Conditional motional heating and recoil averaging

A transition between oscillator energy eigenstates that adds one quantum changes kinetic and potential expectation energies by $\hslash \omega /2$ each. A sudden unobserved momentum kick instead initially adds its energy only to kinetic motion and leaves the position distribution unchanged. Coherent superpositions distinguish these descriptions. Averaging over motional outcomes gives a recoil-scale total energy; time averaging or dephasing gives equal kinetic and potential shares. Broad-line transition rates need not be exactly equal to bare overlap probabilities if finite spectral weights matter.