Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 34E / e / Solution

The allowed Landau level are $E_n=\hslash \omega (n+1/2)$. The commuting $b^{†}$ creates distinct guiding-centre states without changing energy, so every level is infinitely degenerate on the plane; in finite area $A$ the degeneracy is approximately $A/(2\pi r_B^2)$.