Eigenspectrum properties of the confined 3D harmonic oscillator

Abstract

An isotropic 3D harmonic oscillator centrally enclosed in a spherical box with impenetrable walls is treated by analytical methods. It is explicitly shown how imposing the Dirichlet boundary condition on the wavefunctions results in (a) the complete removal of any systematic degeneracy of levels, and (b) the constant energy difference of exactly two harmonic oscillator units between all successive pairs of the confined excited states defined by the orbital quantum numbers l and l+2, specifically when the radius of the spherical box is chosen to be the position of the only radial node in the first excited-state wavefunction of the unconfined harmonic oscillator for a given l. The lowest confined state of the given l is excluded in defining the successive pairs of excited confined states.