Coherent-state oscillator expectation values ($⟨x{⟩}_t=\sqrt{2\hslash /(m\omega )}\mathrm{Re}(\alpha e^{-i\omega t})$)

A normalized oscillator coherent state obeys $a|\alpha ⟩=\alpha |\alpha ⟩$, so its mean occupation number is $N=|\alpha {|}^2$. Under the quantum harmonic oscillator Hamiltonian operator $\hslash \omega (a^{†}a+1/2)$, its evolution is $|\alpha (t)⟩=e^{-i\omega t/2}|\alpha e^{-i\omega t}⟩$. Inserting $x=\sqrt{\hslash /(2m\omega )}(a+a^{†})$ and $p=-i\sqrt{m\hslash \omega /2}(a-a^{†})$ givesThese mean values obey the classical harmonic oscillator equations. The displacement amplitude is $\sqrt{2\hslash N/(m\omega )}$, whose classical orbit has energy $\hslash \omega N$. The full quantum mean energy has the additional zero-point energy $\hslash \omega /2$. Indeed the variances are $\hslash /(2m\omega )$ and $m\hslash \omega /2$, and their kinetic and potential contributions sum to $\hslash \omega /2$.