Defect in the Joint Spectrum of Hydrogen due to Monodromy

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.

5 pages, 5 figures

References in corpus (3)

• Excited state quantum phase transitions in many-body systems
• Non-uniqueness of phase shift in central scattering due to monodromy
• Fractional bidromy in the vibrational spectrum of HOCl

Cited by in corpus (10)

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• Symplectic classification of coupled angular momenta
• Monodromy and chaos for condensed bosons in optical lattices
• Orthogonal separation of variables for spaces of constant curvature
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• Algebra of the spinor invariants and the relativistic hydrogen atom
• Scattering invariants in Euler's two-center problem
• Survey on recent developments in semitoric systems
• Monodromy in Prolate Spheroidal Harmonics
• Quantum Monodromy in the Isotropic 3-Dimensional Harmonic Oscillator