On WKB-Quantization for Kepler Problem in Euclide, Riemann and Lobachevsky 3-Space

TL;DR

This paper extends the WKB quantization method to the quantum Kepler problem in curved 3-space models, deriving exact energy levels and analyzing their accuracy across Euclidean, Riemann, and Lobachevsky geometries.

Contribution

It develops a generalized WKB approach for curved spaces and proves the exactness of energy levels for the Schrödinger and Klein-Fock hydrogen atoms.

Findings

Exact energy levels are obtained for the quantum Kepler problem in curved spaces.

The exactness of these energy levels is demonstrated through WKB-series analysis.

Approximate solutions for the Dirac equation are also discussed.

Abstract

Quantum mechanical WKB-method is elaborated for the known quantum Kepler problem in curved 3-space models Euclide, Riemann and Lobachevsky in the framework of the complex variable function theory. Generalized Schr\"{o}dinger, Klein-Fock hydrogen atoms are considered. Exact energy levels are found and their exactness is proved on the base of exploration into $n$-degree terms of the WKB-series. Dirac equation is solved too, but only approximate energy spectrum is established.