Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 306 / 2 / ii / Solution

For $n>0$, define occupation numbers $N_n,{\tilde{N}}_n\in {\mathbb{N}}_0$. A basis iswhere $\hat{p}|p;0,0⟩=p|p;0,0⟩$ and positive oscillator modes annihilate the vacuum. Since ${\alpha }_{-n}{\alpha }_n$ has eigenvalue $n{N}_n$, the simultaneous eigenvalues are