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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Path integral for scattering theory

Bruce L. Nelson

Published Feb 15, 1983• 4 citations
Publisher

Abstract

We present a well-defined path integral for scattering theory. This path integral is particularly well suited for evaluating the semiclassical expansion for the cross section. As an example, we show how the semiclassical cross section for rainbow scattering can be obtained in a systematic fashion. We indicate how the path-integral approach to semiclassical scattering theory may be more general and more reliable than other approaches.

DOI
10.1103/physrevd.27.841

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We present a well-defined path integral for scattering theory. This path integral is particularly well suited for evaluating the semiclassical expansion for the cross section. As an example, we show how the semiclassical cross section for rainbow scattering can be obtained in a systematic fashion. We indicate how the path-integral approach to semiclassical scattering theory may be more general and more reliable than other approaches.
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.
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.
This paper defines path integrals in phase space without using a time-division approach followed by a limiting process, thereby generalizing a similar procedure used in configuration space. This is useful since the path integral approach cannot always be formulated in configuration space (e.g., when the Hamiltonian is arbitrary) but can always be formulated in phase space. The most general Gaussian measure, absorbing the quadratic portion of the functional to be integrated, is constructed, and large classes of path integrals are evaluated with respect to it. Applications are given to the perturbation expansion and the semiclassical (WKB) expansion for arbitrary Hamiltonians.
A new formulation of nonrelativistic scattering theory is developed which expresses the SSS matrix as a path integral. This formulation appears to have at least two advantages: (1) A closed formula is obtained for the SSS matrix in terms of the potential, not involving a series expansion; (2) the energy-conserving $\ensuremath{\delta}$ function can be explicitly extracted using a technique analogous to that of Faddeev and Popov, thereby yielding a closed path-integral expression for the TTT matrix. The introduction of the concept of the classical interaction picture provides considerable physical insight into this formulation. This formulation also suggests a successionof improvements to the eikonal approximation, the first of which is discussed explicitly.