= Conditional motional heating and recoil averaging
A transition between oscillator <energy> <eigenstates> that adds one quantum changes kinetic and potential expectation energies by $\hbar\omega/2$ 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.
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