The Klein Paradox: A New Treatment

TL;DR

This paper presents a new solution to the Dirac equation with a step potential, resolving the Klein paradox by accounting for charge sign quantum number and revealing a localized positron amplitude at the potential threshold.

Contribution

It introduces a fundamentally different treatment of the Dirac equation with a step potential, eliminating paradoxical energy values and revealing a localized positron component.

Findings

No paradoxical energy values in the new solution

Presence of a localized positron amplitude at the step threshold

Reconciliation of Klein paradox phenomena with charge quantum number considerations

Abstract

The Dirac equation requires a treatment of the step potential that differs fundamentally from the traditional treatment, because the Dirac plane waves, besides momentum and spin, are characterized by a quantum number with the physical meaning of sign of charge. Since the Hermitean operator corresponding to this quantum number does not commute with the step potential, the time displacement parameter used in the ansatz of the stationary state does not have the physical meaning of energy. Therefore there are no paradoxal values of the energy. The new solution of the Dirac equation with a step potential is obtained. This solution, again, allows for phenomena of the Klein paradox type, but in addition it contains a positron amplitude localized at the threshold point of the step potential.