On the WKB Approximation in Time-Dependent Scattering Theory Including Rearrangement Processes

The classical correspondence of the S matrix for scattering processes is examined with a view to obtain the classical approximation and also to clarify the nature of the approximations required for the classical formula. In this discussion the Schrödinger equation in the momentum representation is employed instead of the usual coordinate representation in order to provide a more direct route to the classical limit of the S matrix. The S matrix thus obtained essentially corresponds to the WKB approximation, and, with a further approximation, leads to Miller's expression except for a multiplicative factor and an additional phase which are in general not equal to 1 and 0, respectively. The classical limit is also discussed in terms of Wigner-Moyal phase space distribution function, and when the classical limit of this function is taken, an additional factor and phase appear in the S matrix.