Solving Klein's paradox

TL;DR

This paper resolves Klein's paradox by analyzing Dirac and Klein-Gordon equations with step potentials, demonstrating that reflection coefficients do not exceed unity and exploring implications for massless or nearly massless particles.

Contribution

It provides a piecewise solution approach to Klein's paradox for Dirac and Klein-Gordon equations with step potentials, clarifying the reflection behavior.

Findings

Reflection coefficient never exceeds 1

Solutions differ for regions with E > V and E < V

Applications to massless and nearly massless particles

Abstract

We figure out the famous Klein's paradox arising from the reflection problem when a Dirac particle encounters a step potential with infinite width. The key is to piecewise solve Dirac equation in such a way that in the region where the particle's energy E is greater (less) than the potential V, the solution of the positive (negative) energy branch is adopted. In the case of Klein-Gordon equation with a piecewise constant potential, the equation is decoupled to positive and negative energy equations, and reflection problem is solved in the same way. Both infinitely and finitely wide potentials are considered. The reflection coefficient never exceeds 1. The results are applied to discuss the transmissions of particles with no mass or with very small mass.