The Woods-Saxon Potential in the Dirac Equation

TL;DR

This paper applies a two-component approach to solve the one-dimensional Dirac equation with the Woods-Saxon potential, deriving scattering and bound state solutions, and analyzing phenomena like transmission resonances and supercriticality.

Contribution

It introduces a novel application of the two-component method to the Woods-Saxon potential, deriving explicit conditions for transmission resonance and supercriticality in this context.

Findings

Derived explicit solutions for scattering and bound states.

Identified conditions for transmission resonance and supercriticality.

Demonstrated the relation between potential barriers and zero-momentum transmission resonance.

Abstract

The two-component approach to the one-dimensional Dirac equation is applied to the Woods-Saxon potential. The scattering and bound state solutions are derived and the conditions for a transmission resonance (when the transmission coefficient is unity) and supercriticality (when the particle bound state is at E=-m) are then derived. The square potential limit is discussed. The recent result that a finite-range symmetric potential barrier will have a transmission resonance of zero-momentum when the corresponding well supports a half-bound state at E=-m is demonstrated.