For $n>0$, define occupation numbers $N_n,\widetilde N_n\in\mathbb N_0$. A basis is
$$
\boxed{|p;\{N_n\},\{\widetilde N_n\}\rangle
\propto\prod_{n\geq1}(\alpha_{-n})^{N_n}
(\widetilde\alpha_{-n})^{\widetilde N_n}|p;0,0\rangle},
$$
where $\widehat p|p;0,0\rangle=p|p;0,0\rangle$ and positive oscillator modes annihilate the vacuum. Since $\alpha_{-n}\alpha_n$ has eigenvalue $nN_n$, the simultaneous eigenvalues are
$$
\boxed{H=\frac{p^2}{2}-\frac1{12}
+\sum_{n=1}^\infty n(N_n+\widetilde N_n)},
$$
$$
\boxed{P=\sum_{n=1}^\infty n(\widetilde N_n-N_n)}.
$$


Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-306/2/ii/solution