Semiclassical Theory of Atom–Diatom Collisions: Path Integrals and the Classical S Matrix

The aim of this work is to show how one can use exact solutions of the classical equations of motion (numerically obtained trajectories) to construct the corresponding classical approximation to the time-independent S-matrix elements for use in quantum mechanical expressions for cross sections; it is argued that this should accurately describe many quantum effects in heavy particle collisions. The expression for the S matrix in terms of the classical trajectory is given for systems of any number of degrees of freedom, and the matter is pursued in detail for the A + BC collision system. It is shown that within this classical limit the magnitude of an S-matrix element is explicitly determined by its phase. Constancy of total angular momentum is used throughout to reduce the 12 first-order differential equations of the A + BC system in its center of mass to eight equations. A practical method is also given (in Appendix B) for further reducing the number of coupled equations to six, the minimum number possible.