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Título del libro: A Closer Look At The Hydrogen Atom
Título del capítulo: Analysis of the degeneracy of the two-dimensional hydrogen atom

Autores UNAM:
EUGENIO LEY KOO; GREGORIO SAINZ LOMBARDO;

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Autores externos:

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Idioma:

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Año de publicación:
2024

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Palabras clave:

Confined hydrogen atom; Constants of motion; Coordinate systems; D dimension; Degeneracy; Free hydrogen atom; Geometries; Position and momentum representations; Schrodinger differential and integral equations; Superintegrability; Symmetries; Unitary and rotation groups

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Resumen:

The explanation of the accidental degeneracy of the hydrogen atom was provided by the works of Fock in 1935, and Bargmann in 1936, respectively. The latter recognized that the Schrödinger differential equation in the position representation is separable and integrable in spherical coordinates with the magnitude and axial component of the angular momentum as constants of motion; and also in paraboloidal coordinates, taking the form of two harmonic oscillators in circular coordinates with the magnitude of the Laplace-Runge-Lenz vector in the transverse direction and the z-component of angular momentum as constants of motion. Fock formulated his solution using the momentum representation of the Schrödinger integral equation, associated with the inverse radial dependence of the Coulomb potential energy. He also introduced the stereographic projection from the two-dimensional sphere S2 in R3 space to the three-dimensional sphere S3 in R4 space, in which the equation is identified as the four-dimensional Laplace equation, with harmonic solutions, which are Gegenbauer polynomials, with degeneracy. The Poisson equation in R4 and its Green function can also be constructed and applied to identify the radial solutions inside and outside S3 . Their stereographic projections back to R3 yield the momentum eigenfunctions in the R3 momentum space. In this contribution, the corresponding analysis is applied to the two-dimensional hydrogen atom. The corresponding results are simpler because of the reduction in the dimensionality. This problem has been investigated in the successive decades, illustrating not only the differences with respect to three-dimensional case, but also their common features and connections, taking into account their symmetry, geometry and superintegrability, which also extend to higher dimensions. The main results are restricted to the free hydrogen atom. References are also provided for confinement in coordinate boundaries, introducing symmetry breakings in which the constants of motion persist, but the quantum labels are no longer integers. © 2024 Nova Science Publishers, Inc. All rights reserved.

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Entidades citadas de la UNAM:

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Fuente:


ISBN:
9798895301609
Editorial:
Nova Science Publishers, Inc. Nombre de la publicación:
Analysis of the degeneracy of the two-dimensional hydrogen atom
Página inicial:
23
Página final:
58

Tipo de producto:
Capítulo de un Libro