Reconciling Semiclassical and Bohmian Mechanics: II. Scattering states for discontinuous potentials

TL;DR

This paper extends a bipolar decomposition approach to stationary scattering states in discontinuous potentials, resulting in a classical-like, well-behaved Bohmian mechanics formulation that accurately computes scattering properties.

Contribution

It modifies the bipolar decomposition scheme to handle discontinuous potentials, enabling exact numerical calculation of scattering states and probabilities within a Bohmian framework.

Findings

Bipolar quantum potential is zero except at discontinuities.

The method provides an exact numerical approach for scattering states.

It yields classical-like trajectories even with oscillatory wavefunctions.

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. Moreover, by applying the Madelung-Bohm ansatz to the components rather than to Psi itself, the resultant bipolar Bohmian mechanical formulation satisfies the correspondence principle. As a result, the bipolar quantum trajectories are classical-like and well-behaved, even when Psi has many nodes, or is wildly oscillatory. In this paper, the previous decomposition scheme is modified in order to achieve the same desirable properties for stationary scattering states. Discontinuous potential systems are considered (hard wall, step, square barrier/well), for which the bipolar quantum potential is found to be zero everywhere, except at the discontinuities. This approach leads to an exact numerical method for computing stationary scattering states of any desired boundary conditions, and reflection and transmission probabilities. The continuous potential case will be considered in a future publication.