- #1

- 11

- 0

## Homework Statement

The energy eigenvalues of an s-dimensional harmonic oscillator is:

[tex]\epsilon_j = (j+\frac{s}{2})\hbar\omega[/tex]

show that the jth energy level has multiplicity [tex] \frac{(j + s - 1)!}{j!(s - 1)!}[/tex]

## Homework Equations

partition function: [tex]Z = \Sigma e^{-( (j+\frac{1}{2})\hbar\omega)/kt)}[/tex]

there should be a Sum over j there, but its not showing up.

## The Attempt at a Solution

Besides for drawing a picture or writing out an expansion, I can't come up with a way to calculate this. Infact, no one that I have spoken to has had a good method of calculating this.

Im just wondering how I can get to this answer mathematically. Plenty of resources just state this as the degeneracy of an s dimensional oscillator, but I have yet to see how to calculate it.

Thanks in advanced for any help.

Last edited: