# Multiplicity of s-dimensional Harmonic oscillator

1. May 20, 2010

### cp51

1. The problem statement, all variables and given/known data
The energy eigenvalues of an s-dimensional harmonic oscillator is:

$$\epsilon_j = (j+\frac{s}{2})\hbar\omega$$

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

2. Relevant equations
partition function: $$Z = \Sigma e^{-( (j+\frac{1}{2})\hbar\omega)/kt)}$$
there should be a Sum over j there, but its not showing up.

3. 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: May 20, 2010
2. May 23, 2010

### turin

I don't know how you might use the partition function for this. Anyway, how many different ways are there to excite the dimensional modes to obtain a given energy? For example, if the energy level is j=3 in 3-D, then there are three "sub-j"s, j1, j2 and j3, one for each dimension, and they must satisfy j1+j2+j3=j. So, you can have:
j1=0, j2=0, j3=3
j1=0, j2=1, j3=2
etc.
So, this is a basic combinatorics problem with a constraint: You must choose s numbers, and they must add up to j.