= Solution
The <quantum harmonic oscillator> Hamiltonian is
$H=\hbar\omega(N+\tfrac12)$. Therefore
$$
e^{-iHt/\hbar}|z\rangle
=e^{-i\omega t/2}|ze^{-i\omega t}\rangle,
$$
so the evolved state remains an $A$-eigenstate, with eigenvalue
$\boxed{ze^{-i\omega t}}$.
The coherent-state overlap gives
$$
\left|\langle z|ze^{-i\omega t}\rangle\right|^2
=\exp\!\left[-2|z|^2(1-\cos\omega t)\right]
=\exp\!\left[-4|z|^2\sin^2\frac{\omega t}{2}\right].
$$
For the classical period $T=2\pi/\omega$, this probability equals one whenever $t=kT$.
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