Inelastic Scattering: Coupled-State Approach

Abstract

The semi classical method for elastic scattering described in the previous chapter can be generalized to obtain the wave functions for inelastic transitions in a semiclassical approximation. We approach the problem from the viewpoint of the time-independent theory. The question then centers around how one might obtain the WKB wave functions for a system of coupled Schrödinger equations and construct the S matrix therefrom. There are a number of possible ways to go about it; they may be classified into two basically distinctive approaches. In one method the coupled Schrödinger equations are converted into a set of coupled (nonlinear) Riccati equations [5.1–5] generalizing the Riccati equation arising in the phase-amplitude method for one-dimensional Schrödinger equations. Alternatively, they are converted into coupled integral equations for generalized phases [5.6,7]. In the other method, the coupled second-order equations are converted into coupled first-order linear differential equations. When they are exactly solved, both methods should give the same results, but since that is not the case, there may be numerical differences in their results. Although there has never been a numerical comparison made between the two approaches, the latter approach seems to have an advantage in that it is easier to deal with linear equations than nonlinear ones. For this reason and also for lack of space we shall consider only the linear differential equation approach.