Relativistic time-dependent quantum dynamics across supercritical barriers for Klein-Gordon and Dirac particles

TL;DR

This paper studies relativistic wavepacket dynamics across supercritical barriers for Klein-Gordon and Dirac particles, revealing charge creation phenomena and explaining the Klein paradox through a time-dependent scattering approach.

Contribution

It introduces a time-dependent multiple scattering expansion method that captures charge creation for Klein-Gordon particles and provides a comparative analysis with finite-difference solutions.

Findings

Charge creation occurs at barriers for Klein-Gordon particles.

The multiple scattering expansion diverges for Klein-Gordon but converges for Dirac particles.

The approach explains the Klein paradox within a first quantized framework.

Abstract

We investigate wavepacket dynamics across supercritical barriers for the Klein-Gordon and Dirac equations. Our treatment is based on a multiple scattering expansion (MSE). For spin-0 particles, the MSE diverges, rendering invalid the use of the usual connection formulas for the scattering basis functions. In a time-dependent formulation, the divergent character of the MSE naturally accounts for charge creation at the barrier boundaries. In the Dirac case, the MSE converges and no charge is created. We show that this time-dependent charge behavior dynamics can adequately explain the Klein paradox in a first quantized setting. We further compare our semi-analytical wavepacket approach to exact finite-difference solutions of the relativistic wave equations.