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Defect in the Joint Spectrum of Hydrogen due to Monodromy

Holger R. Dullin and Holger Waalkens
Phys. Rev. Lett. 120, 020507 – Published 12 January 2018
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Abstract

In addition to the well-known case of spherical coordinates, the Schrödinger equation of 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, the z component of the angular momentum, and an operator involving the z component of the quantum Laplace-Runge-Lenz vector obtained from separation in prolate spheroidal coordinates has quantum monodromy for energies sufficiently close to the ionization threshold. The precise value of the energy above which monodromy is observed depends on the distance of the focus points of the spheroidal coordinates. The presence of monodromy 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 multivalued.

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  • Received 31 August 2017

DOI:https://doi.org/10.1103/PhysRevLett.120.020507

© 2018 American Physical Society

Physics Subject Headings (PhySH)

Nonlinear DynamicsAtomic, Molecular & Optical

Authors & Affiliations

Holger R. Dullin*

  • School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia

Holger Waalkens

  • Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, 9700 AK Groningen, The Netherlands

  • *holger.dullin@sydney.edu.au
  • h.waalkens@rug.nl

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Issue

Vol. 120, Iss. 2 — 12 January 2018

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