(i) Each particle in a system of $N$ identical fermions has a set of energy levels $E_{i}$ with degeneracy $g_{i}$, where $i=1,2, \ldots$. Derive the expression

$$\bar{N}_{i}=\frac{g_{i}}{e^{\beta\left(E_{i}-\mu\right)}+1},$$

for the mean number of particles $\bar{N}_{i}$ with energy $E_{i}$. Explain the physical significance of the parameters $\beta$ and $\mu$.

(ii) The spatial eigenfunctions of energy for an electron of mass $m$ moving in two dimensions and confined to a square box of side $L$ are

$$\psi_{n_{1} n_{2}}(\mathbf{x})=\frac{2}{L} \sin \left(\frac{n_{1} \pi x}{L}\right) \sin \left(\frac{n_{2} \pi y}{L}\right)$$

where $n_{i}=1,2, \ldots(i=1,2)$. Calculate the associated energies.

Hence show that when $L$ is large the number of states in energy range $E \rightarrow E+d E$ is

$$\frac{m L^{2}}{2 \pi \hbar^{2}} d E$$

How is this formula modified when electron spin is taken into account?

The box is filled with $N$ electrons in equilibrium at temperature $T$. Show that the chemical potential $\mu$ is given by

$$\mu=\frac{1}{\beta} \log \left(e^{\beta \pi \hbar^{2} \rho / m}-1\right)$$

where $\rho$ is the number of particles per unit area in the box.

What is the value of $\mu$ in the limit $T \rightarrow 0$ ?

Calculate the total energy of the lowest state of the system of particles as a function of $N$ and $L$.