Reconciling Semiclassical and Bohmian Mechanics: III. Scattering states for continuous potentials

TL;DR

This paper develops an exact quantum scattering method using classical trajectories by generalizing bipolar decomposition techniques from bound states to continuous potentials, improving the analysis of complex quantum systems.

Contribution

It introduces a generalized bipolar decomposition for continuous potentials, enabling an exact quantum scattering approach with classical trajectories, including a constant velocity variant for tunneling.

Findings

Bipolar decomposition approach is extended to continuous potentials.

The method provides classical-like trajectories for quantum scattering states.

A constant velocity trajectory version simplifies tunneling analysis.

Abstract

In a previous paper [J. Chem. Phys. 121 4501 (2004)] a unique bipolar decomposition, Psi = Psi1 + Psi2 was presented for stationary bound states Psi of the one-dimensional Schroedinger equation, such that the components Psi1 and Psi2 approach their semiclassical WKB analogs in the large action limit. The corresponding bipolar quantum trajectories, as defined in the usual Bohmian mechanical formulation, are classical-like and well-behaved, even when Psi has many nodes, or is wildly oscillatory. A modification for discontinuous potential stationary stattering states was presented in a second paper [J. Chem. Phys. 124 034115 (2006)], whose generalization for continuous potentials is given here. The result is an exact quantum scattering methodology using classical trajectories. For additional convenience in handling the tunneling case, a constant velocity trajectory version is also developed.