Perturbation theory

Perturbation theory (quantum mechanics)

Inquantum mechanics, perturbation theory is a set of

approximation schemes directly related to mathematical perturbationfor describing a complicated quantum sys-tem 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 quanti-ties associated with the perturbed system (e.g. itsenergy levelsandeigenstates) can be expressed as “corrections” to those of the simple system. These corrections, be-ing small compared to the size of the quantities them-selves, can be calculated using approximate methods such asasymptotic series. The complicated system can there-fore be studied based on knowledge of the simpler one.

1

Approximate Hamiltonians

Perturbation theory is an important tool for describing real quantum systems, as it turns out to be very diffi-cult to find exact solutions to theSchrödinger equationfor Hamiltoniansof even moderate complexity. The Hamil-tonians to which we know exact solutions, such as the hydrogen atom, thequantum harmonic oscillatorand 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.

2

Applying perturbation theory

Perturbation theory is applicable if the problem at hand cannot be solved exactly, but can be formulated by adding a “small” term to the mathematical description of the ex-actly solvable problem.

For example, by adding a perturbativeelectric potential to the quantum mechanical model of the hydrogen atom, tiny shifts in thespectral linesof hydrogen caused by the presence of anelectric field(theStark effect) can be cal-culated. This is only approximate because the sum of a Coulomb potentialwith a linear potential is unstable (has no true bound states) although thetunneling time(decay rate) is very long. This instability shows up as a broad-ening of the energy spectrum lines, which perturbation theory fails to reproduce entirely.

The expressions produced by perturbation theory are not

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 finitepower seriesin 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 usuallydivergent(beingasymptotic series). There exist ways to convert them into convergent series, which can be evaluated for large-expansion parameters, most efficiently byVariational method.

In the theory of quantum electrodynamics (QED), in which theelectron–photoninteraction is treated pertur-batively, the calculation of the electron’s magnetic mo-menthas been found to agree with experiment to eleven decimal places.[1]_{In QED and otherquantum field}

the-ories, special calculation techniques known asFeynman diagramsare used to systematically sum the power series terms.

2.1 Limitations

2.1.1 Large perturbations

Under some circumstances, perturbation theory is an in-valid approach to take. This happens when the system we wish to describe cannot be described by a small per-turbation imposed on some simple system. Inquantum chromodynamics, for instance, the interaction ofquarks with thegluonfield cannot be treated perturbatively at low energies because thecoupling constant(the expansion pa-rameter) becomes too large.

2.1.2 Non-adiabatic states

Perturbation theory also fails to describe states that are not generatedadiabaticallyfrom the “free model”, includ-ingbound statesand various collective phenomena such assolitons. Imagine, for example, that we have a sys-tem of free (i.e. non-interacting) particles, to which an attractive interaction is introduced. Depending on the form of the interaction, this may create an entirely new set of eigenstates corresponding to groups of particles bound to one another. An example of this phenomenon may be found in conventionalsuperconductivity, in which thephonon-mediated attraction betweenconduction elec-tronsleads to the formation of correlated electron pairs known asCooper pairs. When faced with such systems, one usually turns to other approximation schemes, such

as thevariational methodand theWKB approximation. This is because there is no analogue of abound parti-clein 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}

perturba-tion parameter g. Perturbaperturba-tion theory can only detect so-lutions “close” to the unperturbed solution, even if there are other solutions for which the perturbative expansion is not valid.

2.1.3 Difficult computations

The problem of non-perturbative systems has been some-what alleviated by the advent of modern computers. It has become practical to obtain numerical non-perturbative solutions for certain problems, using meth-ods such asdensity 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 inparticle physicsfor generating theoretical results that can be com-pared with experiment.

3

Time-independent perturbation

theory

Time-independent perturbation theory is one of two cat-egories of perturbation theory, the other being time-dependent perturbation (see next section). In time-independent perturbation theory the perturbation Hamil-tonian 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 ofLord Rayleigh,[3]

who investigated harmonic vibrations of a string per-turbed by small inhomogeneities. This is why this perturbation theory is often referred to as Rayleigh–

Schrödinger perturbation theory.

3.1

First order corrections

We begin[4]_{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 time-independentSchrödinger equation:

H0n(0)

⟩

= E_n(0)n(0) ⟩

, n = 1, 2, 3,· · ·

For simplicity, we have assumed that the energies are dis-crete. The (0) superscripts denote that these quantities

are associated with the unperturbed system. Note the use ofbra–ket notation.

We now introduce a perturbation to the Hamiltonian. Let V be a Hamiltonian representing a weak physical distur-bance, 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

The energy levels and eigenstates of the perturbed Hamil-tonian 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 per-turbation is sufficiently weak, we can write them aspower seriesin λ: En = E(0)n + λE (1) n + λ 2_E(2) n +· · · |n⟩ =n(0)⟩ + λn(1) ⟩ + λ2n(2) ⟩ +· · · where En(k)= 1 k! dk_E n dλk  n(k)⟩₌ 1 k! dk_|n⟩ dλk

When λ = 0, these reduce to the unperturbed values, which are the first term in each series. Since the perturba-tion 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

(H0+ λV )(n(0) ⟩ + λn(1) ⟩ +· · · ) = ( E_n(0)+ λE_n(1)+ λ2E_n(2)+· · ·) (n(0) ⟩ + λn(1) ⟩ +· · · ) Expanding this equation and comparing coefficients

of each power of λ results in an infinite series of simultaneous equations. The zeroth-order equation is simply the Schrödinger equation for the unperturbed sys-tem. The first-order equation is

H0n(1) ⟩ + Vn(0) ⟩ = E(0)_n n(1) ⟩ + E_n(1)n(0) ⟩

3.2 Second-order and higher corrections 3

Operating through by⟨n(0)_{| . The first term on the}

left-hand side cancels the first term on the right-left-hand side. (Recall, the unperturbed Hamiltonian isHermitian). This leads to the first-order energy shift:

E_n(1)= ⟨

n(0) V n(0)

⟩

This is simply theexpectation valueof the perturbation Hamiltonian while the system is in the unperturbed state. This result can be interpreted in the following way: sup-pose 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 per-turbation causes the average energy of this state to in-crease 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.

Before we compute the corrections to the energy eigen-state, we need to address the issue of normalization. We may suppose

⟨

n(0) n(0)

⟩ = 1,

but perturbation theory assumes we also have⟨n|n⟩ = 1

. It follows that at first order in λ, we must have

(⟨ n(0) + λ ⟨ n(1)) (n(0) ⟩ + λn(1) ⟩) = 1 ⟨ n(0) n(1) ⟩ + ⟨ n(1) n(0) ⟩ = 0.

Since the overall phase is not determined in quantum mechanics, without loss of generality, we may assume

⟨n(0)_{|n⟩ is purely real. Therefore,}

⟨ n(0) n(1) ⟩ =− ⟨ n(0) n(1) ⟩ , and we deduce ⟨ n(0) n(1) ⟩ = 0.

To obtain the first-order correction to the energy eigen-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 resolution of the identity,

V n(0) ⟩ =  ∑ k_̸=n  k(0)⟩ ⟨_k(0)   V _n(0)⟩₊(_n(0)⟩ ⟨_n(0))_Vn(0)⟩ =∑ k̸=n  k(0)⟩ ⟨_k(0) _{V n}(0)⟩_{+ E}(1) n n (0)⟩_,

where the |k(0)_{⟩ are in the orthogonal complement of}

|n(0)_{⟩ . The first-order equation may thus be expressed}

as ( E_n(0)− H0) n(1) ⟩ =∑ k̸=n  k(0)⟩ ⟨_k(0) _{V n}(0)⟩

For the moment, suppose that the zeroth-order energy level is not degenerate, i.e. there is no eigenstate of

H0in the orthogonal complement of|n(0)⟩ with the

en-ergy En(0) . After renaming the summation dummy

in-dex above as k′ , we can pick any k̸= n , and multiply

through by⟨k(0)_{| giving} ( E_n(0)− E_k(0) ) ⟨ k(0)n(1) ⟩ = ⟨ k(0) V n(0) ⟩ We see that the above⟨k(0)_|n(1)_{⟩ also gives us the}

com-ponent of the first-order correction along|k(0)_{⟩ .}

Thus in total we get,

 n(1)⟩ =∑ k̸=n ⟨ k(0)_{V n}(0)⟩ E(0)n − E_k(0)  k(0)⟩

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-turbation mixes eigenstate n with eigenstate k; it is also inversely proportional to the energy difference between eigenstates k and n, which means that the perturbation de-forms the eigenstate to a greater extent if there are more eigenstates at nearby energies. We see also that the ex-pression is singular if any of these states have the same energy as state n, which is why we assumed that there is no degeneracy.

3.2 Second-order and higher corrections

We can find the higher-order deviations by a similar pro-cedure, though the calculations become quite tedious with our current formulation. Our normalization prescription gives that 2 ⟨ n(0) n(2) ⟩ + ⟨ n(1) n(1) ⟩ = 0.

Up to second order, the expressions for the energies and (normalized) eigenstates are:

En(λ) = En(0)+λ ⟨ n(0) V n(0) ⟩ +λ2∑ k̸=n ⟨k(0)_{V n}(0)⟩2 En(0)− E_k(0) +O(λ3)

|n(λ)⟩ =n(0)⟩_{+ λ}∑ k̸=n  k(0)⟩ ⟨k (0)_{V n}(0)⟩ En(0)− E (0) k + λ2∑ k̸=n ∑ ℓ̸=n  k(0)⟩ ⟨k (0)_{V ℓ}(0)⟩ ⟨_ℓ(0)_{V n}(0)⟩ ( En(0)− E(0)_k ) ( En(0)− E_ℓ(0) ) − λ2∑ k̸=n  k(0)⟩ ⟨n (0)_Vn(0)⟩ ⟨_k(0)_Vn(0)⟩ ( En(0)− E (0) k )2 − 1 2λ 2_n(0)⟩ ∑ k̸=n ⟨ n(0)_{V k}(0)⟩ ⟨_k(0)_{V n}(0)⟩ ( En(0)− E (0) k )2 + O(λ 3 ).

Extending the process further, the third-order energy cor-rection can be shown to be[5]

E_n(3)=∑ k_̸=n ∑ m_̸=n ⟨n(0)_{|V |m}(0)_⟩⟨m(0)_{|V |k}(0)_⟩⟨k(0)_{|V |n}(0)_⟩ ( Em(0)− E (0) n ) ( E(0)_k − En(0) ) −⟨n(0)_{|V |n}(0)_⟩∑ m_̸=n |⟨n(0)_{|V |m}(0)_⟩|2 ( Em(0)− E (0) n )2.

Corrections to fifth order (energies) and fourth order (states) in compact notation

If we introduce the notation,

Vnm≡ ⟨n(0)|V |m(0)⟩

Enm≡ En(0)− E

(0)

m

then the energy corrections to fifth order can be written

E(1)n = Vnn E(2)_n =|Vnk2| 2 Enk2 E(3)_n =Vnk3Vk3k2Vk2n Enk2Enk3 − Vnn|V nk3| 2 E2 nk3 E(4)_n =Vnk4Vk4k3Vk3k2Vk2n Enk2Enk3Enk4 −|Vnk4| 2 E2 nk4 |Vnk2| 2 Enk2 − Vnn Vnk4Vk4k3Vk3n E2 nk3Enk4 − Vnn Vnk4Vk4k2Vk2n Enk2E 2 nk4 + V_nn2 |Vnk4| 2 E3 nk4 =Vnk4Vk4k3Vk3k2Vk2n Enk2Enk3Enk4 − E(2) n |Vnk4| 2 E2 nk4 − 2Vnn Vnk4Vk4k3Vk3n E2 nk3Enk4 + V_nn2 |Vnk4| 2 E3 nk4 E(5)_n =Vnk5Vk5k4Vk4k3Vk3k2Vk2n Enk2Enk3Enk4Enk5 −Vnk5Vk5k4Vk4n E2 nk4Enk5 |Vnk2| 2 Enk2 −Vnk5Vk5k2Vk2n Enk2E 2 nk5 |Vnk2| 2 Enk2 −|Vnk5| 2 E2 nk5 Vnk3Vk3k2Vk2n Enk2Enk3 − Vnn Vnk5Vk5k4Vk4k3Vk3n E2 nk3Enk4Enk5 − Vnn Vnk5Vk5k4Vk4k2Vk2n Enk2E 2 nk4Enk5 − Vnn Vnk5Vk5k3Vk3k2Vk2n Enk2Enk3E 2 nk5 + Vnn|V nk5| 2 E2 nk5 |Vnk3| 2 E2 nk3 + 2Vnn|V nk5| 2 E3 nk5 |Vnk2| 2 Enk2 + V_nn2 Vnk5Vk5k4Vk4n E3 nk4Enk5 + V_nn2 Vnk5Vk5k3Vk3n E2 nk3E 2 nk5 + V_nn2 Vnk5Vk5k2Vk2n Enk2E 3 nk5 − V3 nn |Vnk5| 2 E4 nk5 =Vnk5Vk5k4Vk4k3Vk3k2Vk2n Enk2Enk3Enk4Enk5 − 2E(2) n Vnk5Vk5k4Vk4n E2 nk4Enk5 −|Vnk5| 2 E2 nk5 Vnk3Vk3k2Vk2n Enk2Enk3 − 2Vnn ( Vnk5Vk5k4Vk4k3Vk3n E2 nk3Enk4Enk5 −Vnk5Vk5k4Vk4k2Vk2n Enk2E 2 nk4Enk5 +|Vnk5| 2 E2 nk5 |Vnk3| 2 E2 nk3 + 2E_n(2)|Vnk5| 2 E3 nk5 ) + Vnn2 ( 2Vnk5Vk5k4Vk4n E3 nk4Enk5 +Vnk5Vk5k3Vk3n E2 nk3E 2 nk5 ) − V3 nn |Vnk5| 2 E4 nk5

and the states to fourth order can be written

|n(1)_{⟩ =} Vk1n Enk1 |k(0) 1 ⟩ |n(2)_{⟩ =} ( Vk1k2Vk2n Enk1Enk2 −VnnVk1n E2 nk1 ) |k(0) 1 ⟩ − 1 2 Vnk1Vk1n E2 k1n |n(0)_⟩ |n(3)_{⟩ =} [ −Vk1k2Vk2k3Vk3n Ek1nEnk2Enk3 +VnnVk1k2Vk2n Ek1nEnk2 ( 1 Enk1 − 1 Enk2 ) +|Vnn| 2_V k1n E3 k1n ] |k(0) 1 ⟩ − [ Vnk2Vk2k1Vk1n+ Vk2nVk1k2Vnk1 E2 nk2Enk1 +|Vnk1| 2_V nn E3 nk1 ] |n(0)_⟩ |n(4)_{⟩ =} [ Vk1k2Vk2k3Vk3k4Vk4k2+ Vk3k2Vk1k2Vk4k3Vk2k4 2Ek1nE 2 k2k3Ek2k4 −Vk2k3Vk3k4Vk4nVk1k2 Ek1nEk2nEnk3Enk4 +Vk1k2 Ek1n ( |Vk2k3| 2_V k2k2 E3 k2k3 −|Vnk3| 2_V k2n E2 k3nEk2n ) +VnnVk1k2Vk3nVk2k3 Ek1nEnk3Ek2n ( 1 Enk3 + 1 Ek2n + 1 Ek1n ) +|Vk2n| 2_V k1k3 Enk2Ek1n ( Vk3n Enk1Enk3 − Vk3k1 E2 k3k1 ) −Vnn(Vk3k2Vk1k3Vk2k1+ Vk3k1Vk2k3Vk1k2) 2Ek1nE 2 k1k3Ek1k2 +|Vnn| 2 Ek1n ( Vk1nVnn E3 k1n +Vk1k2Vk2n E3 k2n ) −|Vk1k2| 2_V nnVk1n Ek1nE 3 k1k2 ] |k(0) 1 ⟩ + 1 2 [ Vnk1Vk1k2 Enk1E 2 k2n ( Vk2nVnn Ek2n −Vk2k3Vk3n Enk3 ) −Vk1nVk2k1 E2 k1nEnk2 ( Vk3k2Vnk3 Enk3 +VnnVnk2 Enk2 ) +|Vnk1| 2 E2 k1n ( 3|Vnk2| 2 4E2 k2n −2|Vnn|2 E2 k1n ) −Vk2k3Vk3k1|Vnk1| 2 E2 nk3Enk1Enk2 ] |n(0)_⟩

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 de-fined, 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 differ-ent 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

E_n(0)− H0

does not have a well-defined inverse.

Let D denote the subspace spanned by these degener-ate eigenstdegener-ates. No matter how small the perturbation is, in the degenerate subspace D the energy differences between the eigenstates H0 are zero, so complete

mix-ing of at least some of these states is assured. Typi-cally the eigenvalues will split and the eigenspaces will become simple (one-dimensional), or at least get smaller dimension than D. The successful perturbations will not be “small” relative to a poorly chosen basis of D. Instead, we consider the perturbation “small” if the new eigenstate is close to the subspace D. The new Hamiltonian must be diagonalized in D, or a slight variation of D, so to speak. These correct perturbed eigenstates in D are now the basis for the perturbation expansion:

|n⟩ =∑

k∈D

3.4 Generalization to multi-parameter case 5

For the first-order perturbation we need to solve the per-turbed Hamiltonian restricted to the degenerate subspace D

V|k(0)⟩ = ϵk|k(0)⟩ + small ∀|k(0)⟩ ∈ D.

simultaneously for all the degenerate eigenstates, where

ϵkare first-order corrections to the degenerate energy

lev-els, and “small” is a small vector orthogonal to D. This is equivalent to diagonalizing the matrix

⟨k(0)_{|V |l}(0)_{⟩ = V}

kl ∀ |k(0)⟩, |l(0)⟩ ∈ D.

This procedure is approximate, since we neglected states outside the D subspace. The splitting of degenerate ener-gies ϵkis 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 inElectron Spin Resonanceexperiments. Higher-order corrections due to other eigenstates can be found in the same way as for the non-degenerate case

( En(0)− H0 ) |n(1)_{⟩ =}∑ k̸∈D ( ⟨k(0)_{|V |n}(0)_⟩)_|k(0)_⟩.

The operator on the left hand side is not singular when applied to eigenstates outside D, so we can write

|n(1)_{⟩ =}∑ k_̸∈D ⟨k(0)_{|V |n}(0)_⟩ E(0)n − E (0) k |k(0)_⟩,

but the effect on the degenerate states is minuscule, pro-portional to the square of the first-order correction ϵk.

Near-degenerate states should also be treated in the above manner, since the original Hamiltonian won't be larger than the perturbation in the near-degenerate subspace. An application is found in thenearly free electron model, where near-degeneracy treated properly gives rise to an energy gap even for small perturbations. Other eigen-states will only shift the absolute energy of all near-degenerate states simultaneously.

3.4

Generalization

to

multi-parameter

case

The generalization of the time-independent perturbation theory to the case where there are multiple small param-eters xµ _{= (x}1_{, x}2_{,· · · ) in place of λ can be formulated}

more systematically using the language ofdifferential ge-ometry, which basically defines the derivatives of the quantum states and calculates the perturbative corrections by taking derivatives iteratively at the unperturbed point.

3.4.1 Hamiltonian and force operator

From the differential geometric point of view, a parame-terized Hamiltonian is considered as a function defined on the parametermanifoldthat maps each particular set of parameters (x1_{, x}2_{,· · · ) to an Hermitian operator H(x}μ₎

that acts on the Hilbert space. The parameters here can be external field, interaction strength, or driving parameters in thequantum phase transition. Let En(xμ_{) and|n(x}µ_)⟩

be the n-th eigenenergy and eigenstate of H(xμ₎

respec-tively. In the language of differential geometry, the states

|n(xµ_{)⟩ form avector bundleover the parameter}

mani-fold, on which derivatives of these states can be defined. The perturbation theory is to answer the following ques-tion: given En(x

µ

0)and|n(x

µ

0)⟩ at an unperturbed

refer-ence point xµ0 , how to estimate the En(xμ) and|n(xµ)⟩

at xμ_{close to that reference point.}

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

H(xµ) = H(0) + xµFµ.

If the parameters xμ_{are considered as generalized}

coor-dinates, then Fμ should be identified as the generalized force operators related to those coordinates. Different indices μ label the different forces along different direc-tions in the parameter manifold. For example, if xμ

de-notes the external magnetic field in the μ-direction, then

Fμ should be the magnetization in the same direction. 3.4.2 Perturbation theory as power series expansion

The validity of the perturbation theory lies on the adia-batic assumption, which assumes the eigenenergies and eigenstates of the Hamiltonian are smooth functions of parameters such that their values in the vicinity region can be calculated in power series (likeTaylor expansion) of the parameters: En(xµ) = En+ xµ∂µEn+ 1 2!x µ_xν_∂ µ∂νEn+· · · |n(xµ_{)⟩ = |n⟩ + x}µ_|∂ µn⟩ + 1 2!x µ_xν_|∂ µ∂νn⟩ + · · ·

Here ∂μ denotes the derivative with respect to xμ_{. When}

applying to the state|∂µn⟩ , it should be understood as the

covariant derivativeif the vector bundle is equipped with non-vanishing connection. All the terms on the right-hand-side of the series are evaluated at xμ_{= 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 con-verge slowly or even not concon-verge when the energy levels

are close to each other. The adiabatic assumption breaks down when there is energy level degeneracy, and hence the perturbation theory is not applicable in that case.

3.4.3 Hellman–Feynman theorems

The above power series expansion can be readily eval-uated if there is a systematic approach to calculate the derivates to any order. Using thechain rule, the deriva-tives can be broken down to the single derivative on either the energy or the state. TheHellmann–Feynman theo-remsare used to calculate these single derivatives. The first Hellmann–Feynman theorem gives the derivative of the energy,

∂µEn=⟨n|∂µH|n⟩

The second Hellmann–Feynman theorem gives the derivative of the state (resolved by the complete basis with m ≠ n), ⟨m|∂µn⟩ = ⟨m|∂ µH|n⟩ En− Em , ⟨∂µm|n⟩ = ⟨m|∂ µH|n⟩ Em− En .

For the linearly parameterized Hamiltonian, ∂μH simply stands for the generalized force operator Fμ.

The theorems can be simply derived by applying the dif-ferential operator ∂μ to both sides of the Schrödinger equationH|n⟩ = En|n⟩, which reads

∂µH|n⟩ + H|∂µn⟩ = ∂µEn|n⟩ + En|∂µn⟩.

Then overlap with the state⟨m| from left and make use of the Schrödinger equation⟨m|H = ⟨m|Emagain,

⟨m|∂µH|n⟩+Em⟨m|∂µn⟩ = ∂µEn⟨m|n⟩+En⟨m|∂µn⟩.

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 re-arranging the terms. With the differential rules given by the Hellmann–Feynman theorems, the perturbative cor-rection to the energies and states can be calculated sys-tematically.

3.4.4 Correction of energy and state

To the second order, the energy correction reads

En(xµ) =⟨n|H|n⟩+⟨n|∂µH|n⟩xµ+ℜ ∑ m̸=n ⟨n|∂νH|m⟩⟨m|∂µH|n⟩ En− Em xµxν+· · · ,

whereℜ denotes the real partfunction. The first order derivative ∂μEn is given by the first Hellmann–Feynman 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|∂νH|n⟩ , which

reads

∂µ∂νEn =⟨∂µn|∂νH|n⟩+⟨n|∂µ∂νH|n⟩+⟨n|∂νH|∂µn⟩.

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∂νEn =

∑

m

(⟨∂µn|m⟩⟨m|∂νH|n⟩ + ⟨n|∂νH|m⟩⟨m|∂µn⟩) ,

then all parts can be calculated using the Hellmann– Feynman theorems. In terms of Lie derivatives,

⟨∂µn|n⟩ = ⟨n|∂µn⟩ = 0 according to the definition

of the connection for the vector bundle. Therefore, the case m = n can be excluded from the summation, which avoids the singularity of the energy denominator. The same procedure can be carried on for higher order deriva-tives, from which higher order corrections are obtained. The same computational scheme is applicable for the cor-rection of states. The result to the second order is as fol-lows |n (xµ_{)⟩ = |n⟩ +} ∑ m̸=n ⟨m|∂µH|n⟩ En− Em |m⟩x µ +  ∑ m̸=n ∑ l̸=n ⟨m|∂µH|l⟩⟨l|∂νH|n⟩ (En− Em)(En− El) |m⟩ − ∑ m̸=n ⟨m|∂µH|n⟩⟨n|∂νH|n⟩ (En− Em)2 |m⟩ −1 2 ∑ m̸=n ⟨n|∂µH|m⟩⟨m|∂νH|n⟩ (En− Em)2 |m⟩   xµ_xν_{+· · · .}

Both energy derivatives and state derivatives will be in-volved in deduction. Whenever a state derivative is en-countered, resolve it by inserting the complete set of ba-sis, then the Hellmann-Feynman theorem is applicable. Because differentiation can be calculated systematically, the series expansion approach to the perturbative correc-tions can be coded on computers with symbolic process-ing software likeMathematica.

3.4.5 Effective Hamiltonian

Let H(0) be the Hamiltonian completely restricted either in the low-energy subspaceHLor in the high-energy

sub-spaceHH, such that there is no matrix element in H(0)

connecting the low- and the high-energy subspaces, i.e.

⟨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 out, the effective Hamiltonian in the low energy subspace reads[6]

4.1 Method of variation of constants 7 H_mneff (xµ) =⟨m|H|n⟩+δnm⟨m|∂µH|n⟩xµ+ 1 2! ∑ l∈HH ( ⟨m|∂µH|l⟩⟨l|∂νH|n⟩ Em− El +⟨m|∂νH|l⟩⟨l|∂µH|n⟩ En− El ) xµxν+· · · . Here m, n ∈ HL are restricted in the low energy

sub-space. The above result can be derived by power series expansion of⟨m|H(xµ_{)|n⟩ .}

In a formal way it is possible to define an effective Hamil-tonian that gives exactly the low-lying energy states and wavefunctions.[7] _{In practice, some kind of}

approxima-tion (perturbaapproxima-tion theory) is generally required.

4

Time-dependent

perturbation

theory

4.1

Method of variation of constants

Time-dependent perturbation theory, developed byPaul 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 are its energy levels and eigenstates. Thus, the goals of time-dependent perturbation theory are slightly different from time-independent perturbation theory. One is inter-ested in the following quantities:

• The time-dependentexpectation valueof some ob-servable A, for a given initial state.

• The time-dependent amplitudes of those quantum

states that are energy eigenkets (eigenvectors) in the unperturbed system.

The first quantity is important because it gives rise to theclassical result of an A measurement performed on a macroscopic number of copies of the perturbed sys-tem. For example, we could take A to be the displace-ment in the x-direction of the electron in a hydrogen atom, in which case the expected value, when multiplied by an appropriate coefficient, gives the time-dependent dielectric polarizationof a hydrogen gas. With an appro-priate choice of perturbation (i.e. an oscillating electric potential), this allows one to calculate the ACpermittivity of the gas.

The second quantity looks at the time-dependent proba-bility of occupation for each eigenstate. This is partic-ularly useful inlaserphysics, where one is interested in the populations of different atomic states in a gas when a time-dependent electric field is applied. These probabil-ities are also useful for calculating the “quantum broad-ening” ofspectral lines(seeline broadening) andparticle decayinparticle physicsandnuclear physics.

We will briefly examine the method behind Dirac’s for-mulation of time-dependent perturbation theory. Choose

an energy basis|n⟩ for the unperturbed system. (We drop

the (0) superscripts for the eigenstates, because it is not useful to speak of energy levels and eigenstates for the perturbed system.)

If the unperturbed system is in eigenstate|j⟩ at time t = 0,

its state at subsequent times varies only by aphase(in the Schrödinger picture, where state vectors evolve in time and operators are constant),

|j(t)⟩ = e−iEjt/ℏ|j⟩ .

Now, introduce a time-dependent perturbing Hamilto-nian V(t). The HamiltoHamilto-nian of the perturbed system is

H = H0+ V (t) .

Let|ψ(t)⟩ denote the quantum state of the perturbed

sys-tem at time t. It obeys the time-dependent Schrödinger equation,

H|ψ(t)⟩ = iℏ∂

∂t|ψ(t)⟩ .

The quantum state at each instant can be expressed as a linear combination of the complete eigenbasis of|n⟩ : |ψ(t)⟩ =∑

n

cn(t)e−iEnt/ℏ|n⟩ ,

where the cn(t)s are to be determinedcomplexfunctions of t which we will refer to as amplitudes (strictly speak-ing, they are the amplitudes in theDirac picture). We have explicitly extracted the exponential phase fac-tors exp(−iEnt/ℏ) on the right hand side. This is only

a matter of convention, and may be done without loss of generality. The reason we go to this trouble is that when the system starts in the state|j⟩ and no perturbation is

present, the amplitudes have the convenient property that, for all t, cj(t) = 1 and cn(t) = 0 if n ≠ j.

The square of the absolute amplitude cn(t) is the proba-bility that the system is in state n at time t, since

|cn(t)|2=|⟨n|ψ(t)⟩|2 .

Plugging into the Schrödinger equation and using the fact that ∂/∂t acts by achain rule, one obtains

∑ n ( iℏ∂cn ∂t − cn(t)V (t) ) e−iEnt/ℏ|n⟩ = 0 .

By resolving the identity in front of V, this can be re-duced to a set of coupleddifferential equations for the amplitudes,

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

The matrix elements of V play a similar role as in time-independent perturbation theory, being proportional to the rate at which amplitudes are shifted between states. Note, however, that the direction of the shift is modi-fied by the exponential phase factor. Over times much longer than the energy difference Ek − En, the phase winds around 0 several times. If the time-dependence of V is sufficiently slow, this may cause the state ampli-tudes to oscillate. ( E.g., such oscillations are useful for managing radiative transitions in alaser.)

Up to this point, we have made no approximations, so this set of differential equations is exact. By supplying appropriate initial values cn(t), we could in principle find an exact (i.e., non-perturbative) solution. This is easily done when there are only two energy levels (n = 1, 2), and this solution is useful for modelling systems like the ammoniamolecule.

However, exact solutions are difficult to find when there are many energy levels, and one instead looks for pertur-bative solutions. These may be obtained by expressing the equations in an integral form,

cn(t) = cn(0)+ −i ℏ ∑ k ∫ t 0 dt′⟨n|V (t′)|k⟩ ck(t′) e−i(Ek−En)t ′_/ℏ .

Repeatedly substituting this expression for c back into right hand side, yields an iterative solution,

cn(t) = c(0)n + c

(1)

n + c

(2)

n +· · ·

where, for example, the first-order term is

c(1)_n (t) =−i ℏ ∑ k ∫ t 0 dt′ ⟨n|V (t′)|k⟩ ck(0) e−i(Ek−En)t ′_/ℏ .

Several further results follow from this, such asFermi’s golden rule, which relates the rate of transitions between quantum states to the density of states at particular en-ergies; or theDyson series, obtained by applying the it-erative method to thetime evolution operator, which is one of the starting points for the method ofFeynman di-agrams.

4.2

Method of Dyson series

Time-dependent perturbations can be reorganized through the technique of the Dyson series. The Schrödinger equation

H(t)|ψ(t)⟩ = iℏ∂|ψ(t)⟩ ∂t

has the formal solution

|ψ(t)⟩ = T exp [ −_ℏi ∫ t t0 dt′H(t′) ] |ψ(t0)⟩ ,

where T is the time ordering operator,

T A(t1)A(t2) =

{

A(t1)A(t2) t1> t2

A(t2)A(t1) t2> t1

.

Thus, the exponential represents the followingDyson se-ries, |ψ(t)⟩ = [ 1− i ℏ ∫ t t0 dt1H(t1)− 1 ℏ2 ∫ t t0 dt1 ∫ t1 t0 dt2H(t1)H(t2) + . . . ] |ψ(t0)⟩ .

Note that in the second term, the 1/2! factor exactly can-cels the double contribution due to the time-ordering op-erator, etc.

Consider the following perturbation problem

[H0+ λV (t)]|ψ(t)⟩ = iℏ

∂|ψ(t)⟩

∂t ,

assuming that the parameter λ is small and that the prob-lem H0|n⟩ = En|n⟩ has been solved.

Perform the following unitary transformation to the interaction picture(or Dirac picture),

|ψ(t)⟩ = e−i

ℏH0(t−t0)|ψ

I(t)⟩ .

Consequently, theSchrödinger equationsimplifies to

λeℏiH0(t−t0)_{V (t)e}−_ℏiH0(t−t0)|ψ

I(t)⟩ = iℏ

∂|ψI(t)⟩

∂t ,

so it is solved through the aboveDyson series,

|ψI(t)⟩ = [ 1−iλ ℏ ∫ t t0 dt1e i ℏH0(t1−t0)_{V (t} 1)e− i ℏH0(t1−t0)−λ 2 ℏ2 ∫ t t0 dt1 ∫ t1 t0 dt2e i ℏH0(t1−t0)_{V (t} 1)e− i ℏH0(t1−t0)_eℏiH0(t2−t0)_{V (t} 2)e− i ℏH0(t2−t0)_{+ . . .} ] |ψ(t0)⟩ ,

as a perturbation series with small λ.

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,

|ψI(t)⟩ = [ 1−iλ ℏ ∑ m ∑ n ∫ t t0 dt1⟨m|V (t1)|n⟩e− i ℏ(En−Em)(t1−t0)|m⟩⟨n| + . . . ] |ψ(t0)⟩ .

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 t0 dt1⟨β|V (t1)|α⟩e− i ℏ(Eα−Eβ)(t1−t0)_, as detailed in the previous section——while the corre-sponding transition probability to a continuum is fur-nished byFermi’s golden rule.

As an aside, note that time-independent perturbation the-ory is also organized inside this time-dependent pertur-bation theory Dyson series. To see this, write the unitary evolution operator, obtained from the aboveDyson series, as U (t) = 1−iλ ℏ ∫ t t0 dt1e i ℏH0(t1−t0)_{V (t} 1)e− i ℏH0(t1−t0)−λ 2 ℏ2 ∫ t t0 dt1 ∫ t1 t0 dt2e i ℏH0(t1−t0)_{V (t} 1)e− i ℏH0(t1−t0)_eℏiH0(t2−t0)_{V (t} 2)e− i ℏH0(t2−t0)₊· · · and take the perturbation V to be time-independent.

Using the identity resolution

∑

n

|n⟩⟨n| = 1

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

U (t) = 1− { iλ ℏ ∫ t t0 dt1 ∑ m ∑ n ⟨m|V |n⟩e−i ℏ(En−Em)(t1−t0)|m⟩⟨n| } − { λ2 ℏ2 ∫ t t0 dt1 ∫ t1 t0 dt2 ∑ m ∑ n ∑ q

e−ℏi(En−Em)(t1−t0)⟨m|V |n⟩⟨n|V |q⟩e−_ℏi(Eq−En)(t2−t0)|m⟩⟨q| }

+· · · It is evident that, at second order, one must sum on all the

intermediate states. Assume t0 = 0and the asymptotic

limit of larger times. This means that, at each contri-bution of the perturbation series, one has to add a mul-tiplicative factor e−ϵtin the integrands for ε arbitrarily small. Thus the limit t → ∞ gives back the final state of the system by eliminating all oscillating terms, but keep-ing the secular ones. The integrals are thus computable, and, separating the diagonal terms from the others yields

U (t) = 1−iλ ℏ ∑ n ⟨n|V |n⟩t −iλ2 ℏ ∑ m_̸=n ⟨n|V |m⟩⟨m|V |n⟩ En− Em t−1 2 λ2 ℏ2 ∑ m,n ⟨n|V |m⟩⟨m|V |n⟩t2_{+ . . .} + λ∑ m̸=n ⟨m|V |n⟩ En− Em |m⟩⟨n| + λ2 ∑ m̸=n ∑ q̸=n ∑ n ⟨m|V |n⟩⟨n|V |q⟩ (En− Em)(Eq− En) |m⟩⟨q| + . . .

where the time secular series yields the eigenvalues of the perturbed problem specified above, recursively; whereas the remaining time-constant part yields the corrections to

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.

5 Strong perturbation theory

In a similar way as for small perturbations, it is possible to develop a strong perturbation theory. Let us consider as usual theSchrödinger equation

H(t)|ψ(t)⟩ = iℏ∂|ψ(t)⟩ ∂t

and we consider the question if a dual Dyson series ex-ists that applies in the limit of a perturbation increasingly large. This question can be answered in an affirmative way[8]_{and the series is the well-known adiabatic series.}[9]

This approach is quite general and can be shown in the following way. Let us consider the perturbation problem

[H0+ λV (t)]|ψ(t)⟩ = iℏ

∂|ψ(t)⟩ ∂t

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

|ψ⟩ = |ψ0⟩ +

1

λ|ψ1⟩ +

1

λ2|ψ2⟩ + . . .

but a direct substitution into the above equation fails to produce useful results. This situation can be adjusted making a rescaling of the time variable as τ = λt pro-ducing the following meaningful equations

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 orderequation. But we know that in this case we can use the adiabatic approximation. When V (t) does not depend on time one gets theWigner-Kirkwood series that is often used instatistical mechanics. Indeed, in this case we introduce the unitary transformation

|ψ(t)⟩ = e−i

ℏλV (t−t0)|ψ

F(t)⟩

that defines a free picture as we are trying to eliminate the interaction term. Now, in dual way with respect to

the small perturbations, we have to solve theSchrödinger equation eℏiλV (t−t0)_H 0e− i ℏλV (t−t0)|ψ F(t)⟩ = iℏ ∂|ψF(t)⟩ ∂t

and we see that the expansion parameter λ appears only into the exponential and so, the correspondingDyson se-ries, a dual Dyson sese-ries, is meaningful at large λs and is |ψF(t)⟩ = [ 1− i ℏ ∫ t t0 dt1e i ℏλV (t1−t0)_H 0e− i ℏλV (t1−t0)− 1 ℏ2 ∫ t t0 dt1 ∫ t1 t0 dt2e i ℏλV (t1−t0)_H 0e− i ℏλV (t1−t0)_eℏiλV (t2−t0)_H 0e− i ℏλV (t2−t0)_{+ . . .} ] |ψ(t0)⟩.

After the rescaling in time τ = λt we can see that this is indeed a series in 1/λ justifying in this way the name of dual Dyson series. The reason is that we have obtained this series simply interchanging H0 and

V and we can go from one to another applying this ex-change. This is called duality principle in perturba-tion theory. The choice H0 = p2/2myields, as already

said, aWigner-Kirkwood seriesthat is a gradient expan-sion. TheWigner-Kirkwood seriesis a semiclassical se-ries with eigenvalues given exactly as forWKB approxi-mation.[10]

6

Examples

6.1

Example of first order perturbation

theory – ground state energy of the

quartic oscillator

Let us consider the quantum harmonic oscillator with the quartic potential perturbation and the Hamiltonian

H =−ℏ 2 2m ∂2 ∂x2 + mω2_x2 2 + λx 4

The ground state of the harmonic oscillator is

ψ0= (_α π )1 4 e−αx2/2

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

E₀(0)= 1₂ℏω.

Using the first order correction formula we get

E₀(1)= λ (_α π )1 2∫ e−αx2/2x4e−αx2/2dx = λ (_α π )1 2 ∂2 ∂α2 ∫ e−αx2dx or E₀(1)= λ (_α π )1 2 ∂2 ∂α2 (_π α )1 2 = λ3 4 1 α2 = 3 4 ℏ2_λ m2_ω2

6.2 Example of first and second order

per-turbation theory – quantum pendulum

Consider the quantum mathematical pendulum with the Hamiltonian H =− ℏ 2 2ma2 ∂2 ∂ϕ2 − λ cos ϕ

with the potential energy−λ cos ϕ taken as the

perturba-tion i.e.

V =− cos ϕ

The unperturbed normalized quantum wave functions are those of the rigid rotor and are given by

ψn(ϕ) =

einϕ √

2π and the energies

E_n(0)= ℏ

2_n2

2ma2

The first order energy correction to the rotor due to the potential energy is

E_n(1)=− 1 2π

∫

e−inϕcos ϕeinϕ₌₋ 1

2π ∫

cos ϕ = 0

Using the formula for the second order correction one gets E_n(2)= ma 2 2π2_ℏ2 ∑ k

∫e−ikϕcos ϕeinϕ2

n2_{− k}2 or En(2)= ma2 2ℏ2 ∑ k |(δn,1−k+ δn,−1−k)| 2 n2− k2 or E_n(2)= ma 2 2ℏ2 ( 1 2n− 1 + 1 −2n − 1 ) = ma 2 ℏ2 1 4n2− 1

11

7

References

[1] Aoyama, Tatsumi; Hayakawa, Masashi; Kinoshita, To-ichiro; Nio, Makiko (2012). “Tenth-order QED lepton anomalous magnetic moment: Eighth-order vertices con-taining 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 Eigen-wertproblem” [Quantification of the eigen value problem]. Annalen der Physik (in German). 80 (13): 437–490. Bibcode:1926AnP...385..437S.

doi:10.1002/andp.19263851302.

[3] Rayleigh, J. W. S. (1894). Theory of Sound. I (2nd ed.). London: Macmillan. pp. 115–118. ISBN 1-152-06023-6.

[4] Sakurai, J.J., and Napolitano, J. (1964,2011). Modern quantum mechanics (2nd ed.), Addison WesleyISBN 978-0-8053-8291-4. Chapter 5

[5] Landau, L. D.; Lifschitz, E. M. Quantum Mechanics: Non-relativistic Theory (3rd ed.).ISBN 0-08-019012-X. [6] Bir, Gennadiĭ Levikovich; Pikus, Grigoriĭ Ezekielevich

(1974). “Chapter 15: Perturbation theory for the degen-erate case”.Symmetry and Strain-induced Effects in Semi-conductors.ISBN 978-0-470-07321-6.

[7] Soliverez, Carlos E. (1981). “General The-ory of Effective Hamiltonians”. Physical Re-view A. 24: 4–9. Bibcode:1981PhRvA..24....4S.

doi:10.1103/PhysRevA.24.4– via Academia.Edu. [8] Frasca, M. (1998). “Duality in Perturbation

The-ory and the Quantum Adiabatic Approximation”. Physical Review A. 58 (5): 3439. arXiv: hep-th/9801069 . Bibcode:1998PhRvA..58.3439F.

doi:10.1103/PhysRevA.58.3439.

[9] Mostafazadeh, A. (1997). “Quantum adia-batic approximation and the geometric phase,”. Physical Review A. 55 (3): 1653. arXiv: hep-th/9606053 . Bibcode:1997PhRvA..55.1653M.

doi:10.1103/PhysRevA.55.1653.

[10] Frasca, Marco (2007). “A strongly perturbed quan-tum system is a semiclassical system”. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 463 (2085): 2195. arXiv: hep-th/0603182 . Bibcode:2007RSPSA.463.2195F.

8

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