Variational Methods for Path Integral Scattering

TL;DR

This thesis introduces a variational approach to non-relativistic potential scattering using path integrals, deriving classical equations of motion, and demonstrating improved accuracy over existing methods through numerical tests.

Contribution

It develops a novel variational approximation scheme for path integral scattering, deriving classical equations of motion and including first-order corrections, with demonstrated improvements over previous methods.

Findings

The approximation captures leading and next-to-leading eikonal terms.

Numerical tests show substantial improvements over previous approximations.

The method provides a new way to analyze high-energy scattering behavior.

Abstract

In this master thesis, a new approximation scheme to non-relativistic potential scattering is developed and discussed. The starting points are two exact path integral representations of the T-matrix, which permit the application of the Feynman-Jensen variational method. A simple Ansatz for the trial action is made, and, in both cases, the variational procedure singles out a particular one-particle classical equation of motion, given in integral form. While the first is real, in the second representation this trajectory is complex and evolves according to an effective, time dependent potential. Using a cumulant expansion, the first correction to the variational approximation is also evaluated. The high energy behavior of the approximation is investigated, and is shown to contain exactly the leading and next-to-leading order of the eikonal expansion, and parts of higher terms. Our results are then numerically tested in two particular situations where others approximations turned out to be unsatisfactory. Substantial improvements are found.