Exact solutions for the D-dimensional spherical isotropic confined harmonic oscillator

TL;DR

This paper provides exact and highly accurate solutions for the energy levels of a D-dimensional isotropic harmonic oscillator confined within a finite box, using two different methods for dimensions 1 through 5.

Contribution

It introduces two approaches—exact solution and series expansion—to compute energy eigenvalues for confined harmonic oscillators in multiple dimensions, with unprecedented numerical precision.

Findings

Numerical results are highly accurate, reported with 50 decimal places.

Both methods agree closely across dimensions 1 to 5.

The approaches effectively handle boundary confinement in quantum oscillators.

Abstract

We study the size effect on the energy levels of the D-dimensional isotropic harmonic oscillator confined within a box of radius $r_c$ with impenetrable walls. Two different approaches are used to obtain the energy eigenvalues and eigenfunctions for D=1,2,...,5. In the first we solve the Schroedinger equation exactly. In the second we use a series expansion of the wave function. The numerical results obtained are extremely accurate; these values are reported with 50 decimal places.