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A system has non-degenerate energy levels with energy

  1. Oct 19, 2014 #1
    1. The problem statement, all variables and given/known data
    A system has non-degenerate energy levels with energy[tex] \epsilon=(n+1/2)\hbar\omega [/tex] where h-bar*omega=1.4*10^-23J and n positive integer zero what is the probability that it is in n=1 state with a heat bath of temperature 1K
    2. Relevant equations
    [tex]
    Z=\exp^\frac{-E_i}{k_b T} \\
    p_r=\frac{\exp^\frac{-E_i}{k_b T}}{\sum^N_j \exp^\frac{-E_j}{k_b T}}
    [/tex]

    3. The attempt at a solution
    I'm not really sure what to do now, I dont know how to sum the total number of states to get the fraction of states in the n=1 state
     
    Last edited by a moderator: Oct 19, 2014
  2. jcsd
  3. Oct 19, 2014 #2

    mfb

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    2016 Award

    Staff: Mentor

    The sum is a sum over q^i for some q<1, this has an analytic result. You can just plug in all values and calculate the result.
     
  4. Oct 19, 2014 #3
    Ok I think I might have gotten it, to deal with the infinite sum use a geometric series,

    [tex]

    \sum_0^\inf e^\frac{-(n+\frac{1}{2})}{k_b T} \\

    =e^\frac{-\hbar\omega}{2k_b T}\sum_0^\inf e^\frac{-n}{k_b T}\\

    =\frac{e^\frac{-\hbar\omega}{2k_b T}}{1-e^\frac{-\hbar\omega}{k_b T}

    [/tex]

    then evaluate using the pr as stated before.

    Also I don't know why my LaTeX is not displaying correctly.
     
  5. Oct 20, 2014 #4

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    Some error, probably with brackets.
    Yes the approach is good.
     
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