Exact quantum revivals for the Dirac equation

TL;DR

This paper fully characterizes exact quantum revivals of relativistic fermion wave functions on a torus, generalizing the Talbot effect to relativistic particles with non-zero mass using arithmetic methods.

Contribution

It provides a complete characterization of all quantum states exhibiting exact revivals in the relativistic Dirac equation, extending previous non-relativistic results.

Findings

All revival states are fully characterized.

Revivals are exact without non-relativistic limits.

Plots illustrate the revival phenomena.

Abstract

In the present work, the results obtained in [1] about the revivals of a relativistic fermion wave function on a torus are considerably enlarged. In fact, all the possible quantum states exhibiting revivals are fully characterized. The revivals are exact, that is, are true revivals without taking any particular limit such as the non relativistic one. The present results are of interest since they generalize the Talbot effect and the revivals of the Schr\"odinger equation to a relativistic situation with non zero mass. This makes the problem nontrivial, as the dispersion relation is modified and is not linear. The present results are obtained by the use of arithmetic tools which are described in certain detail. In addition, several plots of the revivals are presented, which are useful for exemplifying the procedure proposed along the text.