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Physical Review A

Semiclassical normalization of a path integral for a multichannel scattering problem

Alvin P. Penner, R. Wallace

Published Jan 1, 1975• 10 citations
Publisher

Abstract

The evaluation of a Feynman path integral is discussed for a scattering problem in which some degrees of freedom of the system are treated classically, while others are treated quantum mechanically. The path integral is of the type $K(0, {t}^{\ensuremath{'}\ensuremath{'}}; 0, {t}^{\ensuremath{'}})$, and is of interest in a multichannel semiclassical collision theory previously developed by Pechukas. It is shown that this path integral can be evaluated precisely in practice, despite the presence of noncausal terms in the integrand.

DOI
10.1103/physreva.11.149

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The evaluation of a Feynman path integral is discussed for a scattering problem in which some degrees of freedom of the system are treated classically, while others are treated quantum mechanically. The path integral is of the type K(0,t\ensuremath′\ensuremath′;0,t\ensuremath′)K(0, {t}^{\ensuremath{'}\ensuremath{'}}; 0, {t}^{\ensuremath{'}})K(0,t\ensuremath′\ensuremath′;0,t\ensuremath′), and is of interest in a multichannel semiclassical collision theory previously developed by Pechukas. It is shown that this path integral can be evaluated precisely in practice, despite the presence of noncausal terms in the integrand.
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.
A numerical approach in scattering theory based on the path-integration technique is considered. An evaluation of a discretized Feynman path integral with an accurate Gaussian expansion of the integrand was used for the calculation of the differential cross sections for the scattering of a wave packet by a three-dimensional spherically symmetric Gaussian potential. The results are compared with the precise calculations using the phase functions method. This reveals separately discretization errors of the path integral, which depends on the single parameter---the number of discrete path points. High accuracy of the approach is demonstrated.
We present a detailed discussion of the semiclassical approximations to path integrals when there exist caustics in the relevant family of classical trajectories. We show carefully how the Airy functions arise from the post-semiclassical approximation of the path integral and interpret the coefficients in the Airy functions in terms of the flow of classical trajectories. Finally, we present a configuration-space path integral for the scattering matrix and extend our discussion of caustics to the case of rainbow scattering.