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Inquantum mechanics,perturbation theory is a set of approximation schemes directly related to mathematicalperturbation for describing a complicatedquantum 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. itsenergy levels andeigenstates) 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 asasymptotic series. The complicated system can therefore be studied based on knowledge of the simpler one. In effect, it is describing a complicated unsolved system using a simple, solvable system.

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

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 exactly solvable problem.

For example, by adding a perturbativeelectric potential to the quantum mechanical model of thehydrogen atom, tiny shifts in thespectral lines of hydrogen caused by the presence of anelectric field (theStark effect) can be calculated. This is only approximate because the sum of aCoulomb potential with a linear potential is unstable (has no true bound states) although thetunneling time (decay rate) is very long. This instability shows up as a broadening 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, sayα, is very small. Typically, the results are expressed in terms of finitepower series inα that seem to converge to the exact values when summed to higher order. After a certain ordern ~ 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 by thevariational method. In practice, convergent perturbation expansions often converge slowly while divergent perturbation expansions sometimes give good results, c.f. the exact solution, at lower order.^[1]

In the theory ofquantum electrodynamics (QED), in which theelectron–photon interaction is treated perturbatively, the calculation of the electron'smagnetic moment has been found to agree with experiment to eleven decimal places.^[2] In QED and otherquantum field theories, special calculation techniques known asFeynman diagrams are used to systematically sum the power series terms.

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. Inquantum chromodynamics, for instance, the interaction ofquarks with thegluon field cannot be treated perturbatively at low energies because thecoupling constant (the expansion parameter) becomes too large, violating the requirement that corrections must be small.

Perturbation theory also fails to describe states that are not generatedadiabatically from the "free model", includingbound states and various collective phenomena such assolitons.^{[citation needed]} Imagine, for example, that we have a system 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 electrons leads 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 method and theWKB approximation. This is because there is no analogue of abound particle in the unperturbed model and the energy of a soliton typically goes as theinverse of the expansion parameter. However, if we "integrate" over the solitonic phenomena, the nonperturbative corrections in this case will be tiny; of the order ofexp(−1/g) orexp(−1/g²) in the perturbation parameterg. 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.^{[citation needed]}

The problem of non-perturbative systems has been somewhat alleviated by the advent of moderncomputers. It has become practical to obtain numerical non-perturbative solutions for certain problems, using methods such asdensity functional theory. These advances have been of particular benefit to the field ofquantum chemistry.^[3] Computers have also been used to carry out perturbation theory calculations to extraordinarily high levels of precision, which has proven important inparticle physics for generating theoretical results that can be compared with experiment.

Time-independent perturbation theory is one of two categories of perturbation theory, the other being time-dependent perturbation (see next section). In time-independent perturbation theory, the perturbation Hamiltonian is static (i.e., possesses no time dependence). Time-independent perturbation theory was presented byErwin Schrödinger in a 1926 paper,^[4] shortly after he produced his theories in wave mechanics. In this paper Schrödinger referred to earlier work ofLord Rayleigh,^[5] who investigated harmonic vibrations of a string perturbed by small inhomogeneities. This is why this perturbation theory is often referred to asRayleigh–Schrödinger perturbation theory.^[6]

The process begins with an unperturbed HamiltonianH₀, which is assumed to have no time dependence.^[7] It has known energy levels andeigenstates, arising from the time-independentSchrödinger equation:

$${\displaystyle H_0|n^{(0)}⟩=E_n^{(0)}|n^{(0)}⟩,n=1,2,3,\cdots }$$

For simplicity, it is assumed that the energies are discrete. The(0) superscripts denote that these quantities are associated with the unperturbed system. Note the use ofbra–ket notation.

A perturbation is then introduced to the Hamiltonian. LetV be a Hamiltonian representing a weak physical disturbance, such as a potential energy produced by an external field. Thus,V is formally aHermitian operator. Letλ be a dimensionless parameter that can take on values ranging continuously from 0 (no perturbation) to 1 (the full perturbation). The perturbed Hamiltonian is:

$${\displaystyle H=H_0+\lambda V}$$

The energy levels and eigenstates of the perturbed Hamiltonian are again given by the time-independent Schrödinger equation,$${\displaystyle \left(H_0+\lambda V\right)|n⟩=E_n|n⟩.}$$

The objective is to expressE_n and${\displaystyle |n⟩}$ in terms of the energy levels and eigenstates of the old Hamiltonian. If the perturbation is sufficiently weak, they can be written as a (Maclaurin)power series inλ,where

Whenk = 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 the order is increased.

Substituting the power series expansion into the Schrödinger equation produces:

$${\displaystyle \left(H_0+\lambda V\right)\left(|n^{(0)}⟩+\lambda |n^{(1)}⟩+\cdots \right)=\left(E_n^{(0)}+\lambda E_n^{(1)}+\cdots \right)\left(|n^{(0)}⟩+\lambda |n^{(1)}⟩+\cdots \right).}$$

Expanding this equation and comparing coefficients of each power ofλ results in an infinite series ofsimultaneous equations. The zeroth-order equation is simply the Schrödinger equation for the unperturbed system,$${\displaystyle H_0|n^{(0)}⟩=E_n^{(0)}|n^{(0)}⟩.}$$

The first-order equation is$${\displaystyle H_0|n^{(1)}⟩+V|n^{(0)}⟩=E_n^{(0)}|n^{(1)}⟩+E_n^{(1)}|n^{(0)}⟩.}$$

Operating through by${\displaystyle ⟨n^{(0)}|}$, the first term on the left-hand side cancels the first term on the right-hand side. (Recall, the unperturbed Hamiltonian isHermitian). This leads to the first-order energy shift,$${\displaystyle E_n^{(1)}=⟨n^{(0)}|V|n^{(0)}⟩.}$$This is simply theexpectation value of the perturbation Hamiltonian while the system is in the unperturbed eigenstate.

This result can be interpreted in the following way: supposing that the perturbation is applied, but the system is kept in the quantum state${\displaystyle |n^{(0)}⟩}$, which is a valid quantum state though no longer an energy eigenstate. The perturbation causes the average energy of this state to increase by${\displaystyle ⟨n^{(0)}|V|n^{(0)}⟩}$. However, the true energy shift is slightly different, because the perturbed eigenstate is not exactly the same as${\displaystyle |n^{(0)}⟩}$. These further shifts are given by the second and higher order corrections to the energy.

Before corrections to the energy eigenstate are computed, the issue of normalization must be addressed. Supposing that$${\displaystyle ⟨n^{(0)}|n^{(0)}⟩=1,}$$but perturbation theory also assumes that${\displaystyle ⟨n|n⟩=1}$.

Then at first order inλ, the following must be true:$${\displaystyle \left(⟨n^{(0)}|+\lambda ⟨n^{(1)}|\right)\left(|n^{(0)}⟩+\lambda |n^{(1)}⟩\right)=1}$$$${\displaystyle ⟨n^{(0)}|n^{(0)}⟩+\lambda ⟨n^{(0)}|n^{(1)}⟩+\lambda ⟨n^{(1)}|n^{(0)}⟩+\cancel{{\lambda }^2⟨n^{(1)}|n^{(1)}⟩}=1}$$$${\displaystyle ⟨n^{(0)}|n^{(1)}⟩+⟨n^{(1)}|n^{(0)}⟩=0.}$$

Since the overall phase is not determined in quantum mechanics,without loss of generality, in time-independent theory it can be assumed that${\displaystyle ⟨n^{(0)}|n^{(1)}⟩}$ is purely real. Therefore,$${\displaystyle ⟨n^{(0)}|n^{(1)}⟩=⟨n^{(1)}|n^{(0)}⟩=-⟨n^{(1)}|n^{(0)}⟩,}$$leading to$${\displaystyle ⟨n^{(0)}|n^{(1)}⟩=0.}$$

To obtain the first-order correction to the energy eigenstate, the expression for the first-order energy correction is inserted back into the result shown above, equating the first-order coefficients ofλ. Then by using theresolution of the identity:where the${\displaystyle |k^{(0)}⟩}$ are in theorthogonal complement of${\displaystyle |n^{(0)}⟩}$, i.e., the other eigenvectors.

The first-order equation may thus be expressed as$${\displaystyle \left(E_n^{(0)}-H_0\right)|n^{(1)}⟩=\sum _{k\ne n}|k^{(0)}⟩⟨k^{(0)}|V|n^{(0)}⟩.}$$

Suppose that the zeroth-order energy level is notdegenerate, i.e. that there is no eigenstate ofH₀ in the orthogonal complement of${\displaystyle |n^{(0)}⟩}$ with the energy${\displaystyle E_n^{(0)}}$. After renaming the summation dummy index above as${\displaystyle k^{\prime }}$, any${\displaystyle k\ne n}$ can be chosen and multiplying the first-order equation through by${\displaystyle ⟨k^{(0)}|}$ gives$${\displaystyle \left(E_n^{(0)}-E_k^{(0)}\right)⟨k^{(0)}|n^{(1)}⟩=⟨k^{(0)}|V|n^{(0)}⟩.}$$

The above${\displaystyle ⟨k^{(0)}|n^{(1)}⟩}$ also gives us the component of the first-order correction along${\displaystyle |k^{(0)}⟩}$.

Thus, in total, the result is,$${\displaystyle |n^{(1)}⟩=\sum _{k\ne n}\frac{⟨k^{(0)}|V|n^{(0)}⟩}{E_n^{(0)}-E_k^{(0)}}|k^{(0)}⟩.}$$

The first-order change in then-th energy eigenket has a contribution from each of the energy eigenstatesk ≠n. Each term is proportional to the matrix element${\displaystyle ⟨k^{(0)}|V|n^{(0)}⟩}$, which is a measure of how much the perturbation mixes eigenstaten with eigenstatek; it is also inversely proportional to the energy difference between eigenstatesk andn, which means that the perturbation deforms the eigenstate to a greater extent if there are more eigenstates at nearby energies. The expression is singular if any of these states have the same energy as staten, which is why it was assumed that there is no degeneracy. The above formula for the perturbed eigenstates also implies that the perturbation theory can be legitimately used only when the absolute magnitude of the matrix elements of the perturbation is small compared with the corresponding differences in the unperturbed energy levels, i.e.,${\displaystyle |⟨k^{(0)}|V|n^{(0)}⟩|\ll |E_n^{(0)}-E_k^{(0)}|.}$

We can find the higher-order deviations by a similar procedure, though the calculations become quite tedious with our current formulation. Our normalization prescription gives that

$${\displaystyle 2⟨n^{(0)}|n^{(2)}⟩+⟨n^{(1)}|n^{(1)}⟩=0.}$$

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

$${\displaystyle E_n(\lambda )=E_n^{(0)}+\lambda ⟨n^{(0)}|V|n^{(0)}⟩+{\lambda }^2\sum _{k\ne n}\frac{{\left|⟨k^{(0)}|V|n^{(0)}⟩\right|}^2}{E_n^{(0)}-E_k^{(0)}}+O({\lambda }^3)}$$

If an intermediate normalization is taken (in other words, if it is required that${\displaystyle ⟨n^{(0)}|n(\lambda )⟩=1}$), then we obtain a nearly identical expression for the second-order correction to the correction given immediately above. To be precise, for an intermediate normalization, the last term would be omitted.

Extending the process further, the third-order energy correction can be shown to be^[8]

$${\displaystyle E_n^{(3)}=\sum _{k\ne n}\sum _{m\ne n}\frac{⟨n^{(0)}|V|m^{(0)}⟩⟨m^{(0)}|V|k^{(0)}⟩⟨k^{(0)}|V|n^{(0)}⟩}{\left(E_n^{(0)}-E_m^{(0)}\right)\left(E_n^{(0)}-E_k^{(0)}\right)}-⟨n^{(0)}|V|n^{(0)}⟩\sum _{m\ne n}\frac{|⟨n^{(0)}|V|m^{(0)}⟩{|}^2}{{\left(E_n^{(0)}-E_m^{(0)}\right)}^2}.}$$

It is possible to relate thek-th order correction to the energyE_n to thek-pointconnected correlation function of the perturbationV in the state${\displaystyle |n^{(0)}⟩}$. For${\displaystyle k=2}$, one has to consider the inverseLaplace transform${\displaystyle {\rho }_{n,2}(s)}$ of the two-point correlator:$${\displaystyle ⟨n^{(0)}|V(\tau )V(0)|n^{(0)}⟩-⟨n^{(0)}|V|n^{(0)}{⟩}^2=:{\int }_{\mathbb{R}}ds{\rho }_{n,2}(s)e^{-(s-E_n^{(0)})\tau }}$$where${\displaystyle V(\tau )=e^{H_0\tau }V{e}^{-H_0\tau }}$ is the perturbing operatorV in the interaction picture, evolving in Euclidean time. Then$${\displaystyle E_n^{(2)}=-{\int }_{\mathbb{R}}\frac{ds}{s-E_n^{(0)}}{\rho }_{n,2}(s).}$$

Similar formulas exist to all orders in perturbation theory, allowing one to express${\displaystyle E_n^{(k)}}$ in terms of the inverse Laplace transform${\displaystyle {\rho }_{n,k}}$ of the connected correlation function$${\displaystyle ⟨n^{(0)}|V({\tau }_1+\dots +{\tau }_{k-1})\cdots V({\tau }_1+{\tau }_2)V({\tau }_1)V(0)|n^{(0)}{⟩}_{\text{conn}}=⟨n^{(0)}|V({\tau }_1+\dots +{\tau }_{k-1})\cdots V({\tau }_1+{\tau }_2)V({\tau }_1)V(0)|n^{(0)}⟩-\text{subtractions}.}$$

To be precise, if we write$${\displaystyle ⟨n^{(0)}|V({\tau }_1+\dots +{\tau }_{k-1})\cdots V({\tau }_1+{\tau }_2)V({\tau }_1)V(0)|n^{(0)}{⟩}_{\text{conn}}={\int }_{\mathbb{R}}\prod _{i=1}^{k-1}d{s}_i{e}^{-(s_i-E_n^{(0)}){\tau }_i}{\rho }_{n,k}(s_1,\dots ,s_{k-1})}$$then thek-th order energy shift is given by^[9]

$${\displaystyle E_n^{(k)}=(-1{)}^{k-1}{\int }_{\mathbb{R}}\prod _{i=1}^{k-1}\frac{d{s}_i}{s_i-E_n^{(0)}}{\rho }_{n,k}(s_1,\dots ,s_{k-1}).}$$

Suppose that two or more energy eigenstates of the unperturbed Hamiltonian aredegenerate. 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$${\displaystyle E_n^{(0)}-H_0}$$does not have a well-defined inverse.

LetD denote the subspace spanned by these degenerate eigenstates. No matter how small the perturbation is, in the degenerate subspaceD the energy differences between the eigenstates ofH are non-zero, so complete mixing of at least some of these states is assured. Typically, the eigenvalues will split, and the eigenspaces will become simple (one-dimensional), or at least of smaller dimension thanD.

The successful perturbations will not be "small" relative to a poorly chosen basis ofD. Instead, we consider the perturbation "small" if the new eigenstate is close to the subspaceD. The new Hamiltonian must be diagonalized inD, or a slight variation ofD, so to speak. These perturbed eigenstates inD are now the basis for the perturbation expansion,$${\displaystyle |n⟩=\sum _{k\in D}{\alpha }_{nk}|k^{(0)}⟩+\lambda |n^{(1)}⟩.}$$

For the first-order perturbation, we need solve the perturbed Hamiltonian restricted to the degenerate subspaceD,$${\displaystyle V|k^{(0)}⟩={ϵ}_k|k^{(0)}⟩+\text{small}\mathrm{\forall }|k^{(0)}⟩\in D,}$$simultaneously for all the degenerate eigenstates, where${\displaystyle {ϵ}_k}$ are first-order corrections to the degenerate energy levels, and "small" is a vector of${\displaystyle O(\lambda )}$ orthogonal toD. This amounts to diagonalizing the matrix$${\displaystyle ⟨k^{(0)}|V|l^{(0)}⟩=V_{kl}\mathrm{\forall }|k^{(0)}⟩,|l^{(0)}⟩\in D.}$$

This procedure is approximate, since we neglected states outside theD subspace ("small"). The splitting of degenerate energies${\displaystyle {ϵ}_k}$ is generally observed. Although the splitting may be small,${\displaystyle O(\lambda )}$, compared to the range of energies found in the system, it is crucial in understanding certain details, such as spectral lines inElectron Spin Resonance experiments.

Higher-order corrections due to other eigenstates outsideD can be found in the same way as for the non-degenerate case,$${\displaystyle \left(E_n^{(0)}-H_0\right)|n^{(1)}⟩=\sum _{k\notin D}\left(⟨k^{(0)}|V|n^{(0)}⟩\right)|k^{(0)}⟩.}$$

The operator on the left-hand side is not singular when applied to eigenstates outsideD, so we can write$${\displaystyle |n^{(1)}⟩=\sum _{k\notin D}\frac{⟨k^{(0)}|V|n^{(0)}⟩}{E_n^{(0)}-E_k^{(0)}}|k^{(0)}⟩,}$$but the effect on the degenerate states is of${\displaystyle O(\lambda )}$.

Near-degenerate states should also be treated similarly, when the original Hamiltonian splits aren't 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 eigenstates will only shift the absolute energy of all near-degenerate states simultaneously.

Let us consider degenerate energy eigenstates and a perturbation that completely lifts the degeneracy to first order of correction.

The perturbed Hamiltonian is denoted as$${\displaystyle \hat{H}={\hat{H}}_0+\lambda \hat{V},}$$where${\displaystyle {\hat{H}}_0}$ is the unperturbed Hamiltonian,${\displaystyle \hat{V}}$ is the perturbation operator, and is the parameter of the perturbation.

Let us focus on the degeneracy of the${\displaystyle n}$-th unperturbed energy${\displaystyle E_n^{(0)}}$. We will denote the unperturbed states in this degenerate subspace as${\displaystyle |{\psi }_{nk}^{(0)}⟩}$ and the other unperturbed states as${\displaystyle |{\psi }_m^{(0)}⟩}$, where${\displaystyle k}$ is the index of the unperturbed state in the degenerate subspace and${\displaystyle m\ne n}$ represents all other energy eigenstates with energies different from${\displaystyle E_n^{(0)}}$. The eventual degeneracy among the other states with${\displaystyle \mathrm{\forall }m\ne n}$ does not change our arguments. All states${\displaystyle |{\psi }_{nk}^{(0)}⟩}$ with various values of${\displaystyle k}$ share the same energy${\displaystyle E_n^{(0)}}$ when there is no perturbation, i.e., when${\displaystyle \lambda =0}$. The energies${\displaystyle E_m^{(0)}}$ of the other states${\displaystyle |{\psi }_m^{(0)}⟩}$ with${\displaystyle m\ne n}$ are all different from${\displaystyle E_n^{(0)}}$, but not necessarily unique, i.e. not necessarily always different among themselves.