Theory of Semiclassical Transition Probabilities (S Matrix) for Inelastic and Reactive Collisions

In the present paper the time-independent Schrödinger equation for inelastic collisions is solved directly in the WKB approximation, using action-angle variables and the method of characteristics. A single wavefunction, consisting of an ingoing and an outgoing term, is thereby derived, describing all collision channels and so avoiding the application of WKB methods to an infinite set of coupled differential equations. An integral is obtained for the S matrix, and asymptotic methods (e.g., steepest descents, stationary phase) are used for its evaluation. The expressions can be calculated using numerical data on classical trajectories or using approximations. To facilitate the latter and to show the connection with approximations in the literature, a canonical perturbation theory is described for the wave phase and amplitude, in a form suited to collisions, and used to relate the theory to those approximations. The topic of collisional selection rules is also considered. The extension of the method employed in the present paper to the direct calculation of differential and total inelastic cross sections, rather than via the S matrix, is briefly described, and the extension to reactive cross sections is also noted. The method can also be used to treat time-dependent problems, and so is not restricted to collisions. These topics and other applications will be described in later papers of this series.