Quantum reflection and Liouville transformations from wells to walls

TL;DR

This paper uses Liouville transformations to relate quantum reflection phenomena from attractive wells to repulsive walls, providing a new way to evaluate reflection probabilities while highlighting differences in semiclassical descriptions.

Contribution

It introduces a novel application of Liouville transformations to connect different scattering scenarios and accurately compute quantum reflection probabilities.

Findings

Quantum reflection probabilities can be transformed between wells and walls.

Semiclassical descriptions differ significantly despite preserved scattering properties.

Liouville transformations provide a rigorous framework for analyzing quantum scattering.

Abstract

Liouville transformations map in a rigorous manner one Schr\"odinger equation into another, with a changed scattering potential. They are used here to transform quantum reflection of an atom on an attractive well into reflection of the atom on a repulsive wall. While the scattering properties are preserved, the corresponding semiclassical descriptions are completely different. A quantitative evaluation of quantum reflection probabilities is deduced from this method.