A free spinless particle moving in two dimensions is confined to a square box of side $L$. By imposing periodic boundary conditions show that the number of states in the energy range $\epsilon \rightarrow \epsilon+d \epsilon$ is $g(\epsilon) d \epsilon$, where

$$g(\epsilon)=\frac{m L^{2}}{2 \pi \hbar^{2}}$$

If, instead, the particle is an electron with magnetic moment $\mu$ moving in a constant external magnetic field $H$, show that

$$g(\epsilon)= \begin{cases}\frac{m L^{2}}{2 \pi \hbar^{2}}, & -\mu H<\epsilon<\mu H \\ \frac{m L^{2}}{\pi \hbar^{2}}, & \mu H<\epsilon\end{cases}$$

Let there be $N$ electrons in the box. Explain briefly how to construct the ground state of the system. Let $\epsilon$ be the Fermi energy. Show that when $\epsilon>\mu H$

$$N=\frac{m L^{2}}{\pi \hbar^{2}} \epsilon .$$

Show also that the magnetic moment $M$ of the system in its ground state is given by

$$M=\frac{\mu^{2} m L^{2}}{\pi \hbar^{2}} H$$

and that the ground state energy is

$$\frac{1}{2} \frac{\pi \hbar^{2}}{m L^{2}} N^{2}-\frac{1}{2} M H$$