Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 34E (Applications of Quantum Mechanics)

Minimal coupling gives $H=(p-qA{)}^2/(2m)$ and $i\hslash {\partial }_t\psi =H\psi$.

Writing $\pi =p-qA$ gives $[{\pi }_x,{\pi }_y]=iq\hslash B$. Hence $[{\rho }_x,{\rho }_y]=i\hslash /(qB)$ and $[R_x,R_y]=-i\hslash /(qB)$ (with signs reversed together if conventions change); all mixed $[R_i,{\rho }_j]$ vanish.

With $r_B^2=\hslash /(qB)$, direct substitution gives $[a,a^{†}]=[b,b^{†}]=1$.

Since ${\pi }^2=(qB{)}^2{\rho }^2$, oscillator algebra gives $H=\hslash \omega (a^{†}a+1/2)$ with cyclotron frequency $\omega =qB/m$.

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 $A/(2\pi r_B^2)$.