Past exam of the physics A course of the University of Cambridge / 2021 / Paper 1 / Section B / B7 / a / Solution

Write $a=a_{-}$ and $a^{†}=a_{+}$. Inverting the annihilation operator and creation operator definitions givesThe ladder operators take $|n⟩$ to orthogonal quantum harmonic oscillator energy eigenstates $|n-1⟩$ and $|n+1⟩$. Thus $⟨n|a|n⟩=⟨n|a^{†}|n⟩=0$, andFor $n=0$, the lowering term is simply zero. The conclusion concerns an individual energy eigenstate; interference between different eigenstates can produce nonzero expectation values.