Barrier Paradox in the Klein Zone

TL;DR

This paper investigates the Klein paradox in the Dirac equation with a barrier potential, revealing multiple reflections, pair production, and a new paradox of growing wave amplitudes.

Contribution

It introduces the concept of antiparticle solutions localized within the barrier, leading to a new paradox of geometrically increasing outgoing wave amplitudes.

Findings

Multiple reflections and transmissions occur in the Klein zone.

Antiparticle solutions within the barrier generate new pairs at each reflection.

Successive outgoing wave amplitudes grow geometrically, indicating a paradox.

Abstract

We study the solutions for a one-dimensional electrostatic potential in the Dirac equation when the incoming wave packet exhibits the Klein paradox (pair production). With a barrier potential we demonstrate the existence of multiple reflections (and transmissions). The antiparticle solutions which are necessarily localized within the barrier region create new pairs with each reflection at the potential walls. Consequently we encounter a new paradox for the barrier because successive outgoing wave amplitudes grow geometrically.