The classical S-matrix: A more detailed study of classically forbidden transitions in inelastic collisions

Procedures are investigated for describing classically forbidden collision processes within the framework of the classical limit of quantum mechanics. The goal is to use exact classical mechanics (numerically computed trajectories) to treat classically forbidden transitions in a complex collision (such as an atom plus diatom) in a manner analogous to the way one uses it to treat barrier transmission by a single particle in one dimension (a classically forbidden process); in this latter example the transmission coefficient is exp (-2 theta), being the classical action integral through the barrier (a classically forbidden region). Numerical application of these procedures is made to the linear atom-diatom collision system (without reaction), and the resulting vibrational transition probabilities for the classically forbidden transitions are in as good agreement with the exact quantum mechanical values as are the transition probabilities for classically allowed transitions. It appears. therefore. that the dynamics of even these classically forbidden transitions is accurately described in this semiclassical framework.