Defect in the Joint Spectrum of Hydrogen due to Monodromy

TL;DR

This paper demonstrates that the joint spectrum of certain quantum operators for the hydrogen atom exhibits monodromy in prolate spheroidal coordinates, preventing a global quantum number assignment near the ionization threshold.

Contribution

It reveals the presence of quantum monodromy in the hydrogen atom's joint spectrum in prolate spheroidal coordinates, a novel insight into atomic spectral structure.

Findings

Quantum monodromy occurs near the ionization threshold.

Global quantum numbers cannot be assigned due to monodromy.

Monodromy affects the joint spectrum of Hamiltonian and angular momentum operators.

Abstract

In addition to the well known case of spherical coordinates the hydrogen atom separates in three further coordinate systems. Separating in a particular coordinate system defines a system of three commuting operators. We show that the joint spectrum of the Hamilton operator, and the $z$-components of the angular momentum and quantum Laplace-Runge-Lenz vectors obtained from separation in prolate spheroidal coordinates has quantum monodromy for energies sufficiently close to the ionization threshold. This means that one cannot globally assign quantum numbers to the joint spectrum. Whereas the principal quantum number $n$ and the magnetic quantum number $m$ correspond to the Bohr-Sommerfeld quantization of globally defined classical actions a third quantum number cannot be globally defined because the third action is globally multi valued.