Semiclassical approximations for scatterings by nonlocal potentials. I The effective-mass method

Abstract

A semiclassical theory for the scattering of a particle by a nonlocal potential is developed with the purpose of applying it to several aspects of atomic collisions. The scheme that we construct is based on the replacement of the Hamiltonian with the nonlocal interactions by an approximate Hamiltonian that includes a certain local potential and an effective mass that depends on the distance of the particle from the origin of the potential. This is followed by the introduction of either the JWKB or the Eikonal approximation in the form familiar in potential scattering theory. The dependence of the approximate semiclassical wavefunctions and phase shifts on the effective mass is studied and effects owing to the variation of the latter quantity with the coordinates are discussed. A treatment is provided for cases where the effective mass has a zero or a pole. Appropriate ``connection formulas'' are derived for the determination of the correct JWKB solution in regions that lie in the neighborhood of points at which such a zero or a singularity occurs. It is pointed out that zeros of the effective mass can lead to pronounced resonances and to barrierlike effects. Since any multichannel collision process can be discussed in terms of equivalent one-body equations involving nonlocal potentials, the semiclassical scheme we develop is applicable to inelastic and rearrangement processes as well as to elastic ones. Applications of this method to atomic collision theory will be described in a forthcoming publication.