Defect in the Joint Spectrum of Hydrogen due to Monodromy

Area of Science:

• Quantum mechanics
• Atomic physics
• Mathematical physics

Background:

• The Schrödinger equation for the hydrogen atom is separable in multiple coordinate systems.
• Separation in a coordinate system defines a set of commuting operators.

Purpose of the Study:

• To investigate the joint spectrum of operators for the hydrogen atom in prolate spheroidal coordinates.
• To identify the presence and characteristics of quantum monodromy in this system.

Main Methods:

• Separation of variables in prolate spheroidal coordinates.
• Analysis of the joint spectrum of the Hamilton operator, angular momentum (z-component), and a quantum Laplace-Runge-Lenz vector operator.
• Examination of quantum monodromy near the ionization threshold.

Main Results:

• Quantum monodromy is observed for energies close to the ionization threshold in prolate spheroidal coordinates.
• The onset energy for monodromy depends on the focal distance of the spheroidal coordinates.
• Monodromy implies that quantum numbers cannot be globally assigned to the energy spectrum.

Conclusions:

• The principal quantum number (n) and magnetic quantum number (m) can be globally defined via Bohr-Sommerfeld quantization.
• A third quantum number cannot be globally defined due to a globally multivalued classical action.
• This finding has implications for understanding quantum systems and their spectral properties.