# Relativistic corrections to Landau levels in the presence of a parallel linear electric field

## TL;DR

This paper derives relativistic corrections to Landau levels for an electron in combined magnetic and linear electric fields using Dirac's equation, revealing how these corrections affect degeneracies and energy level splitting.

## Contribution

It provides the first and second order relativistic corrections to Landau levels in a combined magnetic and linear electric field system, extending non-relativistic results with compact formulas.

## Key findings

- Relativistic corrections lower the energy levels.
- Degeneracies shift due to relativistic effects.
- Energy level splitting occurs at specific frequency ratios.

## Abstract

We consider an electron moving under a constant magnetic field (in the z-direction) and a \textit{linear} electric field parallel to the magnetic field above the z=0 plane and anti-parallel below the plane. Two frequencies characterize the system: the cyclotron frequency $\omega_c$ corresponding to motion along the x-y plane and associated with the usual Landau levels, and a second frequency $\omega_z$ corresponding to motion along the z-direction. In previous work, the non-relativistic energies of this system were obtained, and it was shown that an extra degeneracy (beyond the Landau degeneracy) occurs when the ratio $\text{w}=\omega_c/\omega_z$ is rational. In this paper, we use Dirac's equation to obtain compact formulas for the first and second order relativistic corrections to this system via perturbation theory. The formulas are expressed in terms of the two frequencies $\omega_c$ and $\omega_z$, and two quantum numbers, $n$ and $n_z$, both of which are non-negative integers. The first order correction is negative and lowers the original energies. We plot the energy (zeroth plus first order) versus the ratio $\text{w}$ and there are degeneracies at all points where lines intersect. However, the degeneracy does not occur at the same $\text{w}$ as before. To illustrate this, we show how the first order correction splits the energy levels for the case $\omega_c=\omega_z$.