1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Multiplicity of s-dimensional Harmonic oscillator

  1. May 20, 2010 #1
    1. The problem statement, all variables and given/known data
    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]

    2. Relevant 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.

    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. jcsd
  3. May 23, 2010 #2


    User Avatar
    Homework Helper

    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
    So, this is a basic combinatorics problem with a constraint: You must choose s numbers, and they must add up to j.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook