Exact closed-form solutions of the Dirac equation with a scalar exponential potential

TL;DR

This paper derives exact closed-form solutions for the Dirac equation with an exponential scalar potential in two dimensions, revealing relativistic bound states and zero modes, thus advancing analytical understanding of relativistic quantum systems.

Contribution

It provides the first exact solutions for the Dirac equation with an exponential scalar potential, mapping it to a Morse potential and analyzing zero modes.

Findings

Exact bound solutions for the Dirac equation with exponential potential

Identification of zero modes in the Dirac spinor components

Relativistic nature of the Morse-like effective potential

Abstract

The problem of a fermion subject to a general scalar potential in a two-dimensional world for nonzero eigenenergies is mapped into a Sturm-Liouville problem for the upper component of the Dirac spinor. In the specific circumstance of an exponential potential, we have an effective Morse potential which reveals itself as an essentially relativistic problem. Exact bound solutions are found in closed form for this problem. The behaviour of the upper and lower components of the Dirac spinor is discussed in detail, particularly the existence of zero modes.