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Microstates and oscillators help

  1. Feb 25, 2008 #1
    1. The problem statement, all variables and given/known data
    The number of microstates of a system of N oscillators containing Q quanta of energy hw is given by

    W(N,Q) = (N+Q-1)!/[(N-1)!Q!]

    Show that when one further quantum is added to the system the number of microstates increases by a factor of approximately (1+N/Q), provided that N,Q>>1.


    2. Relevant equations



    3. The attempt at a solution

    So W(N,Q+1) = (N+Q)!/[(N-1)!(Q+1)!]

    My problem class leader showed us how to do the question but im unsure of how he did it, so this is what he did:

    W(N,Q+1)/W(N,Q)= [(N+Q)!(N-1)!Q!]/[(N-1)!(Q+1)!(N+Q-1)!]

    = [(N+Q)!Q!]/[(Q+1)!(N+Q-1)!] by cancelling

    = N+Q/Q+1 = 1+N/Q

    I think he got to the last step by approxiamtion since N,Q>>1 but i don't see how that works. Also i don't know why there aren't factorial signs there, whether he or i forgot to write them in or whether they disappear for some reason.

    Any explanations would be so helpful, thanks
     
  2. jcsd
  3. Feb 25, 2008 #2

    kdv

    User Avatar


    First, notice that (Q+1)!/Q! = Q+1 Do you see why?

    In general, if you have (X+1)!/ X! , this is equal to X+1 (where X can be anything)

    Now, note that [tex] \frac{N+Q}{Q+1} = \frac{1 + N/Q}{1+1/Q} \approx 1+N/Q - 1/Q - N/Q^2 \ldots[/tex] where I have used the binomial expansion and have neglected the corrections of order [itex]1/Q^2, N/Q^2 [/itex] and higher .
     
    Last edited: Feb 25, 2008
  4. Feb 25, 2008 #3
    Yes i see it now, thankyou so much.
     
  5. Feb 25, 2008 #4

    kdv

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    I made a few typos in my post (now corrected)

    I meant to say (Q+1)!/Q! = Q+1

    and (X+1)!/X! = X+1

    Sorry for the typos.

    And you are welcome.
     
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