Perturbation Theory

Abstract

In previous chapters we have seen an important set of systems whose Schrödinger’s equations can be solved analytically. There is, however, an even larger number of quantum systems whose Schrödinger’s equations can not be solved analytically. Generally one has to use approximate methods like the WKB approximation, the perturbation method, and numerous numerical methods. In this chapter we will study only the fundamentals of the perturbation theory, and the interaction representation. The perturbation theory can be used to approach the correct solution, when part of the Hamiltonian is characterized by a small perturbation parameter $\eta$, such that, neglecting the perturbation part $\widehat{V}_p=\eta \widehat{U}$, one is left with a soluble problem for the Hamiltonian ${\widehat{H}_o}={\widehat{H}}-\widehat{V}_p$. When this is the case, the first step of the perturbation method is to solve the problem for ${\widehat{H}_o}$.