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Self-adjoint Extensions for Confined Electrons:from a Particle in a Spherical Cavity to the Hydrogen Atom in a Sphere and on a Cone
- Source :
- Annals of Physics 327 (2012) 1-28
- Publication Year :
- 2012
-
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 lo calized 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.<br />Comment: 22 pages, 9 Figures
- Subjects :
- Quantum Physics
High Energy Physics - Theory
Mathematical Physics
Subjects
Details
- Database :
- arXiv
- Journal :
- Annals of Physics 327 (2012) 1-28
- Publication Type :
- Report
- Accession number :
- edsarx.1204.3434
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1016/j.aop.2012.06.006