Landau Levels in a Gravitational Field: The Schwarzschild Spacetime Case

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 Schrödinger 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.

Spin-1/2 Particles under the Influence of a Uniform Magnetic Field in the Interior Schwarzschild Solution

“…It is clear from this expression that the usual flat-space Landau levels of a charged fermion moving under the influence of a uniform magnetic field are dramatically altered. In fact, instead of the splitting of the levels that occurs due to a weak gravitational field outside [17] or inside a spherical mass [19], what happens here is a complete redistribution of the levels. Both the separation of the latter and their strengths are redefined for each principle quantum number n.…”

“…It is clear that Equation (17) would not be easy to solve exactly. However, we can clearly see now which terms of the equation could safely be dropped out even without making any prior assumption on the orders of magnitude of the various physical quantities involved, such as the energy of the particle and the strengths of the gravitational and magnetic interaction terms.…”

“…In that case, Equation ( 22) reduces to the one that has already been solved exactly in Ref. [17], where the quantized eigenvalues ε n, have been extracted. We will come back to these special cases in Section 3.1.…”

“…The solution to this equation is easily found to be given in terms of the confluent hypergeometric function as follows (see Ref. [17] for the details of the derivation):…”

“…Such a study has recently been conducted in Refs. [17][18][19] based on the Klein-Gordon and Schrödinger equations by considering the gravitational interaction of the particle with a massive sphere as a small perturbation compared to the interaction of the particle with the magnetic field. Although relying on the Schrödinger equation and treating gravity perturbatively as done in these studies might be sufficient for laboratory applications [20][21][22][23], when seeking astrophysical applications, implementing a generalrelativistic and a non-perturbative approach and appealing to the relativistic Dirac equation are highly recommended.…”