CHAPTER 4 - PERTURBATION THEORY

Abstract

This chapter discusses perturbation theory. It describes perturbations independent of time, the secular equation, perturbations depending on time, transitions in the continuous spectrum, intermediate states, the uncertainty relation for energy, and quasi-stationary states. The exact solution of Schrodinger's equation can be found only in a comparatively small number of the simplest cases. The majority of problems in quantum mechanics lead to equations which are too complex to be solved exactly. However, often quantities of different orders of magnitude appear in the conditions of the problem; among them, there may be small quantities such that, when they are neglected, the problem is so much simplified that its exact solution becomes possible. In such cases, the first step in solving the physical problem concerned is to solve exactly the simplified problem, and the second step is to calculate approximately the errors due to the small terms that have been neglected in the simplified problem. There is a general method of calculating these errors; it is called perturbation theory. One of the most important applications of perturbation theory is to calculate the probability of a transition between states of a continuous spectrum under the action of a constant (time-independent) perturbation. This includes various collision processes, where the system in the initial and final states is an assembly of colliding particles, and the perturbation is represented by the interaction between them. The phenomena to which the method described below applies also include processes where a system in a bound state disintegrates into freely moving parts.