Perturbation theory

Perturbation theory (quantum mechanics) In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an additional “perturbing” Hamiltonian representing a weak disturbance to the system. If the disturbance is not too large, the various physical quantities associated with the perturbed system (e.g. its energy levels and eigenstates) can be expressed as “corrections” to those of the simple system. These corrections, being small compared to the size of the quantities themselves, can be calculated using approximate methods such as asymptotic series. The complicated system can therefore be studied based on knowledge of the simpler one.

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exact, but they can lead to accurate results as long as the expansion parameter, say α, is very small. Typically, the results are expressed in terms of finite power series in α that seem to converge to the exact values when summed to higher order. After a certain order n ~ 1/α however, the results become increasingly worse since the series are usually divergent (being asymptotic series). There exist ways to convert them into convergent series, which can be evaluated for large-expansion parameters, most efficiently by Variational method. In the theory of quantum electrodynamics (QED), in which the electron–photon interaction is treated perturbatively, the calculation of the electron’s magnetic moment has been found to agree with experiment to eleven decimal places.[1] In QED and other quantum field theories, special calculation techniques known as Feynman diagrams are used to systematically sum the power series terms.

Approximate Hamiltonians

Perturbation theory is an important tool for describing real quantum systems, as it turns out to be very difficult to find exact solutions to the Schrödinger equation for Hamiltonians of even moderate complexity. The Hamiltonians to which we know exact solutions, such as the hydrogen atom, the quantum harmonic oscillator and the particle in a box, are too idealized to adequately describe most systems. Using perturbation theory, we can use the known solutions of these simple Hamiltonians to generate solutions for a range of more complicated systems.

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2.1 Limitations 2.1.1 Large perturbations Under some circumstances, perturbation theory is an invalid approach to take. This happens when the system we wish to describe cannot be described by a small perturbation imposed on some simple system. In quantum chromodynamics, for instance, the interaction of quarks with the gluon field cannot be treated perturbatively at low energies because the coupling constant (the expansion parameter) becomes too large.

Applying perturbation theory

Perturbation theory is applicable if the problem at hand 2.1.2 Non-adiabatic states cannot be solved exactly, but can be formulated by adding a “small” term to the mathematical description of the ex- Perturbation theory also fails to describe states that are not generated adiabatically from the “free model”, includactly solvable problem. ing bound states and various collective phenomena such For example, by adding a perturbative electric potential as solitons. Imagine, for example, that we have a systo the quantum mechanical model of the hydrogen atom, tem of free (i.e. non-interacting) particles, to which an tiny shifts in the spectral lines of hydrogen caused by the attractive interaction is introduced. Depending on the presence of an electric field (the Stark effect) can be cal- form of the interaction, this may create an entirely new culated. This is only approximate because the sum of a set of eigenstates corresponding to groups of particles Coulomb potential with a linear potential is unstable (has bound to one another. An example of this phenomenon no true bound states) although the tunneling time (decay may be found in conventional superconductivity, in which rate) is very long. This instability shows up as a broad- the phonon-mediated attraction between conduction elecening of the energy spectrum lines, which perturbation trons leads to the formation of correlated electron pairs theory fails to reproduce entirely. known as Cooper pairs. When faced with such systems, The expressions produced by perturbation theory are not one usually turns to other approximation schemes, such 1

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as the variational method and the WKB approximation. This is because there is no analogue of a bound particle in the unperturbed model and the energy of a soliton typically goes as the inverse of the expansion parameter. However, if we “integrate” over the solitonic phenomena, the nonperturbative corrections in this case will be tiny; of the order of exp(−1/g) or exp(−1/g2 ) in the perturbation parameter g. Perturbation theory can only detect solutions “close” to the unperturbed solution, even if there are other solutions for which the perturbative expansion is not valid.

TIME-INDEPENDENT PERTURBATION THEORY

are associated with the unperturbed system. Note the use of bra–ket notation. We now introduce a perturbation to the Hamiltonian. Let V be a Hamiltonian representing a weak physical disturbance, such as a potential energy produced by an external field. (Thus, V is formally a Hermitian operator.) Let λ be a dimensionless parameter that can take on values ranging continuously from 0 (no perturbation) to 1 (the full perturbation). The perturbed Hamiltonian is

H = H0 + λV 2.1.3

Difficult computations

The problem of non-perturbative systems has been somewhat alleviated by the advent of modern computers. It has become practical to obtain numerical nonperturbative solutions for certain problems, using methods such as density functional theory. These advances have been of particular benefit to the field of quantum chemistry. Computers have also been used to carry out perturbation theory calculations to extraordinarily high levels of precision, which has proven important in particle physics for generating theoretical results that can be compared with experiment.

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Time-independent perturbation theory

Time-independent perturbation theory is one of two categories of perturbation theory, the other being timedependent perturbation (see next section). In timeindependent perturbation theory the perturbation Hamiltonian is static (i.e., possesses no time dependence). Time-independent perturbation theory was presented by Erwin Schrödinger in a 1926 paper,[2] shortly after he produced his theories in wave mechanics. In this paper Schrödinger referred to earlier work of Lord Rayleigh,[3] who investigated harmonic vibrations of a string perturbed by small inhomogeneities. This is why this perturbation theory is often referred to as Rayleigh– Schrödinger perturbation theory.

3.1

The energy levels and eigenstates of the perturbed Hamiltonian are again given by the Schrödinger equation:

(H0 + λV ) |n⟩ = En |n⟩. Our goal is to express E and |n⟩ in terms of the energy levels and eigenstates of the old Hamiltonian. If the perturbation is sufficiently weak, we can write them as power series in λ:

En = En(0) + λEn(1) + λ2 En(2) + · · · ⟩ ⟩ ⟩ |n⟩ = n(0) + λ n(1) + λ2 n(2) + · · · where

1 dk En k! dλk ⟩ 1 dk |n⟩ (k) = n k! dλk En(k) =

When λ = 0, these reduce to the unperturbed values, which are the first term in each series. Since the perturbation is weak, the energy levels and eigenstates should not deviate too much from their unperturbed values, and the terms should rapidly become smaller as we go to higher order. Substituting the power series expansion into the Schrödinger equation, we obtain

First order corrections

⟩ ) ( ( ⟩ (1) (0) + · · · = En(0) + λEn(1) + λ2 En(2) + · · (H + λV ) + λ n n 0 We begin with an unperturbed Hamiltonian H 0 , which is also assumed to have no time dependence. It has known energy levels and eigenstates, arising from the Expanding this equation and comparing coefficients of each power of λ results in an infinite series of time-independent Schrödinger equation: simultaneous equations. The zeroth-order equation is simply the Schrödinger equation for the unperturbed sys ⟩ ⟩ (0) tem. The first-order equation is (0) (0) = En n , n = 1, 2, 3, · · · H0 n [4]

⟩ ⟩ ⟩ ⟩ For simplicity, we have assumed that the energies are dis (1) + V n(0) = En(0) n(1) + En(1) n(0) crete. The (0) superscripts denote that these quantities H0 n

3.2

Second-order and higher corrections

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Operating through by ⟨n(0) | . The first term on the left- where the |k (0) ⟩ are in the orthogonal complement of hand side cancels the first term on the right-hand side. |n(0) ⟩ . The first-order equation may thus be expressed (Recall, the unperturbed Hamiltonian is Hermitian). This as leads to the first-order energy shift:

En(1)

⟩ = n(0) V n(0) ⟨

This is simply the expectation value of the perturbation Hamiltonian while the system is in the unperturbed state. This result can be interpreted in the following way: suppose the perturbation is applied, but we keep the system in the quantum state |n(0) ⟩ , which is a valid quantum state though no longer an energy eigenstate. The perturbation causes the average energy of this state to increase by ⟨n(0) |V |n(0) ⟩ . However, the true energy shift is slightly different, because the perturbed eigenstate is not exactly the same as |n(0) ⟩ . These further shifts are given by the second and higher order corrections to the energy.

(

) ⟩ ∑ ⟩⟨ ⟩ (0) En(0) − H0 n(1) = k (0) V n(0) k k̸=n

For the moment, suppose that the zeroth-order energy level is not degenerate, i.e. there is no eigenstate of H 0 in the orthogonal complement of |n(0) ⟩ with the en(0) ergy En . After renaming the summation dummy index above as k ′ , we can pick any k ̸= n , and multiply through by ⟨k (0) | giving (

(0)

En(0) − Ek

)⟨ ⟩ ⟩ ⟨ k (0) n(1) = k (0) V n(0)

We see that the above ⟨k (0) |n(1) ⟩ also gives us the component of the first-order correction along |k (0) ⟩ .

Before we compute the corrections to the energy eigenstate, we need to address the issue of normalization. We Thus in total we get, may suppose ⟨

⟩ n(0) n(0) = 1,

⟩ ⟩ ∑ ⟨k (0) V n(0) ⟩ (0) (1) = k n (0) (0) k̸=n En − Ek

but perturbation theory assumes we also have ⟨n|n⟩ = 1 . It follows that at first order in λ, we must have

The first-order change in the n-th energy eigenket has a contribution from each of the energy eigenstates k ≠ n. Each term is proportional to the matrix element ) ( (⟨ ⟨ ⟩ ⟩) ⟨k (0) |V |n(0) ⟩ , which is a measure of how much the per n(0) + λ n(1) n(0) + λ n(1) =1 turbation mixes eigenstate n with eigenstate k; it is also ⟨ ⟩ ⟨ ⟩ inversely proportional to the energy difference between n(0) n(1) + n(1) n(0) = 0. eigenstates k and n, which means that the perturbation deSince the overall phase is not determined in quantum forms the eigenstate to a greater extent if there are more mechanics, without loss of generality, we may assume eigenstates at nearby energies. We see also that the expression is singular if any of these states have the same ⟨n(0) |n⟩ is purely real. Therefore, energy as state n, which is why we assumed that there is no degeneracy. ⟨ ⟩ ⟨ ⟩ (0) (1) (0) (1) n n =− n n , and we deduce

3.2 Second-order and higher corrections

We can find the higher-order deviations by a similar procedure, though the calculations become quite tedious with our current formulation. Our normalization prescription To obtain the first-order correction to the energy eigen- gives that state, we insert our expression for the first-order energy correction back into the result shown above of equating ⟩ ⟩ ⟨ ⟨ the first-order coefficients of λ. We then make use of the 2 n(0) n(2) + n(1) n(1) = 0. resolution of the identity, ⟨

⟩ n(0) n(1) = 0.

Up to second order, the expressions for the energies and  eigenstates are: ) ⟩ ⟩ ( ⟩ ⟨ (normalized) ⟩⟨ ⟩ ∑ (0) (0) (0) (0) (0) (0)  (0)  + n n V n k V n = V n k k̸=n ⟨ ⟩ ⟩ ⟩ ⟩⟨ ⟩ ⟨ ∑ ∑ k (0) V n(0) 2 (0) (0) (1) (0) (0) (0) 2 (0) (0) + En n , En (λ) = En +λ n V n = k V n +λ +O(λ3 ) k (0) (0) E − E n k̸=n k̸=n k 

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TIME-INDEPENDENT PERTURBATION THEORY

⟩ ⟩ ⟨k (0) V n(0) ⟩ ⟩ ⟨ (0) V ℓ(0) ⟩ ⟨ℓ(0) V n(0) ⟩ ∑ ∑ ∑ (0) (0) 2 (0) ( k )( ) |n(λ)⟩ = n +λ +λ k k (0) (0) (0) (0) (0) (0) E − E E − E E − E n V n n k̸=n k̸=n ℓ̸= n k k n (0) ℓ |n(1) ⟩ = 1 |k1 ⟩ k ⟨ ⟩ ⟨n(0) V n(0) ⟩ ⟨k (0) V n(0) ⟩ E1nk1 ⟩ ∑ n(0) V k (0) ⟩ ⟨k (0) V n(0) ⟩ ∑ ( ) − λ2 − λ2 n(0) + O(λ3 ). k (0) ( )2 V Vnn Vk1 n( (0)(0) (0)1)V2nk1 Vk1 n (0) k k 2 1 2 Vk2 n (0) (0) (2) |n ⟩ = |n ⟩ k̸=n k̸− =n En − Ek En|k1 −⟩E−k 2 Enk1 Enk2 Enk 2 Ek21 n 1 [ ( ) Extending the process further, the third-order energy corVk1 k2 Vk2 k3 Vk3 n Vnn Vk1 k2 Vk2 n 1 1 |V (3) − + − + |n ⟩ = [5] rection can be shown to be Ek1 n Enk2 Enk3 Ek1 n Enk2 Enk1 Enk2 [ Vk1 k2 Vk2 k3 Vk3 k4 Vk4 k2 + Vk3 k2 Vk1 k2 Vk4 k3 Vk2 k4 Vk k Vk k |n(4) ⟩ = − 2 3 3 2 2E E E Ek1 n Ek2 n ∑ ∑ ⟨n(0) |V |m(0) ⟩⟨m(0) |V |k (0) ⟩⟨k (0) |V |n(0) ⟩ ∑ |⟨n(0)k|V1 n|mk(0) ⟩|2 k2 k4 2 k3 ( )( ) En(3) = . −⟨n(0) |V |n(0) ⟩ ( ) ( (0) 2 ) (0) (0) (0) (0) (0) Vnn Vkm̸ Vk3 E 1 1 |Vk2 n |2 Vk Em − En Ek − En nV k̸=n m̸=n 1 k= 2n 3 En 1 mk2 k− + + + + Ek1 n Enk3 Ek2 n Enk3 Ek2 n Ek1 n Enk2 Ek1 ( ) ] Corrections to fifth order (energies) and fourth order Vk1 k2 Vk2 n |Vk1 k2 |2 Vnn Vk1 n |Vnn |2 Vk1 n Vnn (0) + − |k1 + (states) in compact notation Ek1 n Ek31 n Ek32 n Ek1 n Ek31 k2 ( ( ) Vk1 n Vk2 k1 Vk3 k2 Vnk3 Vnn Vnk2 |Vnk1 |2 3|Vnk2 |2 If we introduce the notation, − 2 + + Ek1 n Enk2 Enk3 Enk2 Ek21 n 4Ek22 n Vnm ≡ ⟨n(0) |V |m(0) ⟩ (0) Enm ≡ En(0) − Em

then the energy corrections to fifth order can be written

En(1) = Vnn |Vnk2 |2 Enk2 Vnk3 Vk3 k2 Vk2 n |Vnk |2 = − Vnn 2 3 Enk2 Enk3 Enk3

En(2) = En(3)

En(4) =

All terms involved k should be summed over k such that the denominator does not vanish.

3.3 Effects of degeneracy Suppose that two or more energy eigenstates are degenerate. The first-order energy shift is not well defined, since there is no unique way to choose a basis of eigenstates for the unperturbed system. The various eigenstates for a given energy will perturb with different energies, or may well possess no continuous family of perturbations at all. This is manifested in the calculation of the perturbed eigenstate via the fact that the operator

2 Vnk4 Vk4 k3 Vk3 k2 Vk2 n |Vnk4 |2 |Vnk2 |2 Vnk4 Vk4 k3 Vk3 n Vnk4 Vk4 k2 Vk2 n 2 |Vnk4 | − − V − V + V nn nn nn 2 2 EH 2 3 EEn(0) Enk2 Enk3 Enk4 Enk Enk2 Enk2 Enk Enk nk04 nk3− 4 4 4

2 Vnk4 Vk4 k3 Vk3 k2 Vk2 n |Vnk |2 Vnk4 Vk4 k3 Vk3 n 2 |Vnk4 | inverse. does not have+a Vwell-defined − En(2) 2 4 − 2Vnn nn 2 3 Enk2 Enk3 Enk4 Enk4 Enk3 Enk4 Enk4 Let 2D denote the subspace spanned by these degener2 2 Vnk5 Vk5 k4 Vk4 k3 Vk3 k2 Vk2 n Vnk5 Vk5 k4 Vk4 n |Vnk Vnk5 Vk5 kNo |Vnkthe nk3 Vk3 k2 Vk2 n 2 | eigenstates. 2 Vkmatter 2 n |Vnkhow 2| 5| V (5) ate small perturbation En = − − − 2 E 2 2 Enk2 Enk3 Enk4 Enk5 Enk Eis, Enk2 Enk Enk Enk Enk2 Enk3 nk2in the degenerate nk5 differences 4 5 subspace2 D the energy 5 the eigenstates Vnk5 Vk5 k4 Vk4 k3 Vk3 n Vnk5 Vk5 k4 Vk4 kbetween Vnk5 Vk5 k3HV0k3are |Vnk5 |2 mix|Vnk3 |2 |Vnk | k2 Vzero, k2 n so complete 2 Vk2 n − Vnn − Vnn − least Vnn some +is Vassured. + 2Vnn 3 5 nn 2 2 2 2 2 ing of at of these states TypiEnk3 Enk4 Enk5 Enk2 Enk4 Enk5 Enk2 Enk3 Enk5 Enk5 Enk3 Enk5 cally the eigenvalues will split and 2 the eigenspaces will k5 k2 Vk2 n 2 Vnk5 Vk5 k4 Vk4 n 2 Vnk5 Vk5 k3 Vk3 n 3 |Vnk5 | or at least get smaller 2 Vnk5 V simple (one-dimensional), + Vnn + Vnn − Vnn + Vbecome nn 3 E 2 E2 3 4 Enk E E E Enk nk2 than nk5 nk3 nk5 nk5D. The successful dimension 4 5 perturbations will not 2 be “small” relative to a poorly chosen basis of D. Instead, V V V |V | V V V Vnk5 Vk5 k4 Vk4 k3 Vk3 k2 Vk2 n nk5 k5 k4 k4 n nk5 nk3 k3 k2 k2 n − 2En(2) − = 2 2 we consider the perturbation “small” if the new eigenstate Enk2 Enk3 Enk4 Enk5 Enk4 Enk5 Enk5 Enk2 Enk3 ( ) must be is close to the subspace D. The new Hamiltonian 2 Vnk5 Vk5 k4 Vk4 k3 Vk3 n Vnk5 Vk5 k4 Vk4 kdiagonalized |Vnk |2 |V |2 variation nk 2 Vk2 n 5D, 3slight 5 |D, so to speak. (2) |Vnk in or a of − 2Vnn − + + 2En 2 E 2 E 2 2 3 Enk Enk2 Enk Enk Enk Enk nk4 Enk5 nk5 correct perturbed are 3 4 These 5 3eigenstates in D 5 now the basis ( ) for the perturbation expansion: 2 Vnk5 Vk5 k4 Vk4 n Vnk5 Vk5 k3 Vk3 n 2 3 |Vnk5 | + Vnn 2 + − Vnn 3 2 4 2 Enk4 Enk5 Enk3 Enk5 Enk5 ∑ |n⟩ = αnk |k (0) ⟩ + λ|n(1) ⟩. and the states to fourth order can be written k∈D

=

3.4

Generalization to multi-parameter case

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For the first-order perturbation we need to solve the per- 3.4.1 Hamiltonian and force operator turbed Hamiltonian restricted to the degenerate subspace From the differential geometric point of view, a parameD terized Hamiltonian is considered as a function defined on the parameter manifold that maps each particular set of (0) (0) (0) parameters (x1 , x2 , · · · ) to an Hermitian operator H(x μ ) V |k ⟩ = ϵk |k ⟩ + small ∀|k ⟩ ∈ D. that acts on the Hilbert space. The parameters here can be simultaneously for all the degenerate eigenstates, where external field, interaction strength, or driving parameters μ µ ϵk are first-order corrections to the degenerate energy lev- in the quantum phase transition. Let En(x ) and |n(x )⟩ μ els, and “small” is a small vector orthogonal to D. This is be the n-th eigenenergy and eigenstate of H(x ) respectively. In the language of differential geometry, the states equivalent to diagonalizing the matrix |n(xµ )⟩ form a vector bundle over the parameter manifold, on which derivatives of these states can be defined. The perturbation theory is to answer the following ques(0) (0) (0) (0) ⟨k |V |l ⟩ = Vkl ∀ |k ⟩, |l ⟩ ∈ D. tion: given En (xµ0 ) and |n(xµ0 )⟩ at an unperturbed referµ μ µ This procedure is approximate, since we neglected states enceμ point x0 , how to estimate the En(x ) and |n(x )⟩ outside the D subspace. The splitting of degenerate ener- at x close to that reference point. gies ϵk is generally observed. Although the splitting may be small compared to the range of energies found in the system, it is crucial in understanding certain details, such as spectral lines in Electron Spin Resonance experiments.

Without loss of generality, the coordinate system can be shifted, such that the reference point xµ0 = 0 is set to be the origin. The following linearly parameterized Hamiltonian is frequently used

Higher-order corrections due to other eigenstates can be found in the same way as for the non-degenerate case H(xµ ) = H(0) + xµ Fµ . (

) ) ∑( En(0) − H0 |n(1) ⟩ = ⟨k (0) |V |n(0) ⟩ |k (0) ⟩.

If the parameters x μ are considered as generalized coordinates, then Fμ should be identified as the generalized k̸∈D force operators related to those coordinates. Different The operator on the left hand side is not singular when indices μ label the different forces along different directions in the parameter manifold. For example, if x μ deapplied to eigenstates outside D, so we can write notes the external magnetic field in the μ-direction, then Fμ should be the magnetization in the same direction. ∑ ⟨k (0) |V |n(0) ⟩ |k (0) ⟩, |n(1) ⟩ = (0) (0) 3.4.2 Perturbation theory as power series expansion − E E n k̸∈D k but the effect on the degenerate states is minuscule, pro- The validity of the perturbation theory lies on the adiabatic assumption, which assumes the eigenenergies and portional to the square of the first-order correction ϵk . eigenstates of the Hamiltonian are smooth functions of Near-degenerate states should also be treated in the above parameters such that their values in the vicinity region manner, since the original Hamiltonian won't be larger can be calculated in power series (like Taylor expansion) than the perturbation in the near-degenerate subspace. of the parameters: An application is found in the nearly free electron model, where near-degeneracy treated properly gives rise to an energy gap even for small perturbations. Other eigen1 states will only shift the absolute energy of all near- En (xµ ) = En + xµ ∂µ En + xµ xν ∂µ ∂ν En + · · · 2! degenerate states simultaneously. 1 µ µ |n(x )⟩ = |n⟩ + x |∂µ n⟩ + xµ xν |∂µ ∂ν n⟩ + · · · 2!

3.4

Generalization case

to

multi-parameter Here ∂μ denotes the derivative with respect to x μ . When

The generalization of the time-independent perturbation theory to the case where there are multiple small parameters xµ = (x1 , x2 , · · · ) in place of λ can be formulated more systematically using the language of differential geometry, which basically defines the derivatives of the quantum states and calculates the perturbative corrections by taking derivatives iteratively at the unperturbed point.

applying to the state |∂µ n⟩ , it should be understood as the covariant derivative if the vector bundle is equipped with non-vanishing connection. All the terms on the righthand-side of the series are evaluated at x μ = 0, e.g. En ≡ En(0) and |n⟩ ≡ |n(0)⟩ . This convention will be adopted throughout this subsection, that all functions without the parameter dependence explicitly stated are assumed to be evaluated at the origin. The power series may converge slowly or even not converge when the energy levels

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3

TIME-INDEPENDENT PERTURBATION THEORY

are close to each other. The adiabatic assumption breaks where ℜ denotes the real part function. The first order down when there is energy level degeneracy, and hence derivative ∂μEn is given by the first Hellmann–Feynman the perturbation theory is not applicable in that case. theorem directly. To obtain the second order derivative ∂μ∂νEn, simply applying the differential operator ∂μ to the result of the first order derivative ⟨n|∂ν H|n⟩ , which 3.4.3 Hellman–Feynman theorems reads The above power series expansion can be readily evaluated if there is a systematic approach to calculate the derivates to any order. Using the chain rule, the derivatives can be broken down to the single derivative on either the energy or the state. The Hellmann–Feynman theorems are used to calculate these single derivatives. The first Hellmann–Feynman theorem gives the derivative of the energy,

∂µ En = ⟨n|∂µ H|n⟩

∂µ ∂ν En = ⟨∂µ n|∂ν H|n⟩+⟨n|∂µ ∂ν H|n⟩+⟨n|∂ν H|∂µ n⟩. Note that for linearly parameterized Hamiltonian, there is no second derivative ∂μ∂νH = 0 on the operator level. Resolve the derivative of state by inserting the complete set of basis,

∂µ ∂ν En =

∑

(⟨∂µ n|m⟩⟨m|∂ν H|n⟩ + ⟨n|∂ν H|m⟩⟨m|∂µ n⟩) ,

m

The second Hellmann–Feynman theorem gives the then all parts can be calculated using the Hellmann– In terms of Lie derivatives, derivative of the state (resolved by the complete basis Feynman theorems. ⟨∂µ n|n⟩ = ⟨n|∂µ n⟩ = 0 according to the definition with m ≠ n), of the connection for the vector bundle. Therefore, the case m = n can be excluded from the summation, which ⟨m|∂µ H|n⟩ ⟨m|∂µ H|n⟩ avoids the singularity of the energy denominator. The ⟨m|∂µ n⟩ = , ⟨∂µ m|n⟩ = . En − Em Em − En same procedure can be carried on for higher order derivatives, from which higher order corrections are obtained. For the linearly parameterized Hamiltonian, ∂μH simply The same computational scheme is applicable for the corstands for the generalized force operator Fμ. rection of states. The result to the second order is as folThe theorems can be simply derived by applying the diflows ferential operator ∂μ to both sides of the Schrödinger equation H|n⟩ = En |n⟩, which reads ∑ ⟨m|∂µ H|n⟩ |n (xµ )⟩ = |n⟩ + |m⟩xµ En − Em m̸=n ∂µ H|n⟩ + H|∂µ n⟩ = ∂µ En |n⟩ + En |∂µ n⟩.  ∑ ∑ ⟨m|∂µ H|l⟩⟨l|∂ν H|n⟩ ∑ ⟨m|∂µ H Then overlap with the state ⟨m| from left and make use + |m⟩ − (En − Em )(En − El ) (En of the Schrödinger equation ⟨m|H = ⟨m|Em again, m̸=n l̸=n m̸=n ⟨m|∂µ H|n⟩+Em ⟨m|∂µ n⟩ = ∂µ En ⟨m|n⟩+En ⟨m|∂µ n⟩.

Both energy derivatives and state derivatives will be involved in deduction. Whenever a state derivative is encountered, resolve it by inserting the complete set of basis, then the Hellmann-Feynman theorem is applicable. Because differentiation can be calculated systematically, the series expansion approach to the perturbative corrections can be coded on computers with symbolic processing software like Mathematica.

Given that the eigenstates of the Hamiltonian always form an orthonormal basis ⟨m|n⟩ = δmn , the cases of m = n and m ≠ n can be discussed separately. The first case will lead to the first theorem and the second case to the second theorem, which can be shown immediately by rearranging the terms. With the differential rules given by the Hellmann–Feynman theorems, the perturbative correction to the energies and states can be calculated sys- 3.4.5 Effective Hamiltonian tematically. Let H(0) be the Hamiltonian completely restricted either in the low-energy subspace HL or in the high-energy sub3.4.4 Correction of energy and state space HH , such that there is no matrix element in H(0) connecting the low- and the high-energy subspaces, i.e. To the second order, the energy correction reads ⟨m|H(0)|l⟩ = 0 if m ∈ HL , l ∈ HH . Let Fμ = ∂μH be the coupling terms connecting the subspaces. Then when the high energy degrees of freedoms are integrated ∑ ⟨n|∂ν H|m⟩⟨m|∂µ H|n⟩ µ µ out, the effective in the low energy subspace En (x ) = ⟨n|H|n⟩+⟨n|∂µ H|n⟩x +ℜ xµ xνHamiltonian +· · · , En −reads Em [6] m̸=n

4.1

Method of variation of constants

7

an energy basis |n⟩ for the unperturbed system. (We drop ( the (0) superscripts for the eigenstates, because)it is not 1 ∑ ⟨m|∂µ H|l⟩⟨l|∂ν H|n⟩ ⟨m|∂ν H|l⟩⟨l|∂µ H|n⟩ eff µ the useful to speak of energy for Hmn (xµ ) = ⟨m|H|n⟩+δnm ⟨m|∂µ H|n⟩xµ + + levels and eigenstates x xν +· · · . 2! E − E E − E m l n l perturbed system.) l∈HH

Here m, n ∈ HL are restricted in the low energy sub- If the unperturbed system is in eigenstate |j⟩ at time t = 0, space. The above result can be derived by power series its state at subsequent times varies only by a phase (in the Schrödinger picture, where state vectors evolve in time expansion of ⟨m|H(xµ )|n⟩ . and operators are constant), In a formal way it is possible to define an effective Hamiltonian that gives exactly the low-lying energy states and wavefunctions.[7] In practice, some kind of approxima|j(t)⟩ = e−iEj t/ℏ |j⟩ . tion (perturbation theory) is generally required. Now, introduce a time-dependent perturbing Hamiltonian V(t). The Hamiltonian of the perturbed system is

4

Time-dependent theory

perturbation

H = H0 + V (t) .

Let |ψ(t)⟩ denote the quantum state of the perturbed system at time t. It obeys the time-dependent Schrödinger Time-dependent perturbation theory, developed by Paul equation, Dirac, studies the effect of a time-dependent perturbation V(t) applied to a time-independent Hamiltonian H0 . ∂ Since the perturbed Hamiltonian is time-dependent, so H|ψ(t)⟩ = iℏ ∂t |ψ(t)⟩ . are its energy levels and eigenstates. Thus, the goals of time-dependent perturbation theory are slightly different The quantum state at each instant can be expressed as a from time-independent perturbation theory. One is inter- linear combination of the complete eigenbasis of |n⟩ : ested in the following quantities: ∑ cn (t)e−iEn t/ℏ |n⟩ , • The time-dependent expectation value of some ob- |ψ(t)⟩ = n servable A, for a given initial state. where the cn(t)s are to be determined complex functions • The time-dependent amplitudes of those quantum of t which we will refer to as amplitudes (strictly speakstates that are energy eigenkets (eigenvectors) in the ing, they are the amplitudes in the Dirac picture). unperturbed system. We have explicitly extracted the exponential phase facThe first quantity is important because it gives rise to tors exp(−iEn t/ℏ) on the right hand side. This is only the classical result of an A measurement performed on a matter of convention, and may be done without loss of a macroscopic number of copies of the perturbed sys- generality. The reason we go to this trouble is that when tem. For example, we could take A to be the displace- the system starts in the state |j⟩ and no perturbation is ment in the x-direction of the electron in a hydrogen present, the amplitudes have the convenient property that, atom, in which case the expected value, when multiplied for all t, cj(t) = 1 and cn(t) = 0 if n ≠ j.

4.1

Method of variation of constants

by an appropriate coefficient, gives the time-dependent The square of the absolute amplitude cn(t) is the probadielectric polarization of a hydrogen gas. With an appro- bility that the system is in state n at time t, since priate choice of perturbation (i.e. an oscillating electric potential), this allows one to calculate the AC permittivity 2 2 of the gas. |cn (t)| = |⟨n|ψ(t)⟩| . The second quantity looks at the time-dependent probaPlugging into the Schrödinger equation and using the fact bility of occupation for each eigenstate. This is particthat ∂/∂t acts by a chain rule, one obtains ularly useful in laser physics, where one is interested in the populations of different atomic states in a gas when a ) time-dependent electric field is applied. These probabil- ∑ ( ∂cn ities are also useful for calculating the “quantum broadiℏ − cn (t)V (t) e−iEn t/ℏ |n⟩ = 0 . ∂t ening” of spectral lines (see line broadening) and particle n decay in particle physics and nuclear physics. By resolving the identity in front of V, this can be reWe will briefly examine the method behind Dirac’s for- duced to a set of coupled differential equations for the mulation of time-dependent perturbation theory. Choose amplitudes,

8

4 TIME-DEPENDENT PERTURBATION THEORY

∂cn −i ∑ = ⟨n|V (t)|k⟩ ck (t) e−i(Ek −En )t/ℏ . ∂t ℏ

H(t)|ψ(t)⟩ = iℏ

∂|ψ(t)⟩ ∂t

k

has the formal solution The matrix elements of V play a similar role as in timeindependent perturbation theory, being proportional to ] [ ∫ i t ′ the rate at which amplitudes are shifted between states. ′ |ψ(t)⟩ = T exp − dt H(t ) |ψ(t0 )⟩ , Note, however, that the direction of the shift is modiℏ t0 fied by the exponential phase factor. Over times much longer than the energy difference Ek − En, the phase where T is the time ordering operator, winds around 0 several times. If the time-dependence { of V is sufficiently slow, this may cause the state amplitudes to oscillate. ( E.g., such oscillations are useful for T A(t )A(t ) = A(t1 )A(t2 ) t1 > t2 . 1 2 A(t2 )A(t1 ) t2 > t1 managing radiative transitions in a laser.)

Up to this point, we have made no approximations, so Thus, the exponential represents the following Dyson sethis set of differential equations is exact. By supplying ries, appropriate initial values cn(t), we could in principle find an exact (i.e., non-perturbative) solution. This is easily [ ] ∫ ∫ t ∫ t1 done when there are only two energy levels (n = 1, 2), i t 1 dt1 H(t1 ) − 2 dt1 dt2 H(t1 )H(t2 ) + . . . | and this solution is useful for modelling systems like the |ψ(t)⟩ = 1 − ℏ ℏ t0 t0 t0 ammonia molecule. Note that in the second term, the 1/2! factor exactly canHowever, exact solutions are difficult to find when there cels the double contribution due to the time-ordering opare many energy levels, and one instead looks for perturerator, etc. bative solutions. These may be obtained by expressing Consider the following perturbation problem the equations in an integral form, −i ∑ cn (t) = cn (0)+ ℏ k

∫

t

′ λV (t)]|ψ(t)⟩ = iℏ [H + dt′ ⟨n|V (t′ )|k⟩ ck (t′ ) e−i(Ek −En0)t /ℏ .

∂|ψ(t)⟩ , ∂t

0

assuming that the parameter λ is small and that the probRepeatedly substituting this expression for c back into lem H0 |n⟩ = En |n⟩ has been solved. right hand side, yields an iterative solution, Perform the following unitary transformation to the interaction picture (or Dirac picture), (1) (2) cn (t) = c(0) n + cn + cn + · · ·

|ψ(t)⟩ = e− ℏ H0 (t−t0 ) |ψI (t)⟩ . i

where, for example, the first-order term is

c(1) n (t) =

−i ∑ ℏ k

∫

t 0

Consequently, the Schrödinger equation simplifies to ′

i ∂|ψI (t)⟩ dt′ ⟨n|V (t′ )|k⟩ ck (0) e−i(Ek −En )t /ℏ . ℏi H0 (t−t0 ) , λe V (t)e− ℏ H0 (t−t0 ) |ψI (t)⟩ = iℏ ∂t

Several further results follow from this, such as Fermi’s so it is solved through the above Dyson series, golden rule, which relates the rate of transitions between [ quantum states to the density of states at particular en∫ ∫ ∫ i iλ t λ2 t H0 (t1 −t0 ) − ℏi H0 (t1 −t0 ) ergies; or the Dyson series, obtained by applying the it- |ψI (t)⟩ = 1 − ℏ dt1 e dt1 V (t1 )e − 2 ℏ t0 ℏ t0 erative method to the time evolution operator, which is one of the starting points for the method of Feynman di- as a perturbation series with small λ. agrams. Using the solution of the unperturbed problem H0 |n⟩ = ∑ En |n⟩ and n |n⟩⟨n| = 1 (for the sake of simplicity assume a pure discrete spectrum), yields, to first order, 4.2 Method of Dyson series

[ Time-dependent perturbations can be reorganized ∫ t i The |ψ (t)⟩ = 1 − iλ ∑ ∑ through the technique of the Dyson series. dt1 ⟨m|V (t1 )|n⟩e− ℏ (En −Em )(t1 −t0 ) |m⟩⟨n I ℏ m n t0 Schrödinger equation

9 Thus, the system, initially in the unperturbed state |α⟩ = |ψ(t0 )⟩ , by dint of the perturbation can go into the state |β⟩ . The corresponding transition probability amplitude to first order is

Aαβ

iλ =− ℏ

∫

t

dt1 ⟨β|V (t1 )|α⟩e− ℏ (Eα −Eβ )(t1 −t0 ) ,

the stationary eigenfunctions also given above ( |n(λ)⟩ = U (0; λ)|n⟩) .) The unitary evolution operator is applicable to arbitrary eigenstates of the unperturbed problem and, in this case, yields a secular series that holds at small times.

i

t0

5 Strong perturbation theory

as detailed in the previous section——while the corresponding transition probability to a continuum is fur- In a similar way as for small perturbations, it is possible nished by Fermi’s golden rule. to develop a strong perturbation theory. Let us consider As an aside, note that time-independent perturbation the- as usual the Schrödinger equation ory is also organized inside this time-dependent perturbation theory Dyson series. To see this, write the unitary ∂|ψ(t)⟩ evolution operator, obtained from the above Dyson series, H(t)|ψ(t)⟩ = iℏ ∂t as and we consider the question if a dual Dyson series exists that applies ∫ ∫ t1 in the limit of a perturbation increasingly 2 ∫ t i i i i i iλ t λ large. This question can be answeredi in an affirmative H (t −t ) − H (t −t ) (t2 −t0 ) U (t) = 1− dt1 e ℏ 0 1 0 V (t1 )e ℏ 0 1 0 − 2 dt1 dt2 e ℏ H0 (t1 −t0 ) V (t1 )e− ℏ H0 (t1 −t0 ) e ℏ H0[9] V (t2 )e− ℏ H0 [8] and the series is the well-known adiabatic series. ℏ t0 ℏ way t0 t0 This approach is quite general and can be shown in the and take the perturbation V to be time-independent. following way. Let us consider the perturbation problem Using the identity resolution ∑

[H0 + λV (t)]|ψ(t)⟩ = iℏ

|n⟩⟨n| = 1

∂|ψ(t)⟩ ∂t

being λ→ ∞. Our aim is to find a solution in the form

n

with H0 |n⟩ = En |n⟩ for a pure discrete spectrum, write {

∫

|ψ⟩ = |ψ0 ⟩ } +

1 1 |ψ1 ⟩ + 2 |ψ2 ⟩ + . . . λ λ

∑∑ i −t0 ) ⟨m|V |n⟩e− ℏ (En −Em )(t1but a|m⟩⟨n| direct substitution into the above equation fails to t0 m n produce useful results. This situation can be adjusted { ∫ } ∫ t1 ∑∑∑ i i variable as τ = λt proλ2 t making a rescaling of the time − dt1 dt2 e− ℏ (En −Em )(t1 −t0 ) ⟨m|V |n⟩⟨n|V |q⟩e− ℏ (Eq −En )(t2 −t0 ) |m⟩⟨q| + · · · ducing the following meaningful equations ℏ2 t0 t0 m n q

U (t) = 1 −

iλ ℏ

t

dt1

It is evident that, at second order, one must sum on all the intermediate states. Assume t0 = 0 and the asymptotic limit of larger times. This means that, at each contribution of the perturbation series, one has to add a multiplicative factor e−ϵt in the integrands for ε arbitrarily small. Thus the limit t → ∞ gives back the final state of the system by eliminating all oscillating terms, but keeping the secular ones. The integrals are thus computable, and, separating the diagonal terms from the others yields

V (t)|ψ0 ⟩ = iℏ

∂|ψ0 ⟩ ∂τ

V (t)|ψ1 ⟩ + H0 |ψ0 ⟩ = iℏ

∂|ψ1 ⟩ ∂τ

.. .

that can be solved once we know the solution of the leading order equation. But we know that in this case we can use the adiabatic approximation. When V (t) does ∑time one gets the Wigner-Kirkwood series not depend iλ2 ∑ ⟨n|V |m⟩⟨m|V |n⟩ 1 λ2 on iλ ∑ ⟨n|V |n⟩t − t − ⟨n|V |m⟩⟨m|Vmechanics. |n⟩t2 + . . .Indeed, in this U (t) = 1 − 2 that is often used in statistical ℏ n ℏ En − Em 2 ℏ m,n m̸=n case we introduce the unitary transformation ∑ ⟨m|V |n⟩ ∑ ∑ ∑ ⟨m|V |n⟩⟨n|V |q⟩ 2 +λ |m⟩⟨n| + λ |m⟩⟨q| + . . . En − Em (En − Em )(Eq − En ) n m̸=n

m̸=n q̸=n

|ψ(t)⟩ = e− ℏ λV (t−t0 ) |ψF (t)⟩ i

where the time secular series yields the eigenvalues of the perturbed problem specified above, recursively; whereas that defines a free picture as we are trying to eliminate the remaining time-constant part yields the corrections to the interaction term. Now, in dual way with respect to

10

6 EXAMPLES

the small perturbations, we have to solve the Schrödinger or equation (1)

e ℏ λV (t−t0 ) H0 e− ℏ λV (t−t0 ) |ψF (t)⟩ = iℏ i

i

∂|ψF (t)⟩ ∂t

E0

=λ

( α ) 12 ∂ 2 ( π ) 12 3 1 3 ℏ2 λ =λ = 2 2 π ∂α α 4α 4 m2 ω 2

and we see that the expansion parameter λ appears only 6.2 Example of first and second order perinto the exponential and so, the corresponding Dyson seturbation theory – quantum pendulum ries, a dual Dyson series, is meaningful at large λs and is Consider the quantum mathematical pendulum with the Hamiltonian

[ ∫ t1 ∫ ∫ t i i i i i i 1 i t λV (t1 −t0 ) dt1 e ℏ λV (t1 −t0 ) H0 e− ℏ λV (t1 −t0 ) − 2 dtℏ12 ∂ dt H0 e− ℏ λV (t1 −t0 ) e ℏ λV (t2 −t0 ) H0 e− ℏ λV (t2 |ψF (t)⟩ = 1 − 2 2e ℏ ℏ t0 ℏ t t 0 H = −0 − λ cos ϕ 2ma2 ∂ϕ2 After the rescaling in time τ = λt we can see that this is indeed a series in 1/λ justifying in this way the with the potential energy −λ cos ϕ taken as the perturbaname of dual Dyson series. The reason is that we tion i.e. have obtained this series simply interchanging H 0 and V and we can go from one to another applying this exchange. This is called duality principle in perturba- V = − cos ϕ tion theory. The choice H0 = p2 /2m yields, as already said, a Wigner-Kirkwood series that is a gradient expan- The unperturbed normalized quantum wave functions are sion. The Wigner-Kirkwood series is a semiclassical se- those of the rigid rotor and are given by ries with eigenvalues given exactly as for WKB approximation.[10] einϕ ψn (ϕ) = √ 2π

6

Examples

6.1

and the energies

Example of first order perturbation 2 2 theory – ground state energy of the En(0) = ℏ n 2 2ma quartic oscillator

The first order energy correction to the rotor due to the Let us consider the quantum harmonic oscillator with the potential energy is quartic potential perturbation and the Hamiltonian ∫ ∫ 1 1 (1) −inϕ inϕ En = − e cos ϕe =− cos ϕ = 0 2π 2π ℏ2 ∂ 2 mω 2 x2 4 H=− + + λx 2m ∂x2 2 Using the formula for the second order correction one The ground state of the harmonic oscillator is

ψ0 =

( α ) 14 π

e

−αx2 /2

gets

En(2)

∫ 2 ma2 ∑ e−ikϕ cos ϕeinϕ = 2π 2 ℏ2 n2 − k 2 k

( α = mω/ℏ ) and the energy of unperturbed ground or state is

(0) E0

=

En(2) =

1 2 ℏω.

k

Using the first order correction formula we get

(1) E0

=λ

( α ) 12 ∫ π

2 ma2 ∑ |(δn,1−k + δn,−1−k )| 2ℏ2 n2 − k 2

e

−αx2 /2 4 −αx2 /2

x e

or

( ) 2 ( α ) 12 ∂ 2 ∫ 1 1 ma2 1 ma 2 (2) −αx + = E = dx = λ e dx n 2 2 2 2 2ℏ 2n − 1 −2n − 1 ℏ 4n − 1 π ∂α

11

7

References

[1] Aoyama, Tatsumi; Hayakawa, Masashi; Kinoshita, Toichiro; Nio, Makiko (2012). “Tenth-order QED lepton anomalous magnetic moment: Eighth-order vertices containing a second-order vacuum polarization”. Physical Review D. American Physical Society. 85 (3): 033007. arXiv:1110.2826 . Bibcode:2012PhRvD..85c3007A. doi:10.1103/PhysRevD.85.033007. [2] Schrödinger, E. (1926). “Quantisierung als Eigenwertproblem” [Quantification of the eigen value problem]. Annalen der Physik (in German). 80 Bibcode:1926AnP...385..437S. (13): 437–490. doi:10.1002/andp.19263851302. [3] Rayleigh, J. W. S. (1894). Theory of Sound. I (2nd ed.). London: Macmillan. pp. 115–118. ISBN 1-152-060236. [4] Sakurai, J.J., and Napolitano, J. (1964,2011). Modern quantum mechanics (2nd ed.), Addison Wesley ISBN 9780-8053-8291-4 . Chapter 5 [5] Landau, L. D.; Lifschitz, E. M. Quantum Mechanics: Nonrelativistic Theory (3rd ed.). ISBN 0-08-019012-X. [6] Bir, Gennadiĭ Levikovich; Pikus, Grigoriĭ Ezekielevich (1974). “Chapter 15: Perturbation theory for the degenerate case”. Symmetry and Strain-induced Effects in Semiconductors. ISBN 978-0-470-07321-6. [7] Soliverez, Carlos E. (1981). “General Theory of Effective Hamiltonians”. Physical Review A. 24: 4–9. Bibcode:1981PhRvA..24....4S. doi:10.1103/PhysRevA.24.4 – via Academia.Edu. [8] Frasca, M. (1998). “Duality in Perturbation Theory and the Quantum Adiabatic Approximation”. Physical Review A. 58 (5): 3439. arXiv:hepth/9801069 . Bibcode:1998PhRvA..58.3439F. doi:10.1103/PhysRevA.58.3439. [9] Mostafazadeh, A. (1997). “Quantum adiabatic approximation and the geometric phase,”. Physical Review A. 55 (3): 1653. arXiv:hepth/9606053 . Bibcode:1997PhRvA..55.1653M. doi:10.1103/PhysRevA.55.1653. [10] Frasca, Marco (2007). “A strongly perturbed quantum system is a semiclassical system”. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 463 (2085): 2195. arXiv:hepth/0603182 . Bibcode:2007RSPSA.463.2195F. doi:10.1098/rspa.2007.1879.

12

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