Exponential approximations in collision theory

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Abstract
The exponential form of the scattering matrix, S=exp (i A/ħ), is used to obtain a quantitative and qualitative formulation of quantal collision theory in the short wavelength regime. It is shown that to the lowest order in ħ and in the strength of the potential, A is determined by the usual first-order perturbation theory integrals, and that the exponential form above provides the proper way of extending perturbation theory in the short wavelength limit. When a classical trajectory for the relative motion can be defined, A can be obtained using semi-classical, first-order perturbation theory. Here A is the matrix representation of the classical action along the collision trajectory. In the weak coupling regime, the exponential form reduces to the usual results of first-order perturbation theory. For limited coupling, it is shown that perturbation theory correctly determines the selection rules and the ratios of the transition probabilities, S≅a0 I+a1 A, where a 0 and a 1 are given numbers, of magnitude below unity. Since the largest cross sections are for those transitions that contribute in the weak and limited coupling regimes, perturbation theory determines the propensity rules. In the dominant coupling regime, statistical considerations apply to the averaged probability, which can be written as a classical sum of terms, the interference terms being quenched out in the averaging process. The action criterion for determining the coupling regime and the magnitude of the transition probability is discussed with an example. Here one examines the ratio of the (classical) action required for the transition to the action actually available during the collision, large ratios corresponding to adiabatic, weakly coupled collisions, while the opposite holds for impulsive, dominantly coupled transitions. A quantitative version of the action criterion provides a direct, simple way of determining the energy (and other parameters) dependence of the inelastic cross sections. In the correspondence principle limit of the theory, it is shown that the change in an observable (say, internal energy) due to collisions can be written as a classical sum (i.e. integral) over the different contributions, without any quantal interference terms.