Development and Numerical Analysis of "Black-box" Counterpropagating Wave Algorithm for Exact Quantum Scattering Calculations

TL;DR

This paper introduces a robust and efficient 'black-box' algorithm based on bipolar counter-propagating wave decomposition for precise quantum scattering calculations in one-dimensional systems.

Contribution

It develops a universal numerical method leveraging bipolar wave components that remain well-behaved even with complex oscillatory wavefunctions.

Findings

Algorithm is numerically stable and efficient.

Accurately computes scattering quantities for quantum systems.

Works for systems with highly oscillatory wavefunctions.

Abstract

In a recent series of papers [J. Chem. Phys. 121 4501 (2004), J. Chem. Phys. 124 034115 (2006), J. Chem. Phys. 124 034116 (2006)] a bipolar counter-propagating wave decomposition, Psi = Psi+ + Psi-, was presented for stationary bound states Psi of the one-dimensional Shrodinger equation, such that the components Psi+- approach their semiclassical WKB analogs in the large action limit. The corresponding bipolar quantum trajectories are classical-like and well-behaved, even when Psi has many nodes, or is wildly oscillatory. In this paper, the earlier results are used to construct a universal ``black-box'' algorithm, numerically robust, stable and efficient, for computing accurate scattering quantities of any quantum dynamical system in one degree of freedom.