJ Image:World lines and world sheet.svg  Source: http://en.wikipedia.org/w/index.php?title=File:World_lines_and_world_sheet.svg  License: Public Domain  Contributors: Kurochka, svg version by Actam Image:Calabi-Yau.png  Source: http://en.wikipedia.org/w/index.php?title=File:Calabi-Yau.png  License: Creative Commons Attribution-Sharealike 2.5  Contributors: en:User:Lunch File:Quantum gravity.png  Source: http://en.wikipedia.org/w/index.php?title=File:Quantum_gravity.png  License: Public Domain  Contributors: Raidr File:Gravity Probe B.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Gravity_Probe_B.jpg  License: Public Domain  Contributors: Emesee, GDK, Pline File:Point&string.png  Source: http://en.wikipedia.org/w/index.php?title=File:Point&string.png  License: Public domain  Contributors: Actam, Kilom691, Kurochka, 2 anonymous edits File:Calabi-Yau.png  Source: http://en.wikipedia.org/w/index.php?title=File:Calabi-Yau.png  License: Creative Commons Attribution-Sharealike 2.5  Contributors: en:User:Lunch File:Spinnetwork.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Spinnetwork.jpg  License: Creative Commons Attribution-Sharealike 3.0,2.5,2.0,1.0  Contributors: Skoglund S File:Hydrogen Density Plots.png  Source: http://en.wikipedia.org/w/index.php?title=File:Hydrogen_Density_Plots.png  License: Public domain  Contributors: PoorLeno (talk)

201

License

License Creative Commons Attribution-Share Alike 3.0 Unported //creativecommons.org/licenses/by-sa/3.0/

202