Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 34D (Applications of Quantum Mechanics)

With the magnetic minimal coupling convention for charge $e$,Here $\nabla \cdot \mathbf{A}=0$, ${\mathbf{A}}^2=B^2(x^2+y^2)/4$, andTherefore in Cartesian coordinatesIn cylindrical coordinates this isThis choice of $\mathbf{A}$ is the symmetric gauge.

The factor $e^{i{k}_{z}z}$ is an eigenstate of longitudinal momentum with $p_z=\hslash k_z$. The integer $n$ is the azimuthal angular momentum quantum number, since $L_z=-i\hslash {\partial }_{\theta }$ has eigenvalue $n\hslash$.

Fordirect differentiation givesRegularity and cancellation of the $r^{-2}$ term require $\alpha =n$ for $n\ge 0$. Cancellation of the $r^2$ terms requiresassuming $eB>0$. The remaining radial energy is independent of $n$:This is the lowest Landau level. If $eB<0$, the same family uses nonpositive angular-momentum quantum numbers and $|eB|$ replaces $eB$.

The radial factoris sharply peaked, for large $n$, nearRequiring the orbit to lie inside a disk of radius $R$ permitsThus the total degeneracy at fixed $k_z$ is, up to the negligible boundary correction,Since the disk has area $\pi R^2$, the degeneracy of a Landau level per unit area isFor either sign of the charge-field product, replace $eB$ by $|eB|$.