# 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}

}