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Partition sum of particle, high/low temperature limits

  1. Feb 29, 2012 #1
    1. The problem statement, all variables and given/known data
    We have a single particle that can be in one of three different microstates, [itex]\epsilon_0[/itex], [itex]\epsilon_1[/itex] or [itex]\epsilon_2[/itex], with [itex]\epsilon_0 < \epsilon_1 < \epsilon_2[/itex]. The particle is in thermal equilibrium with a heat bath at temperature T.

    1) Calculate the canonical partition function.

    2) Give the probabilities [itex]P_j[/itex] that this particle is in state [itex]j[/itex] for [itex]j = 0, 1, 2[/itex].

    3) What is the average energy in the high and low temperature limit?

    3. The attempt at a solution

    1) [tex]Z(\beta) = e^{- \beta \epsilon_0} + e^{- \beta \epsilon_1} + e^{- \beta \epsilon_2} [/tex]

    2)

    [tex]P(0) = \frac{e^{- \beta \epsilon_0}}{Z(\beta)}[/tex]

    [tex]P(1) = \frac{e^{- \beta \epsilon_1}}{Z(\beta)}[/tex]

    [tex]P(2) = \frac{e^{- \beta \epsilon_2}}{Z(\beta)}[/tex]

    3)

    [tex]
    \langle E \rangle = \epsilon_0 \cdot P(0) + \epsilon_1 \cdot P(1) + \epsilon_2 \cdot P(2) \\
    = \frac{\epsilon_0 e^{- \beta \epsilon_0} + \epsilon_1 e^{- \beta \epsilon_1} + \epsilon_2 e^{- \beta \epsilon_2}}{e^{- \beta \epsilon_0} + e^{- \beta \epsilon_1} + e^{- \beta \epsilon_2}}
    [/tex]

    So then I just plug in [itex]\beta = \frac{1}{k_B T}[/itex] and see what happens when T gets very low or very high, and then for high temperatures I get [itex]U = \frac{\epsilon_0 + \epsilon_1 + \epsilon_2}{3}[/itex] and for low temperatures it goes to 0. Is this correct?

    Edit: I meant partition function in the title.
     
    Last edited: Feb 29, 2012
  2. jcsd
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