Harmonic oscillator in a 1D or 2D cavity with general perfectly reflecting walls

Abstract

We investigate the simple harmonic oscillator in a 1D box, and the 2D 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.

Keywords:

• vibrational spectrum in finite volume
• cavity resonance
• self-adjoint extensions

Acknowledgements

The author would like to thank U.-J. Wiese for his careful reading of this article, and for his illuminating suggestions and discussions. This work is supported in parts by the Schweizerischer Nationalfonds (SNF). The author also likes to thank the city of Bern for their support in the framework of the Swiss national qualification program Biomedizin-Naturwissenschaft-Forschung (BNF). The ‘Albert Einstein Center for Fundamental Physics’ at Bern University is supported by the ‘Innovations- und Kooperationsprojekt C-13’ of the Schweizerische Universitätskonferenz (SUK/CRUS).