Hydrogen atom in de Sitter spaces

TL;DR

This paper investigates the behavior of hydrogen atoms in de Sitter and anti de Sitter spaces using wave equations, revealing instability in de Sitter space and stability in anti de Sitter space, with extensions to spin 1/2 particles.

Contribution

It develops a theoretical framework for hydrogen atoms in curved spacetimes, deriving energy levels and decay probabilities, and extends analysis to spin 1/2 particles in these models.

Findings

Hydrogen atom in de Sitter space is quasi-stationary and unstable.

Hydrogen atom in anti de Sitter space is stable.

Approximate energy levels are derived using quasi-classical methods.

Abstract

The hydrogen atom theory is developed for the de Sitter and anti de Sitter spaces on the basis of the Klein-Gordon-Fock wave equation in static coordinates. In both models, after separation of the variables, the problem is reduced to the general Heun equation, a second order linear differential equation having four regular singular points. A qualitative examination shows that the energy spectrum for the hydrogen atom in the de Sitter space should be quasi-stationary, and the atom should be unstable. We derive an approximate expression for energy levels within the quasi-classical approach and estimate the probability of decay of the atom. A similar analysis shows that in the anti de Sitter model the hydrogen atom should be stable in the quantum-mechanical sense. Using the quasi-classical approach, we derive approximate formulas for energy levels for this case as well. Finally, we present the extension to the case of a spin 1/2 particle for both de Sitter models. This extension leads to complicated differential equations with 8 singular points.