The linear potential and the Dirac equation

TL;DR

This paper analyzes the Dirac equation with a linear potential, highlighting how the Lorentz nature influences bound state existence, and provides analytic solutions for specific cases, including bound and quasi-bound states.

Contribution

It offers an analytic solution for the Dirac equation with a linear potential, emphasizing the impact of vector and scalar components on bound state formation.

Findings

Equal vector and scalar potentials yield a strictly bound state.

Larger vector component leads to quasi-bound states with inhibited decay.

Analytic solutions are derived for specific potential configurations.

Abstract

The solution of the Dirac equation for an attractive linear potential is considered. The Lorentz nature of the potential (vector or scalar) affects the existence of bound states. For simplicity, and since it is sufficient for the goals of this study, only the ground state is considered. The case of equal vector and scalar pieces of the linear potential is emphasized because it lends itself to a simple analytic solution. This solution corresponds to a state which is strictly bound. For a linear potential with a larger component of the vector part, we find a state that is only quasi-bound, but its decay can be strongly inhibited by an effective potential barrier.