Low Momentum Scattering in the Dirac Equation

TL;DR

This paper demonstrates that low-momentum Dirac particles are generally fully reflected by finite-range potentials, with exceptions occurring when the potential supports a half-bound state at zero momentum, affecting transmission.

Contribution

It provides a detailed analysis of low-momentum scattering in the Dirac equation, highlighting conditions under which transmission is non-zero or complete.

Findings

Reflection amplitude is -1 for low-momentum particles in general.

Transmission coefficient T=0 unless the potential supports a half-bound state at zero energy.

Symmetric potentials can have T=1, indicating full transmission in specific cases.

Abstract

It is shown that the amplitude for reflection of a Dirac particle with arbitrarily low momentum incident on a potential of finite range is -1 and hence the transmission coefficient T=0 in general. If however the potential supports a half-bound state at k=0 this result does not hold. In the case of an asymmetric potential the transmission coefficient T will be non-zero whilst for a symmetric potential T=1.