10.1 Introduction

Often there are situations in which the solutions to the time-independent Schr dinger equation are known for a particular potential but not for a similar but different potential. Time-independent perturbation theory provides a means of finding approximate solutions using an expansion in the known eigenfunctions.

As an example, consider the one-dimensional, rectangular potential well with infinite barrier energy shown in Fig. 10.1. The width of the well is L, and the potential energy is V( x) = 0 for 0 < x < L and V( x) = ? for 0 ? x ? L .

Fig. 10.1: Sketch of a one-dimensional, rectangular potential well with infinite barrier energy. The width of the well is L, and the potential energy is V(x) = 0 for 0 < x < L and V(x) = ? for 0 ? x ? L. The known eigenfunction for this potential can be used to obtain approximate solutions in the presence of a small deformation in the potential.

The time-independent Schr dinger equation for a particle of mass m in the potential V( x) is

The solutions to this equation have eigenfunctions

and eigenenergy

where

and the value of n takes an integer value so that n = 1 , 2 , 3 ,

We now suppose that the potential V( x) is deformed by the presence of an additional term W(