Consider a large, essentially two-dimensional, rectangular sample of conductor of area $A$, and containing $2 N$ electrons of charge $-e$. Suppose a magnetic field of strength $B$ is applied perpendicularly to the sample. Write down the Landau Hamiltonian for one of the electrons assuming that the electron interacts just with the magnetic field.

[You may ignore the interaction of the electron spin with the magnetic field.]

Find the allowed energy levels of the electron.

Find the total energy of the $2 N$ electrons at absolute zero temperature as a function of $B$, assuming that $B$ is in the range

$$\frac{\pi \hbar N}{e A} \leqslant B \leqslant \frac{2 \pi \hbar N}{e A} .$$

Comment on the values of the total energy when $B$ takes the values at the two ends of this range.