Tunneling dynamics of the relativistic Schrodinger/Salpeter equation

TL;DR

This paper explores the tunneling behavior of relativistic particles described by the Salpeter equation, revealing unique properties such as the absence of Klein tunneling and potential effects on wavepacket propagation outside the light cone.

Contribution

It provides a numerical analysis of tunneling dynamics for the Salpeter equation, highlighting differences from standard relativistic wave equations and addressing the role of pseudo-differential operators.

Findings

No Klein tunneling observed in Salpeter equation

Potential influences wavepacket outside the light cone

Numerical solutions for scattering on potential barriers

Abstract

We investigate potential scattering and tunneling dynamics of a particle wavepacket evolving according to the relativistic Schr\"odinger equation (also known as the Salpeter equation). The tunneling properties of the Salpeter equation differ from those of the standard relativistic wave equations (such as the Klein-Gordon or Dirac equations). In particular, the tunneling solutions must be found by working in momentum space, given that the equation in configuration space contains a pseudo-differential operator. The resulting integral equations are derived and solved numerically for wavepackets scattering on model potential barriers. The solutions are characterized by the absence of Klein tunneling and an effect of the potential on the fraction of the transmitted wavepacket that propagates outside the light cone, a feature that has in the past been well-studied only for free propagation.