Self-adjoint Extensions for Confined Electrons:from a Particle in a Spherical Cavity to the Hydrogen Atom in a Sphere and on a Cone

TL;DR

This paper investigates self-adjoint extensions of the Hamiltonian for confined electrons, exploring bound states at cavity walls, symmetry breaking, and specific geometric conditions in spherical and conical confinements.

Contribution

It provides a detailed analysis of self-adjoint extensions for electrons in spherical and conical geometries, revealing conditions for bound states and symmetry preservation.

Findings

Bound states localized at cavity walls resonate with hydrogen states.

Symmetry generated by the Runge-Lenz vector is broken but can persist at specific radii.

Electron confinement on a cone exhibits similar symmetry considerations.

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.