Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 34B / b / i / Solution

For independent frequencies $1$ and $2$,
$$
H_0=N_A+2N_B+\frac32,
$$
so the eigenspace at energy $k+3/2$ has one basis state $|n,m\rangle$ for each nonnegative solution of $n+2m=k$.


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 Incoming links: Two commensurate quantum harmonic oscillators