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Physical Review

Time-Dependent Semiclassical Scattering Theory. I. Potential Scattering

Philip Pechukas

Published May 5, 1969• 391 citations
Publisher

Abstract

The semiclassical theory of potential scattering is derived from Feynman's path-integral formulation of quantum mechanics. Since the partial-wave expansion is avoided, the results are applicable to scattering by a noncentral field. Special attention is given to the conditions for validity of the semiclassical approximation.
DOI
10.1103/physrev.181.166

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Rather general expressions are derived which represent the semiclassical time-dependent propagator as an integral over initial conditions for classical trajectories. These allow one to propagate time-dependent wave functions without searching for special trajectories that satisfy two-time boundary conditions. In many circumstances, the integral expressions are free of singularities and provide globally valid uniform asymptotic approximations. In special cases, the expressions for the propagators are related to existing semiclassical wave function propagation techniques. More generally, the present expressions suggest a large class of other, potentially useful methods. The behavior of the integral expressions in certain limiting cases is analyzed to obtain simple formulas for the Maslov index that may be used to compute the Van Vleck propagator in a variety of representations.
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The space-time propagator for multichannel scattering is cast in the form of a Feynman path integral where the action is a matrix and the interactions are nonlocal in time. The SSS matrix is evaluated in the semiclassical limit, leading to trajectories which naturally follow individual channel potentials in noninteracting regions and adiabatic-like ones in strongly interacting ones. A numerical example is provided for the curve-crossing problem, and the agreement with exact quantum results is good even when the interaction region is large and/or encompasses classical turning points.