Harmonic Oscillator in a 1D or 2D Cavity with General Perfectly Reflecting Walls

TL;DR

This paper explores the energy spectra of harmonic oscillators confined in 1D and 2D cavities with perfectly reflecting walls, revealing bound states influenced by self-adjoint extension parameters and connecting to generalized uncertainty principles.

Contribution

It introduces a detailed analysis of harmonic oscillators in finite cavities considering self-adjoint extensions, extending the understanding of boundary effects on quantum states.

Findings

Bound states localized at cavity walls for negative extension parameters

Energy spectra depend on self-adjoint extension parameters

Comparison with free particle in a circular cavity

Abstract

We investigate the simple harmonic oscillator in a 1-d box, and the 2-d isotropic harmonic oscillator problem in a circular cavity with perfectly reflecting boundary conditions. The energy spectrum has been calculated as a function of the self-adjoint extension parameter. For sufficiently negative values of the self-adjoint extension parameter, there are bound states localized at the wall of the box or the cavity that resonate with the standard bound states of the simple harmonic oscillator or the isotropic oscillator. A free particle in a circular cavity has been studied for the sake of comparison. This work represents an application of the recent generalization of the Heisenberg uncertainty relation related to the theory of self-adjoint extensions in a finite volume.