Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 33C / d / Solution

Contracting the result of part (c) with the Pauli matrix givesThus the $T^a$ form the Schwinger boson representation of the $\mathfrak{su}(2)$ commutation relations. The second Pauli matrix givesand therefore

On the energy level $E=N+1$, the total occupation is $N$. The $N+1$ states $|n_x,N-n_x⟩$ form the spin-$j=N/2$ irreducible representation. Every component of its angular momentum generator has eigenvaluesSince $L=2T^2$, the possible orbital angular momenta areThere are $N+1$ distinct values, exactly matching the degeneracy found in part (a); hence each occurs once. Restoring $\hslash$ multiplies every $\ell$ by $\hslash$.