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Homework Help: Partition function for harmonic oscillators

  1. Feb 19, 2017 #1
    1. The problem statement, all variables and given/known data
    Calculate the partition function, the entropy and the heat capacity of a system of N independent harmonic oscillators, with hamiltonian ##H = \sum_1^n(p_i^2+\omega^2q_i^2)##
    2. Relevant equations
    ##Z = \sum_E e^{-E/kT}##

    3. The attempt at a solution
    I am not really sure what to do. From what I see this is not a quantum oscillator (and we haven't covered that in class anyway) so the E in the formula for partition function would have any value from 0 to infinity so It would turn into an integral. But this can be argued for a single classical harmonic oscillator, too, so I don't know where to use the fact that there are N of them.
  2. jcsd
  3. Feb 19, 2017 #2


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    It looks like the Hamiltonian for a quantum mechanical HO, I would say. ##m## has been set to 1 for convenience. The link tells you energy levels, which you will need.

    What are you summing over to get your partition function ? I don't see the oscillators (for example if there are 100 oscillators, how does that end up in your summation?)

    Be sure to keep the numbering of the energy levels of a single oscillator and the numbering of the oscillators themselves well distinguished.
  4. Feb 19, 2017 #3


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    Check out https://en.wikipedia.org/wiki/Partition_function_(statistical_mechanics) and read about the canonical partition function. The fact you have ##N## oscillators is reflected in the Hamiltonian.
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