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Partition function problems

  1. Aug 2, 2011 #1
    1. The problem statement, all variables and given/known data

    1. If the system, which has N identical particles, only has two possible energy states

    E=0,e(e is an energy) ,what's the ensemble average of E?

    2. Find the partition function which has two identical Fermion system if the energy states only have

    E=0,e.


    2. Relevant equations

    I think what's different about partition function for identical Boson, Fermion and Boltmann stat particle?


    3. The attempt at a solution

    I made a solution but I'm not sure this is wrong or correct.

    1. Z1 = 1 +exp(-e/kT)

    Then, the total partition function is Z = (Z1)^N / (N!).

    So Ensemble average of energy is <E> = (1/Z) * e * exp(-e/kt).

    This is my solution.

    Is this right or not?

    2. Z1 = 1+exp(-e/kT)

    Because of identical these Fermion -> Z = 1/2 * Z1^2

    .....But it should be changed due to property of Fermion.

    The first identical particle can occupy E=0,e , so Z1 = 1+exp(-e/kT).

    Then, if first particle occupy state which state energy is 0, Z2 can only exp(-e/kT).

    Therefore partition function is (1+exp(-e/kT))*(-e/kT).

    And another sitution is first particle occupy energy e state.

    Then, another partition function is (1+exp(-e/kT)*1.

    I thought this is connected by linearly

    Final partition function is Z = 1/2 * (1/2*((1+exp(-e/kT))*(-e/kT)+ (1+exp(-e/kT)*1))

    But I believe something is wrong this second problem.

    What's the wrong?
     
  2. jcsd
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