Perturbation Theory

Most problems encountered in Quantum Mechanics cannot be solved exactly. The solution of the Schrodinger Equation for real systems is usually quite complicated and most of the time no exact mathematical solution can be found. One way to get around this difficulty is to use numerical methods. This is generally very tedious and also not very illuminating because a ready comparison between slightly different systems is not possible. The other procedure for handling systems which cannot be solved exactly, is using approximation methods. A variety of such methods have been developed. The most powerful approximation method used in Quantum Mechanics is the Perturbation Theory. 

Perturbation Theory is based on the assumption that the problem we wish to solve is, in some sense, only slightly different from a problem that can be solved exactly. Perturbation Theory is a systematic procedure for obtaining approximate solutions to the perturbed problem by building on the known exact solutions to the unperturbed case.

1)

The Schrodinger Equation for hydrogen atom can be solved exactly but if we apply an electric or magnetic field then we'll have to consider an added perturbation due to these fields and the eigen-values and eigen-functions of the total Hamiltonian would then be obtained by the use of Perturbation Theory.

2)

The Schrodinger Equation for Harmonic Oscillator potential can be solved exactly and the influence of the an-harmonic perturbation would be determined by the use of Perturbation Theory.     

Suppose we have solved the (Time independent) Schrodinger Equation for some potential (say the one dimensional infinite square well)  

Obtaining a complete set of ortho-normal eigen-function ψ⁰ₙ

This is the fundamental result of first order Perturbation Theory; as a practical matter, it may well be the most important equation in quantum mechanics. It says that the first order correction to the energy is the expectation value of the perturbation in the unperturbed state. This is given explicitly as 

To find the energy eigenvalues and eigenfunctions of the total Hamiltonian H, consider Eq. (12)

This too is a very important formula, especially since the first order shift frequently vanishes on grounds of symmetry. We may interpret the formula as follows:

The second order energy shift is the sum of terms, whose strength is given by the square of the matrix element connecting the given state (Phi-n)to all other states by the perturbing potential, weighted by the reciprocal of the energy difference between the states. We can draw several conclusions from the formula: