# Dynamics of Majorana fermions in two-dimensions

## TL;DR

This paper explores the behavior of Majorana fermions in 1+1 dimensions under static scalar potentials, employing supersymmetric quantum mechanics to find analytical solutions for shape-invariant cases.

## Contribution

It introduces a novel application of supersymmetric quantum mechanics to Majorana fermion dynamics in external fields, providing analytical solutions for shape-invariant potentials.

## Key findings

- Majorana fermions can have bound states but no stationary states in this setting.
- Analytical solutions are obtained for shape-invariant potentials.
- The method is demonstrated with a linear potential.

## Abstract

A Majorana fermion is the single fermionic particle that is its own antiparticle. Its dynamics is determined by the Majorana equation where the spinor field $(\psi)$ is by definition equal to its charge-conjugate field $(\psi_c)$. Here, we study the dynamics of Majorana fermions in the presence of the most general static external field in $1+1$ dimensions, which is just a scalar potential, by implementing for the first time the methods of supersymmetric quantum mechanics. In particular, for potentials for which shape invariance holds, we show how to obtain analytical solutions. We find that although this equation does have bound states, it does not have stationary states. The approach is illustrated with a linear potential.