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SM J04 2

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Consider a gas of N nonrelativistic fermions with spin 1/2 and mass m initially at zero temperature and confined in a volume $V 0$ and kept at.

a. Express the kinetic energy of the gas in terms of N and $V 0$ .

b. What is the pressure of the gas? You can assume here that the gas is ideal.

c. Now the gas is allowed to expand to the volume $V_{1}\gg V_{0}$ without any energy exchange with the outside world. Calculate the temperature of the gas after it will reach an equilibrium due to weak interactions between the fermions.

d. What is the pressure of the gas in the final state.

Define $L=V_{0}^{1/3}$ as the length of one side of the cube containing the gas. We can write the wavefunction of one of the fermions as:

$\psi=c_{1}e^{i(\mathbf{k}\cdot\mathbf{x})}$

Because we are limited to volume $V 0$ , we must have:

$k_{x}=\frac{\pi n_{x}}{L};\quad k_{y}=\frac{\pi n_{y}}{L};\quad k_{z}=\frac{\pi n_{z}}{L}$

So that the total energy is:

$U=\frac{\hbar^{2}}{2m}\sum_{i=1}^{N}\left(k_{x}^{2}+k_{y}^{2}+k_{z}^{2}\right)=\frac{\hbar^{2}\pi^{2}}{2mL^{2}}\sum_{i=1}^{N}\left(n_{x}^{2}+n_{y}^{2}+n_{z}^{2}\right)$

Since T=0, the fermions are in their minimum occupancy, but since they have spin 1/2 each level can be filled by only two fermions. Thus:

$U=\frac{6\hbar^{2}\pi^{2}}{2mL^{2}}\sum_{n=1}^{N/6}n^{2}\approx\frac{6\hbar^{2}\pi^{2}}{2mL^{2}}\frac{1}{3}\left(\frac{N}{6}\right)^{3}=\frac{\hbar^{2}\pi^{2}}{mV_{0}^{2/3}}\left(\frac{N}{6}\right)^{3}$

Where we used the limit of very large N for the second equality.

We can use the ideal gas equation: