Theory of Inelastic Collisions: Uniform Asymptotic (WKB) Solutions and Semiclassical S-Matrix Elements for Two-Channel Problems

Abstract

Semiclassical scattering matrix elements for two-channel problems are obtained by means of uniform asymptotic (WKB) solutions and solving a set of coupled first-order differential equations generated from the radial Schrödinger equations. By solving such first-order coupled equations, Stueckelberg's mathematically unsatisfactory procedure of tracing solutions in a complex plane to obtain the S-matrix elements is completely removed and the expression thus obtained for the transition probability is devoid of physically unreasonable shortcomings of Stueckelberg's result. The S-matrix elements obtained are given in terms of well-defined quadratures easy to calculate and valid for any types of the interaction matrix elements, Vii (x) and w (x). The transition probability obtained from the S-matrix elements is shown to reduce essentially to the well-known Landau—Zener expression as a limiting case when w (x) is small.