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. In effect, it is describing a complicated unsolved system using a simple, solvable system.
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.
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 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
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. Even convergent perturbations can converge to the wrong answer and divergent perturbations expansions can sometimes give good results at lower order.
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, violating the requirement that corrections must be small.
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. 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 conventional
superconductivity, in which the
phonon-mediated attraction between
conduction electrons leads to the formation of correlated electron pairs known as
Cooper pairs. When faced with such systems, one usually turns to other approximation schemes, such 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.
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.
First order corrections
The process begins with an unperturbed Hamiltonian H0, which is assumed to have no time dependence. It has known energy levels and
eigenstates, arising from the time-independent
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 of
A perturbation is then introduced 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:
The energy levels and eigenstates of the perturbed Hamiltonian are again given by the time-independent Schrödinger equation,
The objective is to express En and 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 λ,
When k = 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:
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 system,
The first-order equation is
Operating through by , the first term on the left-hand side cancels the first term on the right-hand side. (Recall, the unperturbed Hamiltonian is
Hermitian). This leads to the first-order energy shift,
This is simply the
expectation 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 , which is a valid quantum state though no longer an energy eigenstate. The perturbation causes the average energy of this state to increase by . However, the true energy shift is slightly different, because the perturbed eigenstate is not exactly the same as . 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
but perturbation theory also assumes that .
Then at first order in λ, the following must be true:
Since the overall phase is not determined in quantum mechanics,
without loss of generality, in time-independent theory it can be assumed that is purely real. Therefore,
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 λ. Then by using the
resolution of the identity:
Supposing that the zeroth-order energy level is not
degenerate, i.e. that there is no eigenstate of H0 in the orthogonal complement of with the energy . After renaming the summation dummy index above as , any can be chosen and multiplying the first-order equation through by gives
The above also gives us the component of the first-order correction along .
Thus, in total, the result is,
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 , which is a measure of how much the perturbation 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 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 n, 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.,
Second-order and higher-order 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 gives that
Up to second order, the expressions for the energies and (normalized) eigenstates are:
If an intermediate normalization is taken (it means, if we require that ), we obtain the same expression for the second-order correction to the wave function, except for the last term.
Extending the process further, the third-order energy correction can be shown to be 
Corrections to fifth order (energies) and fourth order (states) in compact notation
If we introduce the notation,
then the energy corrections to fifth order can be written
and the states to fourth order can be written
All terms involved kj should be summed over kj such that the denominator does not vanish.
It is possible to relate the k-th order correction to the energy En to the k-point
connected correlation function of the perturbation V in the state . For , one has to consider the inverse
Laplace transform of the two-point correlator:
where is the perturbing operator V in the interaction picture, evolving in Euclidean time. Then
Similar formulas exist to all orders in perturbation theory, allowing one to express in terms of the inverse Laplace transform of the connected correlation function
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.
This is manifested in the calculation of the perturbed eigenstate via the fact that the operator
does not have a well-defined inverse.
Let D denote the subspace spanned by these degenerate eigenstates. No matter how small the perturbation is, in the degenerate subspace D the energy differences between the eigenstates of H 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 D. The new Hamiltonian must be diagonalized in D, or a slight variation of D, so to speak. These perturbed eigenstates in D are now the basis for the perturbation expansion,
For the first-order perturbation, we need solve the perturbed Hamiltonian restricted to the degenerate subspace D,
simultaneously for all the degenerate eigenstates, where are first-order corrections to the degenerate energy levels, and "small" is a vector of orthogonal to D. This amounts to diagonalizing the matrix
This procedure is approximate, since we neglected states outside the D subspace ("small"). The splitting of degenerate energies 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.
Higher-order corrections due to other eigenstates outside D can be found in the same way as for the non-degenerate case,
The operator on the left-hand side is not singular when applied to eigenstates outside D, so we can write
but the effect on the degenerate states is of .
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.
The perturbed Hamiltonian is denoted as
where is the unperturbed Hamiltonian, is the perturbation operator, and is the parameter of the perturbation.
Let us focus on the degeneracy of the -th unperturbed energy . We will denote the unperturbed states in this degenerate subspace as and the other unperturbed states as , where is the index of the unperturbed state in the degenerate subspace and represents all other energy eigenstates with energies different from . The eventual degeneracy among the other states with does not change our arguments. All states with various values of share the same energy when there is no perturbation, i.e., when . The energies of the other states with are all different from , but not necessarily unique, i.e. not necessarily always different among themselves.
By and , we denote the matrix elements of the perturbation operator in the basis of the unperturbed eigenstates. We assume that the basis vectors in the degenerate subspace are chosen such that the matrix elements are diagonal. Assuming also that the degeneracy is completely lifted to the first order, i.e. that if , we have the following formulae for the energy correction to the second order in
and for the state correction to the first order in
Notice that here the first order correction to the state is orthogonal to the unperturbed state,
Generalization to multi-parameter case
The generalization of time-independent perturbation theory to the case where there are multiple small parameters 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 that maps each particular set of parameters 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 the
quantum phase transition. Let En(x μ) and be the n-th eigenenergy and eigenstate of H(x μ) respectively. In the language of differential geometry, the states 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 question: given and at an unperturbed reference point , how to estimate the En(x μ) and at x μ close to that reference point.
Without loss of generality, the coordinate system can be shifted, such that the reference point is set to be the origin. The following linearly parameterized Hamiltonian is frequently used
If the parameters x μ are considered as generalized coordinates, then Fμ should be identified as the generalized force operators related to those coordinates. Different indices μ label the different forces along different directions in the parameter manifold. For example, if x μ denotes the external magnetic field in the μ-direction, then Fμ 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 ∂μ denotes the derivative with respect to x μ. When applying to the state , 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 x μ = 0, e.g. En ≡ En(0) and . 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.
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,
The second Hellmann–Feynman theorem gives the derivative of the state (resolved by the complete basis with m ≠ n),
For the linearly parameterized Hamiltonian, ∂μH simply stands for the generalized force operator Fμ.
The theorems can be simply derived by applying the differential operator ∂μ to both sides of the
Schrödinger equation which reads
Then overlap with the state from left and make use of the Schrödinger equation again,
Given that the eigenstates of the Hamiltonian always form an orthonormal basis , 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 systematically.
Correction of energy and state
To the second order, the energy correction reads
where denotes the
real part function.
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 , which reads
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,
then all parts can be calculated using the Hellmann–Feynman theorems. In terms of Lie derivatives, 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 derivatives, from which higher order corrections are obtained.
The same computational scheme is applicable for the correction of states. The result to the second order is as follows
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
Let H(0) be the Hamiltonian completely restricted either in the low-energy subspace or in the high-energy subspace , such that there is no matrix element in H(0) connecting the low- and the high-energy subspaces, i.e. if . 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
Here are restricted in the low energy subspace. The above result can be derived by power series expansion of .
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, developed by
Paul 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 interested in the following quantities:
expectation value of some observable A, 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 A measurement performed on a macroscopic number of copies of the perturbed system. For example, we could take A to be the displacement 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 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
We will briefly examine the method behind Dirac's formulation of time-dependent perturbation theory. Choose an energy basis 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 an eigenstate (of the Hamiltonian) at time t = 0, its state at subsequent times varies only by a
phase (in the
Schrödinger picture, where state vectors evolve in time and operators are constant),
Now, introduce a time-dependent perturbing Hamiltonian V(t). The Hamiltonian of the perturbed system is
Let denote the quantum state of the perturbed system at time t. It obeys the time-dependent Schrödinger equation,
The quantum state at each instant can be expressed as a linear combination of the complete eigenbasis of :
where the cn(t)s are to be determined
complex functions of t which we will refer to as amplitudes (strictly speaking, they are the amplitudes in the
We have explicitly extracted the exponential phase factors 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 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 probability that the system is in state n at time t, since
Plugging into the Schrödinger equation and using the fact that ∂/∂t acts by a
product rule, one obtains
By resolving the identity in front of V and multiplying through by the
bra on the left, this can be reduced to a set of coupled
differential equations for the amplitudes,
where we have used equation (1) to evaluate the sum on n in the second term, then used the fact that .
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 modified 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 amplitudes to oscillate. ( E.g., such oscillations are useful for managing radiative transitions in a
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
However, exact solutions are difficult to find when there are many energy levels, and one instead looks for perturbative solutions. These may be obtained by expressing the equations in an integral form,
Repeatedly substituting this expression for cn back into right hand side, yields an iterative solution,
where, for example, the first-order term is
To the same approximation, the summation in the above expression can be removed since in the unperturbed state so that we have
Using the solution of the unperturbed problem and (for the sake of simplicity assume a pure discrete spectrum), yields, to first order,
Thus, the system, initially in the unperturbed state , by dint of the perturbation can go into the state . The corresponding transition probability amplitude to first order is
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, as
and take the perturbation V to be time-independent.
Using the identity resolution
with for a pure discrete spectrum, write
It is evident that, at second order, one must sum on all the intermediate states. Assume and the asymptotic limit of larger times. This means that, at each contribution of the perturbation series, one has to add a multiplicative factor 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
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 (.)
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
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 λ→ ∞. Our aim is to find a solution in the form
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 producing the following meaningful equations
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
and we see that the expansion parameter λ appears only into the exponential and so, the corresponding
Dyson series, a dual Dyson series, is meaningful at large λs and is
After the rescaling in time we can see that this is indeed a series in 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 exchange. This is called duality principle in perturbation theory. The choice 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
Example of first-order perturbation theory – ground-state energy of the quartic oscillator
Consider the quantum harmonic oscillator with the quartic potential perturbation and the Hamiltonian
The ground state of the harmonic oscillator is
(), and the energy of unperturbed ground state is
Using the first-order correction formula, we get
Example of first- and second-order perturbation theory – quantum pendulum
Consider the quantum-mathematical pendulum with the Hamiltonian
with the potential energy taken as the perturbation i.e.
The unperturbed normalized quantum wave functions are those of the rigid rotor and are given by
and the energies
The first-order energy correction to the rotor due to the potential energy is
Using the formula for the second-order correction, one gets
Potential energy as a perturbation
When the unperturbed state is a free motion of a particle with kinetic energy , the solution of the
corresponds to plane waves with wavenumber . If there is a weak potential energy present in the space, in the first approximation, the perturbed state is described by the equation