Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 34D / b / Solution

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 $eB$.