Perturbation theory (quantum mechanics)

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.

The energy levels and eigenstates of the perturbed Hamiltonian are again given by the time-independent Schrödinger equation,

Substituting the power series expansion into the Schrödinger equation produces:

This is simply the expectation value of the perturbation Hamiltonian while the system is in the unperturbed eigenstate.

Before corrections to the energy eigenstate are computed, the issue of normalization must be addressed. Supposing that

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:

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

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

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

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

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.

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

Notice that here the first order correction to the state is orthogonal to the unperturbed state,

The validity of the 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:

The second Hellmann–Feynman theorem gives the derivative of the state (resolved by the complete basis with m ≠ n),

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 Mathematica.

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:

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,

Note that in the second term, the 1/2! factor exactly cancels the double contribution due to the time-ordering operator, etc.

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

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

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.

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

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

Example of first and second order perturbation theory – quantum pendulum

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

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