Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 34D (Applications of Quantum Mechanics)

Take $B>0$ and use the Landau gauge $\mathbf{A}=(0,Bx,0)$. With zero $z$-momentum, the Hamiltonian becomesFor a state $e^{i{k}_{y}y}\chi (x)$, this is a quantum harmonic oscillator in $x$, centred atwith cyclotron frequencyThe Landau levels are therefore

Periodicity in $y$ quantizes $k_y=2\pi \ell /R_y$. Requiring the corresponding oscillator centre to lie in an $x$ interval of length $R_x$ leavesallowed centres for each $n$. This is the degeneracy of a Landau level, equal to the magnetic flux through the rectangle divided by the magnetic flux quantum $h/|e|$. Consistent magnetic periodic boundary conditions make $D$ an integer.

Assume $eB>0$; reversing the sign only exchanges the spin labels. Since $\mathbf{B}\cdot \mathbf{\sigma }=B{\sigma }_3$, the Pauli term shifts the spinless energies by $\pm \hslash {\omega }_c/2$. ThusAfter collecting equal energies, the spin splitting of Landau levels isThe level $\ell =0$ contains only $(n,sigma)=(0,-1)$ and has degeneracy $D$. Every level $\ell \ge 1$ contains $(\ell ,-1)$ and $(\ell -1,+1)$ and has degeneracy $2D$.

For $N$ noninteracting electrons, the Pauli exclusion principle fills these states from the bottom. If $0\le N\le D$, every electron fits in the zero-energy level andFor $N>D$, writeThen the first $k$ positive levels are full and $r$ states in level $k+1$ are occupied. The ground-state energy isEquivalently, onThe graph is continuous and piecewise linear: it is flat up to $N=D$, then has slopes $\hslash {\omega }_c,2\hslash {\omega }_c,3\hslash {\omega }_c,\dots$, with kinks at