Liouville transformations and quantum reflection

TL;DR

This paper explores Liouville transformations of Schrödinger equations, introducing a special gauge that maps quantum reflection problems on attractive potentials to those on repulsive walls, enabling quantitative probability evaluations.

Contribution

It introduces a specific Liouville gauge that simplifies quantum reflection analysis by transforming attractive potentials into repulsive ones.

Findings

Mapped quantum reflection on Casimir-Polder potentials to reflection on repulsive walls.

Provided a quantitative evaluation of quantum reflection probabilities.

Linked reflection probabilities to a universal solution of a specific potential.

Abstract

Liouville transformations of Schr\"odinger equations preserve the scattering amplitudes while changing the effective potential. We discuss the properties of these gauge transformations and introduce a special Liouville gauge which allows one to map the problem of quantum reflection of an atom on an attractive Casimir-Polder well into that of reflection on a repulsive wall. We deduce a quantitative evaluation of quantum reflection probabilities in terms of the universal probability which corresponds to the solution of the $V_4=-C_4/z^4$ far-end Casimir-Polder potential.