(i) A spinless quantum mechanical particle of mass $m$ moving in two dimensions is confined to a square box with sides of length $L$. Write down the energy eigenfunctions for the particle and the associated energies.

Show that, for large $L$, the number of states in the energy range $E \rightarrow E+d E$ is $\rho(E) d E$, where

$$\rho(E)=\frac{m L^{2}}{2 \pi \hbar^{2}}$$

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

$$\begin{array}{rlr} \rho(E) & =\frac{m L^{2}}{2 \pi \hbar^{2}}, & -\mu H<E<\mu H \\ & =\frac{m L^{2}}{\pi \hbar^{2}}, & \mu H<E \end{array}$$

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

$$N=\frac{m L^{2}}{\pi \hbar^{2}} E_{F}$$

Show also that the magnetic moment, $M$, of the system in the ground state is

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

and that the ground state energy is

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

Part II