On real solutions of the Dirac equation for a one-dimensional Majorana particle

TL;DR

This paper derives and analyzes real solutions of the (1+1)-dimensional Dirac equation for a Majorana particle under various boundary conditions and potentials, highlighting the importance of the Majorana condition in the solution structure.

Contribution

It provides a systematic construction of real solutions to the Dirac equation with a Lorentz scalar potential under the Majorana condition in (1+1) dimensions, including multiple boundary scenarios.

Findings

Real solutions depend on the Majorana condition and boundary conditions.

Complex solutions are also possible without the Majorana constraint.

The solutions illustrate the nuanced role of the Majorana condition in Dirac equations.

Abstract

We construct general solutions of the time-dependent Dirac equation in (1+1) dimensions with a Lorentz scalar potential, subject to the so-called Majorana condition, in the Majorana representation. In this situation, these solutions are real-valued and describe a one-dimensional Majorana single particle. We specifically obtain solutions for the following cases: a Majorana particle at rest inside a box, a free (i.e., in a penetrable box with the periodic boundary condition), in an impenetrable box with no potential (here we only have four boundary conditions), and in a linear potential. All these problems are treated in a very detailed and systematic way. In addition, we obtain and discuss various results related to real wave functions. Finally, we also wish to point out that, in choosing the Majorana representation, the solutions of the Dirac equation with a Lorentz scalar potential can be chosen to be real but do not need to be real. In fact, complex solutions for this equation can also be obtained. Thus, a Majorana particle cannot be described only with the Dirac equation in the Majorana representation without explicitly imposing the Majorana condition.