2016, Vol.19, No.1, pp.16-29
The hydrogen atom theory is developed for the de Sitter and anti de Sitter spaces of constant negative curvature on the basis of the Klein-Gordon-Fock wave equation in static coordinates. In both models, after separation of variables, the problem is reduced to the general Heun equation, a second order linear differential equation having four regular singular points. 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 the 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.
Key words: Quantum mechanics, hydrogen atom, de Sitter space, anti de Sitter space, WKB-approach, general Heun functions
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