Complex trajectory method in time-dependent WKB

TL;DR

This paper enhances the time-dependent WKB method by superposing crossing trajectories and including higher order terms, improving its accuracy in quantum wavefunction propagation, especially in interference regions.

Contribution

It introduces a superposition approach of crossing trajectories and higher order expansions to improve TDWKB accuracy in interference regions.

Findings

Superposition of crossing trajectories extends TDWKB applicability.

Higher order terms improve approximation accuracy.

Method shows potential as an alternative to existing semiclassical techniques.

Abstract

We present a significant improvement to a time-dependent WKB (TDWKB) formulation developed by Boiron and Lombardi [JCP {\bf108}, 3431 (1998)] in which the TDWKB equations are solved along classical trajectories that propagate in the complex plane. Boiron and Lombardi showed that the method gives very good agreement with the exact quantum mechanical result as long as the wavefunction does not exhibit interference effects such as oscillations and nodes. In this paper we show that this limitation can be overcome by superposing the contributions of crossing trajectories. We also demonstrate that the approximation improves when incorporating higher order terms in the expansion. These improvements could make the TDWKB formulation a competitive alternative to current time-dependent semiclassical methods.