Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 33C (Principles of Quantum Mechanics)

Work in units with $\hslash =1$. For unit mass and frequency, the two-dimensional isotropic harmonic oscillator hasThe product number states are therefore energy eigenvectors:For the level $E=N+1$, the possibilities areso its degeneracy isRestoring units multiplies every energy by $\hslash \omega$.

The planar orbital angular momentum isThe canonical commutation relations giveConsequentlyThus angular momentum preserves each energy eigenspace. More explicitly,soWithin a degenerate energy eigenspace, this argument places no further restriction on the matrix of $L$.

For $T_{ij}=a_i^{†}{a}_j$, expand the commutator and commute one annihilation operator past one creation operator:Hence the oscillator bilinear commutator isSincewe obtainEvery $T_{ij}$ therefore acts within a fixed energy eigenspace.

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 $\hslash$.