155.Aspects of the Quantum Theory

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Number 155

Aspects of Quantum Theory It is often said of quantum physics that if you understand it you are missing the point. That is not quite true, but you do have to put aside much of what you have learned about physics so far and start to see things from a new perspective.

Classical Physics Most of A level physics studies the physics of the world around us. It is about how objects can be expected to behave, following rules like Newton’s Laws of Motion. Everything interacts in a predictable and logical way. With a keen eye for angles, playing pool is simple application of Newton’s laws. Provided that you can control the cue, you can predict exactly what will happen when you strike the cue ball. If you were to play pool with quantum particles however you would find it far less straightforward and predictable.

How is Quantum Physics different? It is not that the laws and rules of ‘classical’ physics do not apply on the small scale, it is more that things are more complicated when you get down to looking at individual particles. There are more rules that have to be obeyed and often particles to not behave in a way which could have been predicted by classical physics. Basically, we do not notice the effects of quantum physics in our everyday lives so it can seem illogical to us. The thing to remember is that it does make sense from the right perspective and more importantly, it is necessary for the universe to work at all! At a very simple level, the differences can be summed up as follows: Measuring Quantities Quantities are things like mass, energy, charge, position -anything you can measure

Classical Physics

Quantum Physics

Quantities can take any value. They are continuous.

Quantities can only take specific values. We say they are 'Quantised'. Quanta literally means discrete bundles or chunks. A single chunk is a Quantum.

E.g. planets can orbit the sun with any value of kinetic energy.

E.g. electrons orbiting in an atom may only have very specific energy values. Predicting Outcomes By 'outcomes' we mean simply 'anything which might happen'. Striking a pool ball is an action. Where the ball goes and what it does on the way is the outcome.

Outcomes are definite.

Outcomes depend on probability.

Events are totally predictable given enough information.

There are a number of possible outcomes from any action and each has an associated probability. The important thing is there is no way to know for certain which outcome will occur.

E.g if you know the speed and angle of a pool ball you can work out its exact trajectory and rebound and the way it will affect other balls on the table.

Wave Particle Duality One of the side effects of quantum physics is something called wave particle duality. This is simply where waves have particle properties and particles have wave properties. These are things that can be proven by experiment:

Example 1 An electron emits a photon as it loses energy. The energy has frequency 6 × 10 14 Hz which is in the optical part of the electromagnetic spectrum. (a) What is the energy carried by the photon? (b) Another electron emits infra red, how does the energy emitted change? [3 marks]

• Photons Although light acts like a wave (e.g. it diffracts), it travels in packets (quanta) of energy called photons. The existence of these has been proven by the photoelectric effect which is an example of when light must be treated like a particle rather than in a wave in order to match the experimental observations.

Answers (a) E = hf = 6.63×10-34 × 6×1014 = 4.0×10-19 J (b) Infra red is lower frequency than visible and therefore the energy will be less.

The energy carried by the photon depends on the frequency: E = hf

E is the energy in Joules h is Planck’s constant, 6.63x10-34 Js f is the frequency in Hertz (Hz).

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Physics Factsheet

155. Aspects of Quantum Theory

De Broglie Wavelengths of Electrons in the Atom

• De Broglie wavelength All particles which are moving have an associated wavelength, which depends upon their momentum, p (momentum = mass × velocity):

Further evidence for the De Broglie wavelength of the electron comes in the form of the spectral lines produced when the electrons jump between energy levels. You should be familiar with emission spectra before continuing.

The De Broglie wavelength of a particle is given by:

Classical physics says that an electron should orbit the nucleus of the atom in much the same way as a planet orbiting the Sun. The range of energies it could have should be continuous. In other words it should be able to have any kinetic energy up to the point where it escapes the atom altogether.

λ = h/p = h / mv λ is the associated de Broglie wavelength h is Planck’s constant, 6.63×10-34 Js p is the momentum ( kgms-1) or mv is mass × velocity

Observations of spectral lines emitted by electrons as they lose energy (emitted at a photon) shows that they only exist in very specific energy levels.

Example 2: (a) Estimate your own de Broglie wavelength when walking. [2 marks] (b) How will the wavelength of an electron change as it is accelerated? [1 mark] (c) A proton and electron are each travelling at speed v. Without calculation explain what you know about the De Broglie wavelengths of each particle. [1 mark] (d) What speed must an electron be travelling at to have the wavelength of 6×10-9 m (me = 9.1 × 10-31 kg)? [2 marks]

This means that only specific energy levels are allowed.

Fig 2a Hydrogen emission spectra 400 A(nm)

500

600

700

Only certain frequencies of light (and therefore energies of photon) are emitted by hydrogen showing that the electrons can only move between certain energy levels.

Answers (a) Pick suitable, easy numbers. v = 1 ms—1, m = 70 kg λ = h/mv = 6.63×10-34/(70×1) = 9.5×10-36 m (Note that this is why on the scale of people and objects you don’t notice the effects of your De Broglie wavelength) (b) As v increases, λ decreases (c) As the mass of the proton is greater it will have a smaller wavelength (d) Rearrange the equation v = h/mλ = 6.63×10-34/(9.1×10-31 × 6×10-9) = 1.2×105 ms-1

To explain this we must think of the electron as a wave. As it is trapped inside the atom it sets up a standing wave (like a guitar string vibrating as it is fixed at both ends). Fig 2b Allowed wavelengths in the atom

nucleus

nucleus n =1

Electron diffraction is evidence for the wave nature of electrons. This is where a beam of electrons create a diffraction pattern when fired through a small hole, similar to that made by a laser when fired through a slit.

nucleus n =2

n =3

In theory there are an infinite number as the wavelengths continue to get smaller.

Fig 1a Diffraction pattern created by electrons

Fig 2b shows that only certain De Broglie wavelengths of the electron can fit inside the atom. As the wavelength depends upon the speed of the atom then this means that only certain speeds and therefore certain kinetic energies are allowed.

Fig 2c. The allowed energy levels correspond to the allowed wavelengths n=∞ n=5 n=4 n=3

Fig 1b Diffraction pattern made by firing a laser through a single slit

n=2

n=1 ground state

This is simply a one dimensional version of the electron diffraction pattern.

Not only does this allow for the discrete energy levels but it also explains the pattern of the energy levels in an atom where the levels get closer and closer together. This corresponds to the differences

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155. Aspects of Quantum Theory Pauli’s Exclusion Principle Each energy level represents a position that the electron may occupy in the atom, however most electrons will remain at the ground state (n=1) unless excited (absorbing energy and moving to a higher state). Even then they will immediately emit the energy as a photon and drop back to the ground state. If the atom has several electrons however, then the exclusion principle comes into effect.

Uncertainty in Position and Speed The first idea is that it is impossible to measure both the speed and position of a particle. This may seem ridiculous, but it is a valid fact. To understand it, you must think about the way that we ‘see’.

There are several states on each energy level. The exact number is Where n is the energy level. given by: No. of states = 2n2

How do you see a ball?

For example on energy level n=1 there are 2×12 = 2 states and on n=2, there are 2×22 = 8 states.

Fig 4. Large scale, classical model of ‘seeing’ a ball

Example 3: (a) What is the maximum number of electrons that can exist on the 5th energy level? [3 marks] (b) What happens to any electrons which are added above this number? [1 mark]

S

Answers (a) Max no. of electrons depends on the energy states, one electron is allowed per state , There are 2n2 = 2×52 = 50 states. So 50 electrons on the 5th energy level. (b) Electrons go onto 6th level until is full then go onto higher levels. Light comes from a source, reflects off the ball and is detected by your eye. As a result of this you can not only see exactly where the ball is, but you can take measurements to determine its exact speed too.

The Pauli exclusion principle states that only one electron can occupy a single state at any one time. Given that an electron will tend to sit at the lowest possible energy level, then an atom with many electrons will fill up the energy levels from the bottom, not moving to the next level until all the states have been filled up.

It makes no real difference whether we use the photon model or the wave model to represent the light, the effect is the same in either case.

Fig.3. The Bohr atom, constructed using the energy levels and placing one electron in each state. This shows possible positions of electrons for the first 4 energy levels.

How do you ‘see’ an electron? Being tiny we could not see them with the naked eye of course, but detecting them uses the same principle as seeing; to shoot something (light or particles) at the object you wish to look at and detect the particles as they bounce off it. It makes sense then that we shine light at the electron.

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nucleus

At this scale we must use the photon model of light. Fig 5. A photon hitting an electron

Heisenberg’s Uncertainty Principle Heisenberg stated that it is impossible to know everything about a situation exactly.

photon E = hf

There are two main ideas here: • Firstly that you cannot know the exact position of a particle and know its velocity. This is explained in more detail below. • The second idea is to do with uncertainty in the energy of particles. All we need to know is that there is an uncertainty in the energy of a photon that is emitted by an atom. As the frequency of the photon is determined by its energy this means that there is an uncertainty in the frequency of the spectral line produced. This explains why spectral lines are slightly thicker than you might expect them to be if only one exact frequency was being emitted. The tiny range of frequencies emitted for each line depends upon the uncertainty.

kinetic energy = hf

Electron

When a photon (carrying energy E = hf) strikes an electron, the energy it carries is absorbed and becomes kinetic energy, causing the electron to change velocity. This means that the act of trying to observe the electron changes its velocity and position.

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155. Aspects of Quantum Theory Schrödinger’s Atom

An alternative is to use a beam of electrons to detect the electron. Of course even this cannot measure both the velocity and position of an electron.

Although we started with an electron looking like a small round ball, we have discovered that it is actually more like a wave at times, especially inside an atom. We have also discovered that it is very hard to pinpoint where an electron is and what it is doing.

Fig 6a observing an electron using electrons

If we consider the De Broglie wavelength that represents an electron and also apply the uncertainty principle, what we get is a probability function in space which represents the probability of finding the electron at a given position.

beam of electrons

Fig 7. The exact position cannot be determined but the probability of finding the electron can be calculated

electron being 'observe'

areas represent probability of finding the electron in that position

Fig. 6b the observed electron is knocked out of position

Schrödinger took this one step further. Instead of imagining the electron as buzzing around somewhere inside the probability function he said that it actually exists at all points inside the wave at the same time.

electron being observed is repelled

Schrödinger said that the electron actually occupies all possible positions and states simultaneously and that only by observing the electron did we force it into one state or another. To help explain he used the famous Schrödinger’s cat analogy.

Fig 8 The cat in the box

beam of electrons is scattered

Fig 6 shows that using an electron microscope to measure position and velocity of an electron is a bit like finding a tennis ball by Note that Schrödinger did not actually conduct this experiment. throwing many tennis balls at it. Neither should you! If you put a cat in a solid box with an open bottle of poison attached to the lid, the cat will be in one of two states. It will either be alive or it will have knocked over the poison and sadly died and there is no way of knowing which without removing the lid and having a look. The idea is that the cat is both alive and dead (occupying all possible states). The only way to be sure is to open the box, but as the poison is attached to the lid, the poison will spill and kill the cat, thereby forcing the cat into one of the two states.

The problem is not that it is a bit complicated to work it out; the point of the uncertainty principle is that the very act of observing the electron changes its velocity and position. Although it would be possible to look at the scattering of the electrons and determine where the observed electron once was, the only thing we can say for certain is that it is no longer there and that we have no idea how fast it is or was going. Example 4: Heisenberg is pulled up by the police. The officer steps out and asks ‘Do you know how fast you were going?’ Heisenberg replies ‘No, but I can tell you where I was.’ You might not find it funny, but what could Heisenberg’s other response have been? [1 mark]

This new idea about electrons leads to a different view of the atom from Bohr’s very clearly defined model.

Fig 9. Schrödinger’s atom

Answer He could have replied ‘Yes, but I’ve no idea where I am.’ (sadly this is even less humorous) In the Schrödinger model of the atom, the electrons do not exist as point particles orbiting the nucleus, but as a ‘cloud’ which represents the electrons occupying their many possible states. When observed, the electrons will be forced to occupy one of the states shown in the Bohr model (fig 3).

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155. Aspects of Quantum Theory Practice Questions 1. (a) The Heisenberg uncertainty principle states that it is possible to know either the exact speed or location of an object, but not both.This is still applicable to large objects, for instance a ball, although it is not as noticeable. Suggest why. [2 marks] (b) Will advancements in technology of microscopes eventually make the uncertainty principle less of a problem? Explain your answer. [2 marks] 2 (a) (b) (c) (d)

Calculate the possible number of energy states that exist in each of the first four energy levels of an atom. [3 marks] What is the maximum number of electrons that can be held in the first 4 levels? [1 mark] Sketch that atom using the Bohr model. [2 marks] Describe one difference between the Bohr and Schrödinger models of the atom. [2 marks]

3 (a) Explain why the Schrödinger’s cat analogy fits in with Heisenberg’s basic idea behind the uncertainty principle. [2 marks] (b) An electron is on the 2nd energy level of an atom. Give one fact about the atom from the point of view of (i) Bohr’s model of the atom (ii) Schrödinger’s model of the atom and (iii) the classical physics view of the atom. [3 marks] 4. State two pieces of evidence that demonstrate that classical physics is incomplete. [2 marks]

Answers 1. (a) The uncertainty in the position of the ball is much smaller than the size of the ball itself and becomes inconsequential. (b) No. No matter how advanced the technology, to observe something you must still use particles to detect it and the ‘observed particle’ will always be affected by the particles fired at it. 2 (a) using 2n2 for each level and add the states in each level together gives: 2×12 + 2×22 + 2×32 + 2×42 = 2 + 8 + 18 + 32 = 60 possible states in the four levels. (b) 60 electrons, one allowed in each state (c) 32

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nucleus

[levels in correct order (n=1 with two states in middle) ] [correct number placed on a separate level.]

(d) In the Bohr model, electrons can occupy any one state on that energy level (one electron per state). electrons occupy all possible states until observed.

In the Schrödinger model

3 (a) The uncertainty principle basically says that you cannot know everything about a situation, in this case the ‘state’ the cat is in Only by observing it, is it forced into a state (being dead) because the poison is knocked over if it wasn’t already. You have no way of knowing what state it was in before you opened the lid. If it was alive then you killed it by opening the lid, there is no way to tell if it was already dead. (b) (i) It is any of 2×22 = 8 possible energy states (or you could have ‘it is about to emit a photon and drop down to the ground state energy level’) (ii) it occupies all 8 possible states (the second statement is also correct for this part too) (iii) a trick question. In classical physics there is no 2nd energy level as the possible energies it can take are continuous not discrete (i.e it can take any value) 4. Anything from: [one mark for correct idea, a second for correct explanation] The observations of the photoelectric effect which show light has particle properties. The diffraction of electrons when fired through a small hole showing they have wave like properties. The fact that emission spectra of atoms show that electrons can only occupy very specific energy states. The existence of states within in an energy level and the fact that only a single electron can occupy that state. Note that there are many more examples including the fact that stars and electron microscopes would not work without the uncertainty and probability laws associated with quantum physics. Acknowledgements: This Physics Factsheet was researched and written by Kieron Nixon The Curriculum Press,Bank House, 105 King Street,Wellington, Shropshire, TF1 1NU Physics Factsheets may be copied free of charge by teaching staff or students, provided that their school is a registered subscriber. No part of these Factsheets may be reproduced, stored in a retrieval system, or transmitted, in any other form or by any other means, without the prior permission of the publisher. ISSN 1351-5136

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