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Homework Help: 2D fermi gas

  1. Mar 13, 2014 #1
    1. The problem statement, all variables and given/known data

    Show that PV = E
    2. Relevant equations

    [tex] E= \int^\infty_0 D(\epsilon)n_{FD}(\epsilon) \epsilon \cdot d\epsilon[/tex]
    [tex]n_{FD}=\frac{1}{(1+ e^{-(\alpha +\beta \epsilon_k)})}[/tex]

    [tex]\psi(\alpha ,\beta, V) =\beta PV =\sum_\vec{k} \ln{(1+e^{-(\alpha +\beta \epsilon_k)}) }[/tex]

    and in an earlier problem I found that

    [tex] D= \frac{A\cdot m \cdot (2s+1)}{2\pi \hbar^2 } [/tex]

    3. The attempt at a solution

    I think I'm supposed to wirte $$ \frac{\psi}{\beta }$$ as an integral over $$\epsilon$$ and then compare it to the integral for E that I gave, but I have no clue on how to do this. I also have no idea on how to get rid of the logarithm.
  2. jcsd
  3. Mar 14, 2014 #2

    Simon Bridge

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    Science Advisor
    Homework Helper

    If ##PV=E##, then, off what you wrote above: $$\frac{1}{\beta}\sum_\vec{k} \ln{(1+e^{-(\alpha +\beta \epsilon_k)}) }=\int^\infty_0 D(\epsilon)n_{FD}(\epsilon) \epsilon \cdot d\epsilon$$ ... so you need to play around with this expression a bit to see if you can make LHS look like the RHS.

    i.e. can you take the derivative of both sides?
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