Uniform approximation for the overlap caustic of a quantum state with its translations

TL;DR

This paper derives a uniform approximation for the overlap of a quantum state with its translations, focusing on the caustic structures in the semiclassical Wigner and chord functions, and describes the transition from oscillatory to evanescent overlaps.

Contribution

It introduces a new uniform approximation for the overlap caustic of a quantum state and its translations, extending semiclassical analysis to the conjugate phase space curves.

Findings

Derived a uniform approximation for the overlap caustic transition.

Analyzed the caustic structure of the Wigner and chord functions.

Described the transition from oscillatory to evanescent overlaps.

Abstract

The semiclassical Wigner function for a Bohr-quantized energy eigenstate is known to have a caustic along the corresponding classical closed phase space curve in the case of a single degree of freedom. Its Fourier transform, the semiclassical chord function, also has a caustic along the conjugate curve defined as the locus of diameters, i.e. the maximal chords of the original curve. If the latter is convex, so is its conjugate, resulting in a simple fold caustic. The uniform approximation through this caustic, that is here derived, describes the transition undergone by the overlap of the state with its translation, from an oscillatory regime for small chords, to evanescent overlaps, rising to a maximum near the caustic. The diameter-caustic for the Wigner function is also treated.