Approximation Methods for Stationary States

Abstract

Although we have succeeded in solving some important and interesting quantum mechanical problems, an exact solution is not possible in complicated situations, and we must then resort to approximation methods. For the calculation of stationary states and energy eigenvalues, these include perturbation theory, the variational method , and the WKB approximation. Perturbation theory is applicable if the problem differs from an exactly solvable problem by a small amount. The variational method is appropriate for the calculation of the ground state energy if one has a qualitative idea of the form of the wave function, and finally the WKB method is applicable in the nearly classical limit. 11.1 Time Independent Perturbation Theory (Rayleigh-Schrödinger) Let the Hamiltonian H consist of two parts, H = H 0 + λH 1. (11.1) Let the eigenvalues E 0 n and eigenfunctions |n 0 of the operator H 0 be known exactly, H 0 |n 0 = E 0 n |n 0 , (11.2) and the "perturbation" λH 1 be, in comparison to H 0 , a small additional term. One seeks the discrete stationary states |n and eigenvalues E n of H, H|n = E n |n. (11.3) We now assume that the eigenvalues and eigenfunctions can be expanded in a power series in the parameter λ E n = E 0 n + λE 1 n + λ 2 E 2 n +. .. , |n = |n 0 + λ|n 1 + λ 2 |n 2 +. .. , (11.4) where in both expansions the first term is the "unperturbed" one.