New exact solution of the one dimensional Dirac Equation for the Woods-Saxon potential within the effective mass case

TL;DR

This paper derives an exact solution to the one-dimensional Dirac equation with a Woods-Saxon potential and position-dependent mass, providing analytical expressions for scattering and bound states, advancing quantum mechanical modeling.

Contribution

It introduces a specific continuous mass function allowing exact hypergeometric solutions for the Dirac equation with Woods-Saxon potential.

Findings

Derived explicit hypergeometric solutions for the Dirac equation.

Obtained exact formulas for reflection and transmission coefficients.

Numerically solved the energy eigenvalues for bound states.

Abstract

We study the one-dimensional Dirac equation in the framework of a position dependent mass under the action of a Woods-Saxon external potential. We find that constraining appropriately the mass function it is possible to obtain a solution of the problem in terms of the hypergeometric function. The mass function for which this turns out to be possible is continuous. In particular we study the scattering problem and derive exact expressions for the reflection and transmission coefficients which are compared to those of the constant mass case. For the very same mass function the bound state problem is also solved, providing a transcendental equation for the energy eigenvalues which is solved numerically.