Revivals and quantum carpets for the relativistic Schr\"odinger equation

TL;DR

This paper studies the dynamics of relativistic wavepackets in a box using the relativistic Schr"odinger equation, deriving solutions, revival times, and quantum carpets, and analyzing level spacing statistics across regimes.

Contribution

It provides well-defined solutions for the relativistic Schr"odinger equation in a box and explores wavepacket revivals, quantum carpets, and spectral statistics across relativistic regimes.

Findings

Derived explicit solutions for the relativistic Schr"odinger equation in a box.

Analyzed wavepacket revival times and quantum carpets in different regimes.

Examined level spacing statistics from non-relativistic to ultra-relativistic limits.

Abstract

We investigate wavepacket dynamics for a relativistic particle in a box evolving according to the relativistic Schr\"odinger (also known as the Salpeter) equation. We derive the solutions for an infinite well -- which contrary to the standard relativistic wave equations (such as the Klein-Gordon or Dirac equations) -- are well defined, and use these solutions to construct wavepackets. We obtain expressions for the wavepacket revival times and explore the corresponding quantum carpets (the space-time probability density plots) for different dynamical regimes. We further analyze level spacing statistics as the dynamics goes from the non-relativistic regime to the ultra-relativistic limit.