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Homework Help: 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


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    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.
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