Solutions of the Dirac equation in one-dimensional variable width potential well

TL;DR

This paper explores solutions to the Dirac equation in a one-dimensional potential well with variable width, revealing complex momentum states and potential Tachyon-like states, extending understanding of relativistic quantum dynamics.

Contribution

It provides the first analysis of Dirac equation solutions in a time-varying potential well, highlighting novel complex momentum states and implications for Fermi acceleration mechanisms.

Findings

Dirac particles can exhibit complex-valued momentum states.

Potential for Tachyon-like state preparation.

Extends quantum dynamics understanding in variable potentials.

Abstract

The Fermi acceleration mechanism is a significant source of cosmic rays. When the width of a potential well changes over time, the velocity of particles within the well also changes. For quantum systems, such dynamics should be described by the Schr\"odinger, Klein-Gordon, and Dirac equations. Previous studies have solved the Schr\"odinger and Klein-Gordon equations under these conditions, but no research has addressed the Dirac equation for spin-$\frac{1}{2}$ particles like electrons. This paper investigates the solutions of the Dirac equation in a dynamically varying potential well and demonstrates that Dirac particles can exhibit complex-valued momentum states via the Fermi acceleration mechanism, enabling Tachyon-like states preparation.