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, known system. The idea is to start with a simple system for which a mathematical solution is known (e.g. the time-independent Schrödinger equation) and add an additional "perturbing" Hamiltonian). If the disturbance is small, the new energy levels and eigenstates of the perturbed system can be expressed as "corrections" to the known energy levels and eigenstates of the simpler system. These corrections can be made at progressively smaller orders of magnitude until an n-th order that is so small a correction that it is negligible to the accuracy of the approximation.

Approximate Hamiltonians

Perturbation theory is an important tool for describing quantum systems without exact solutions. Systems with known exact solutions, such as the hydrogen atom, the quantum harmonic oscillator, and the particle in a box, are idealized and may not adequately describe related systems. Perturbation theory uses known solutions to generate solutions for more complicated systems. The procedure uses approximations to the system Hamiltonian.

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 exactly solvable problem.

For example, by adding a perturbative electric potential to the quantum mechanical model of the hydrogen atom, tiny shifts in the spectral lines of hydrogen caused by the presence of an electric field (the Stark effect) can be calculated. This is only approximate because the sum of a Coulomb potential with a linear potential is unstable (has no true bound states) although the tunneling 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, say , is very small. Typically, the results are expressed in terms of finite power series in that seem to converge to the exact values when summed to higher order. After a certain order 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 the variational 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.

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. In QED and other quantum field theories, special calculation techniques known as Feynman diagrams are used to systematically sum the power series terms.

Limitations

Large perturbations

Some systems cannot be described by a small perturbation imposed on some simple system. Perturbation theory requires small perturbations. 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, violating the requirement that corrections must be small.

Non-adiabatic states

Perturbation theory also fails to describe states that are not generated adiabatically from the "free model", including bound states and various collective phenomena such as solitons. A system of free (i.e. non-interacting) particles, to which an attractive interaction is introduced, 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 conventional superconductivity, in which the phonon-mediated attraction between conduction electrons leads to the formation of correlated electron pairs known as Cooper pairs. Perturbation theory fails 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. Other approximation schemes, such as the variational method and the WKB approximation may apply to these cases.

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 non-perturbative 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.

Time-independent perturbation theory

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 by Erwin Schrödinger in a 1926 paper, shortly after he produced his theories in wave mechanics. In this paper Schrödinger referred to earlier work of Lord Rayleigh, 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. Time-independent perturbation theory can itself be separated into non-degenerate and degenerate perturbation theory.

Non-degenerate perturbation theory

==== First order corrections ==== The process begins with an unperturbed Hamiltonian , which is assumed to have no time dependence. It has known energy levels and eigenstates, arising from the time-independent Schrödinger equation:

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

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

The objective is to express 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 in

When , 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:

Expanding this equation and comparing coefficients of each power of results in an infinite series of simultaneous equations

, the first term on the left-hand side cancels the first term on the right-hand side. (Recall, the unperturbed Hamiltonian is Hermitian This is simply the expectation value of the perturbation Hamiltonian while the system is in the unperturbed eigenstate.

. These further shifts are given by the second and higher order corrections to the energy.

.

Then at first order in

The 2 term is dropped in first-order expansions.

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

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 of .

The first-order correction to the energy eigenstate can also be obtained through the following considerations. By using the resolution of the identity are in the orthogonal complement, i.e., the other eigenvectors.

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

. Thus, in the

The first-order change in the -th energy eigenstate has a contribution from each of the other energy eigenstates , which is a measure of how much the perturbation mixes eigenstate with eigenstate ; it is also inversely proportional to the energy difference between eigenstates and , 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 state ==== Second-order and higher-order corrections ==== Higher-order deviations are calculated by a similar procedure, though the calculations become quite tedious within this formulation. Normalization convention, of the whole state vectors of the perturbed Schrodinger equation being orthonormal, gives

We will bring a phase to the whole state vector of the perturbed Schrodinger equation which will lead to the first term having only a real part. Up to second order, the expressions for the energies and (normalized) eigenstates are:

), 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

It is possible to relate the k-th order correction to the energy to the -point connected correlation function of the perturbation , one has to consider the inverse Laplace transform is the perturbing operator

then the -th order energy shift is given by

Degenerate perturbation theory

Suppose that two or more energy eigenstates of the unperturbed Hamiltonian 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.

does not have a well-defined inverse.

Let denote the subspace spanned by these degenerate eigenstates. No matter how small the perturbation is, in the degenerate subspace the energy differences between the eigenstates of 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 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 . The new Hamiltonian must be diagonalized in , or a slight variation of D, so to speak. These perturbed eigenstates in

For the first-order perturbation, we need solve the perturbed Hamiltonian restricted to the degenerate subspace orthogonal to D

This procedure is approximate, since we neglected states outside the , 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.

Higher-order corrections due to other eigenstates outside

The operator on the left-hand side is not singular when applied to eigenstates outside .

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 the nearly 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.

Degeneracy lifted to first order

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

is the parameter of the perturbation.

, but not necessarily unique, i.e. not necessarily always different among themselves.

Generalization to multi-parameter case

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.

Hamiltonian and force operator

From the differential geometric point of view, a parameterized Hamiltonian is considered as a function defined on the parameter manifold to an Hermitian operator that acts on the Hilbert space. The parameters here can be external field, interaction strength, or driving parameters in the quantum phase transition. Let be the -th eigenenergy and eigenstate of form a vector bundle, how to estimate the at close to that reference point.

If the parameters are considered as generalized coordinates, then should be identified as the generalized force operators related to those coordinates. Different indices label the different forces along different directions in the parameter manifold. For example, if denotes the external magnetic field in the -direction, then should be the magnetization in the same direction.

Perturbation theory as power series expansion

The validity of perturbation theory lies on the adiabatic 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 (like Taylor expansion) of the parameters:

Here denotes the derivative with respect to , it should be understood as the covariant derivative if the vector bundle is equipped with non-vanishing connection. All the terms on the right-hand-side of the series are evaluated at , e.g. . 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 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.

Hellmann–Feynman theorems

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

The second Hellmann–Feynman theorem gives the derivative of the state (resolved by the complete basis with

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

The theorems can be simply derived by applying the differential operator to both sides of the Schrödinger equation which reads

again,

, the cases of and 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 systematically.

Correction of energy and state

To the second order, the energy correction reads

denotes the real part function. The first order derivative is given by the first Hellmann–Feynman theorem directly. To obtain the second order derivative , simply applying the differential operator , which reads

Note that for a linearly parameterized Hamiltonian, there is no second derivative according to the definition of the connection for the vector bundle. Therefore, the case can be excluded from the summation, which avoids the singularity of the energy denominator. The same procedure can be carried on for higher order derivatives, from which higher order corrections are obtained.

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.

Effective Hamiltonian

Let , such that there is no matrix element in . Let 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

.

In a formal way it is possible to define an effective Hamiltonian that gives exactly the low-lying energy states and wavefunctions. In practice, some kind of approximation (perturbation theory) is generally required.

Time-dependent perturbation theory

Method of variation of constants

Time-dependent perturbation theory, initiated by Paul Dirac and further developed by John Archibald Wheeler, Richard Feynman, and Freeman Dyson, studies the effect of a time-dependent perturbation applied to a time-independent Hamiltonian . It is an extremely valuable tool for calculating the properties of any physical system. It is used for the quantitative description of phenomena as diverse as proton-proton scattering, photo-ionization of materials, scattering of electrons off lattice defects in a conductor, scattering of neutrons off nuclei, electric susceptibilities of materials, neutron absorption cross sections in a nuclear reactor, and much more.

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 interested in the following quantities:

• The time-dependent expectation value of some observable , for a given initial state.

• The time-dependent expansion coefficients (w.r.t. a given time-dependent state) of those basis states that are energy eigenkets (eigenvectors) in the unperturbed system.

The first quantity is important because it gives rise to the classical result of an measurement performed on a macroscopic number of copies of the perturbed system. For example, we could take to be the displacement in the -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 polarization of a hydrogen gas. With an appropriate choice of perturbation (i.e. an oscillating electric potential), this allows one to calculate the AC permittivity of the gas.

The second quantity looks at the time-dependent probability of occupation for each eigenstate. This is particularly 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 probabilities are also useful for calculating the "quantum broadening" of spectral lines (see line broadening) and particle decay in particle physics and nuclear physics.

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.)

at time = 0, its state at subsequent times varies only by a phase (in the Schrödinger picture

Now, introduce a time-dependent perturbing Hamiltonian denote the quantum state of the perturbed system at time

The quantum state at each instant can be expressed as a linear combination: where the s are to be determined complex functions of which we will refer to as amplitudes (strictly speaking, they are the amplitudes in the Dirac picture).

and no perturbation is present, the amplitudes have the convenient property that, for all , = 1 and = 0 if .

The square of the absolute amplitude is the probability that the system is in state at time

Plugging into the Schrödinger equation and using the fact that ∂/∂t acts by a product rule

By resolving the identity in front of and multiplying through by the bra on the left, this can be reduced to a set of coupled differential equations

where we have used equation () to evaluate the sum on .

The matrix elements of 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 modified by the exponential phase factor. Over times much longer than the energy difference , the phase winds around 0 several times. If the time-dependence of is sufficiently slow, this may cause the state amplitudes to oscillate. (For example, such oscillations are useful for managing radiative transitions in a laser.)

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

Repeatedly substituting this expression for

Several further results follow from this, such as Fermi's golden rule, which relates the rate of transitions between quantum states to the density of states at particular energies; or the Dyson series, obtained by applying the iterative method to the time evolution operator, which is one of the starting points for the method of Feynman diagrams.

Method of Dyson series

Time-dependent perturbations can be reorganized through the technique of the Dyson series. The Schrödinger equation where Thus, the exponential represents the following Dyson series Note that in the second term, the 1/2! factor exactly cancels the double contribution due to the time-ordering operator, etc.

assuming that the parameter has been solved.

Perform the following unitary transformation to the interaction picture Consequently, the Schrödinger equation so it is solved through the above Dyson series as a perturbation series with small .

as detailed in the previous section——while the corresponding transition probability to a continuum is furnished by Fermi's golden rule.

As an aside, note that time-independent perturbation theory is also organized inside this time-dependent perturbation theory Dyson series. To see this, write the unitary evolution operator, obtained from the above Dyson series and take the perturbation to be time-independent.

in the integrands for arbitrarily small. Thus the limit .)

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.

Strong perturbation theory

In a similar way as for small perturbations, it is possible to develop a strong perturbation theory. Consider as usual the Schrödinger equation

and we consider the question if a dual Dyson series exists that applies in the limit of a perturbation increasingly large. This question can be answered in an affirmative way and the series is the well-known adiabatic series. This approach is quite general and can be shown in the following way. Consider the perturbation problem

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

producing the following meaningful equations

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 does not depend on time one gets the Wigner-Kirkwood series that is often used in statistical mechanics. Indeed, in this case we introduce the unitary transformation

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 the Schrödinger equation

and we see that the expansion parameter appears only into the exponential and so, the corresponding Dyson series, a dual Dyson series, is meaningful at large s and is

justifying in this way the name of dual Dyson series. The reason is that we have obtained this series simply interchanging and and we can go from one to another applying this exchange. This is called duality principle yields, as already said, a Wigner-Kirkwood series that is a gradient expansion. The Wigner-Kirkwood series is a semiclassical series with eigenvalues given exactly as for WKB approximation.

Examples

Example of first-order perturbation theory – ground-state energy of the quartic oscillator

Example of first- and second-order perturbation theory – quantum pendulum

Potential energy as a perturbation

, the solution of the Schrödinger equation whose particular integral is is the Hankel function of the first kind.

Applications

References

External links

• (lecture by Barton Zwiebach)

• Quantum Physics Online - perturbation theory