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Title: Self-adjoint extensions for confined electrons: From a particle in a spherical cavity to the hydrogen atom in a sphere and on a cone

Abstract

In a recent study of the self-adjoint extensions of the Hamiltonian of a particle confined to a finite region of space, in which we generalized the Heisenberg uncertainty relation to a finite volume, we encountered bound states localized at the wall of the cavity. In this paper, we study this situation in detail both for a free particle and for a hydrogen atom centered in a spherical cavity. For appropriate values of the self-adjoint extension parameter, the bound states localized at the wall resonate with the standard hydrogen bound states. We also examine the accidental symmetry generated by the Runge-Lenz vector, which is explicitly broken in a spherical cavity with general Robin boundary conditions. However, for specific radii of the confining sphere, a remnant of the accidental symmetry persists. The same is true for an electron moving on the surface of a finite circular cone, bound to its tip by a 1/r potential. - Highlights: Black-Right-Pointing-Pointer The spectrum of confined electrons and self-adjoint extension parameter. Black-Right-Pointing-Pointer Cavity resonances between hydrogen bound states and states localized at the wall. Black-Right-Pointing-Pointer Accidental symmetry for hydrogen atom confined in a sphere or on a cone.

Authors:
;
Publication Date:
OSTI Identifier:
22157009
Resource Type:
Journal Article
Journal Name:
Annals of Physics (New York)
Additional Journal Information:
Journal Volume: 327; Journal Issue: 11; Other Information: Copyright (c) 2012 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0003-4916
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 74 ATOMIC AND MOLECULAR PHYSICS; ATOMS; BOUND STATE; BOUNDARY CONDITIONS; ELECTRONS; HAMILTONIANS; HYDROGEN; POTENTIALS; RESONANCE; SPHERES; SPHERICAL CONFIGURATION; UNCERTAINTY PRINCIPLE; VECTORS

Citation Formats

Al-Hashimi, M.H., E-mail: hashimi@itp.unibe.ch, and Wiese, U.-J. Self-adjoint extensions for confined electrons: From a particle in a spherical cavity to the hydrogen atom in a sphere and on a cone. United States: N. p., 2012. Web. doi:10.1016/J.AOP.2012.06.006.
Al-Hashimi, M.H., E-mail: hashimi@itp.unibe.ch, & Wiese, U.-J. Self-adjoint extensions for confined electrons: From a particle in a spherical cavity to the hydrogen atom in a sphere and on a cone. United States. doi:10.1016/J.AOP.2012.06.006.
Al-Hashimi, M.H., E-mail: hashimi@itp.unibe.ch, and Wiese, U.-J. Thu . "Self-adjoint extensions for confined electrons: From a particle in a spherical cavity to the hydrogen atom in a sphere and on a cone". United States. doi:10.1016/J.AOP.2012.06.006.
@article{osti_22157009,
title = {Self-adjoint extensions for confined electrons: From a particle in a spherical cavity to the hydrogen atom in a sphere and on a cone},
author = {Al-Hashimi, M.H., E-mail: hashimi@itp.unibe.ch and Wiese, U.-J.},
abstractNote = {In a recent study of the self-adjoint extensions of the Hamiltonian of a particle confined to a finite region of space, in which we generalized the Heisenberg uncertainty relation to a finite volume, we encountered bound states localized at the wall of the cavity. In this paper, we study this situation in detail both for a free particle and for a hydrogen atom centered in a spherical cavity. For appropriate values of the self-adjoint extension parameter, the bound states localized at the wall resonate with the standard hydrogen bound states. We also examine the accidental symmetry generated by the Runge-Lenz vector, which is explicitly broken in a spherical cavity with general Robin boundary conditions. However, for specific radii of the confining sphere, a remnant of the accidental symmetry persists. The same is true for an electron moving on the surface of a finite circular cone, bound to its tip by a 1/r potential. - Highlights: Black-Right-Pointing-Pointer The spectrum of confined electrons and self-adjoint extension parameter. Black-Right-Pointing-Pointer Cavity resonances between hydrogen bound states and states localized at the wall. Black-Right-Pointing-Pointer Accidental symmetry for hydrogen atom confined in a sphere or on a cone.},
doi = {10.1016/J.AOP.2012.06.006},
journal = {Annals of Physics (New York)},
issn = {0003-4916},
number = 11,
volume = 327,
place = {United States},
year = {2012},
month = {11}
}