What is an ideal gas? Explain how the microstates of an ideal gas of indistinguishable particles can be labelled by a set of integers. What range of values do these integers take for (a) a boson gas and (b) a Fermi gas?

Let $E_{i}$ be the energy of the $i$-th one-particle energy eigenstate of an ideal gas in thermal equilibrium at temperature $T$ and let $p_{i}\left(n_{i}\right)$ be the probability that there are $n_{i}$ particles of the gas in this state. Given that

$$p_{i}\left(n_{i}\right)=e^{-\beta E_{i} n_{i}} / Z_{i} \quad\left(\beta=\frac{1}{k T}\right)$$

determine the normalization factor $Z_{i}$ for (a) a boson gas and (b) a Fermi gas. Hence obtain an expression for $\bar{n}_{i}$, the average number of particles in the $i$-th one-particle energy eigenstate for both cases (a) and (b).

In the case of a Fermi gas, write down (without proof) the generalization of your formula for $\bar{n}_{i}$ to a gas at non-zero chemical potential $\mu$. Show how it leads to the concept of a Fermi energy $\epsilon_{F}$ for a gas at zero temperature. How is $\epsilon_{F}$ related to the Fermi momentum $p_{F}$ for (a) a non-relativistic gas and (b) an ultra-relativistic gas?

In an approximation in which the discrete set of energies $E_{i}$ is replaced with a continuous set with momentum $p$, the density of one-particle states with momentum in the range $p$ to $p+d p$ is $g(p) d p$. Explain briefly why

$$g(p) \propto p^{2} V$$

where $V$ is the volume of the gas. Using this formula, obtain an expression for the total energy $E$ of an ultra-relativistic gas at zero chemical potential as an integral over $p$. Hence show that

$$\frac{E}{V} \propto T^{\alpha},$$

where $\alpha$ is a number that you should compute. Why does this result apply to a photon gas?

Using the formula $(*)$ for a non-relativistic Fermi gas at zero temperature, obtain an expression for the particle number density $n$ in terms of the Fermi momentum and provide a physical interpretation of this formula in terms of the typical de Broglie wavelength. Obtain an analogous formula for the (internal) energy density and hence show that the pressure $P$ behaves as

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

where $\gamma$ is a number that you should compute. [You need not prove any relation between the pressure and the energy density you use.] What is the origin of this pressure given that $T=0$ by assumption? Explain briefly and qualitatively how it is relevant to the stability of white dwarf stars.