(a) Consider an ideal gas of Fermi particles obeying the Pauli exclusion principle with a set of one-particle energy eigenstates $E_{i}$. Given the probability $p_{i}\left(n_{i}\right)$ at temperature $T$ that there are $n_{i}$ particles in the eigenstate $E_{i}$ :

$$p_{i}\left(n_{i}\right)=\frac{e^{\left(\mu-E_{i}\right) n_{i} / k T}}{Z_{i}},$$

determine the appropriate normalization factor $Z_{i}$. Use this to find the average number $\bar{n}_{i}$ of Fermi particles in the eigenstate $E_{i}$.

Explain briefly why in generalizing these discrete eigenstates to a continuum in momentum space (in the range $p$ to $p+d p$ ) we must multiply by the density of states

$$g(p) d p=\frac{4 \pi g_{s} V}{h^{3}} p^{2} d p$$

where $g_{s}$ is the degeneracy of the eigenstates and $V$ is the volume.

(b) With the energy expressed as a momentum integral

$$E=\int_{0}^{\infty} E(p) \bar{n}(p) d p$$

consider the effect of changing the volume $V$ so slowly that the occupation numbers do not change (i.e. particle number $N$ and entropy $S$ remain fixed). Show that the momentum varies as $d p / d V=-p / 3 V$ and so deduce from the first law expression

$$\left(\frac{\partial E}{\partial V}\right)_{N, S}=-P$$

that the pressure is given by

$$P=\frac{1}{3 V} \int_{0}^{\infty} p E^{\prime}(p) \bar{n}(p) d p .$$

Show that in the non-relativistic limit $P=\frac{2}{3} U / V$ where $U$ is the internal energy, while for ultrarelativistic particles $P=\frac{1}{3} E / V$.

(c) Now consider a Fermi gas in the limit $T \rightarrow 0$ with all momentum eigenstates filled up to the Fermi momentum $p_{\mathrm{F}}$. Explain why the number density can be written as

$$n=\frac{4 \pi g_{s}}{h^{3}} \int_{0}^{p_{\mathrm{F}}} p^{2} d p \propto p_{\mathrm{F}}^{3}$$

From similar expressions for the energy, deduce in both the non-relativistic and ultrarelativistic limits that the pressure may be written as

$$P \propto n^{\gamma}$$

where $\gamma$ should be specified in each case.

(d) Examine the stability of an object of radius $R$ consisting of such a Fermi degenerate gas by comparing the gravitational binding energy with the total kinetic energy. Briefly point out the relevance of these results to white dwarfs and neutron stars.