Variational Approximations in a Path-Integral Description of Potential Scattering

TL;DR

This paper develops new variational approximations within a path integral framework for potential scattering, incorporating classical trajectories and quantum effects, and demonstrates improved accuracy over traditional methods at high energies and large angles.

Contribution

It introduces a novel variational approach using a quadratic ansatz in the path integral representation for the T-matrix, combining classical and quantum elements, with numerical solutions and corrections.

Findings

Better agreement with exact results at large scattering angles

Improved accuracy over eikonal approximations at high energies

Inclusion of quantum wave spreading effects

Abstract

Using a recent path integral representation for the T-matrix in nonrelativistic potential scattering we investigate new variational approximations in this framework. By means of the Feynman-Jensen variational principle and the most general ansatz quadratic in the velocity variables -- over which one has to integrate functionally -- we obtain variational equations which contain classical elements (trajectories) as well as quantum-mechanical ones (wave spreading).We analyse these equations and solve them numerically by iteration, a procedure best suited at high energy. The first correction to the variational result arising from a cumulant expansion is also evaluated. Comparison is made with exact partial-wave results for scattering from a Gaussian potential and better agreement is found at large scattering angles where the standard eikonal-type approximations fail.