Landau levels in a gravitational field: The Schwarzschild spacetime case

TL;DR

This paper explores how a gravitational field affects Landau levels, showing that gravity removes the degeneracy of energy levels in a magnetic field and discussing implications for testing gravity through quantum effects.

Contribution

It provides a detailed analysis of Landau levels in a gravitational field, comparing multiple approaches and clarifying previous literature on the quantization condition in curved spacetime.

Findings

Gravity removes Landau level degeneracy.

Two independent methods yield similar results.

Relativistic effects are also analyzed.

Abstract

We investigate the gravitational effect on Landau levels. We show that the familiar infinite Landau degeneracy of the energy levels of a quantum particle moving inside a uniform and constant magnetic field is removed by the interaction of the particle with a gravitational field. Two independent approaches are used to solve the relevant Schr\"odinger equation within the Newtonian approximation. It is found that both approaches yield qualitatively similar results within their respective approximations. With the goal of clarifying some results found in the literature concerning the use of a third independent approach for extracting the quantization condition based on a similar differential equation, we show that such an approach cannot yield a general and yet consistent result. We point out to the more accurate, but impractical, way to use such an approach; a way which does in principle yield a consistent quantization condition. We discuss how our results could be used to contribute in a novel way to the existing methods for testing gravity at the tabletop experiments level as well as at the astrophysical observational level by deriving the corrections brought by Yukawa-like and power-law deviations from the inverse-square law. The full relativistic regime is also examined in detail.