# 2D fermi gas

1. Mar 13, 2014

### dumbperson

1. The problem statement, all variables and given/known data

Show that PV = E
2. Relevant equations

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

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

and in an earlier problem I found that

$$D= \frac{A\cdot m \cdot (2s+1)}{2\pi \hbar^2 }$$

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. Mar 14, 2014

### Simon Bridge

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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