Common trajectory methods for the calculation of differential cross sections for inelastic transitions in atom(ion)-atom collisions

Starting from a partial wave decomposition of the quantum-mechanical equations, a systematic development of the semiclassical theory of inelastic transitions in slow atom(ion)-atom collisions is presented. Both atomic and molecular representations are discussed. It is shown that the introduction of an average potential reduces the semiclassical equations to the well known impact parameter equations; the notions of the classical trajectory and the classical collision plane arise as a natural consequence of the semiclassical approximation. Expressions for the differential scattering cross sections for inelastic transitions are subsequently derived. The connection with the straight line eikonal approximation is discussed.