Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 34A (Principles of Quantum Mechanics)

From the number operator commutator $[N,A]=-A$,so $A|n⟩$ is proportional to $|n-1⟩$. Its squared norm isWith the conventional phase choice,

Orthogonality of the energy eigenstates givesUsing the lowering-operator action from part (a) and shifting the summation index,Thus this coherent state is a normalized eigenstate of $A$ with eigenvalue $z$.

The number probabilities area Poisson distribution of parameter $|z{|}^2$. HenceThe relative uncertainty is $1/|z|=1/\sqrt{⟨N⟩}$, which tends to zero as the mean occupation tends to infinity.

The quantum harmonic oscillator Hamiltonian is
$H=\hslash \omega (N+\frac{1}{2})$. Thereforeso the evolved state remains an $A$-eigenstate, with eigenvalue
$\boxed{z{e}^{-i\omega t}}$.

The coherent-state overlap givesFor the classical period $T=2\pi /\omega$, this probability equals one whenever $t=kT$.