Quantum occupation scale of a classical oscillator ($\bar{n}\simeq m\omega A^2/(2\hslash )$)

A classical oscillator of amplitude $A$ has energy $m{\omega }^2{A}^2/2$. Comparing with the oscillator spacing $\hslash \omega$ gives the large-occupation scale displayed. A coherent state with this classical amplitude has exactly this mean excitation occupation and a Poisson number distribution; its total mean energy also includes the zero-point term. An observed macroscopic amplitude does not, by itself, establish a pure coherent state.