Harmonic-oscillator recoil matrix element ($⟨n|e^{-ikx}|0⟩=e^{-{\eta }^2/2}(-i\eta {)}^n/\sqrt{n!}$)

For normalized oscillator states and $x=\sqrt{\hslash /(2m\omega )}(a+a^{†})$, define $\eta =k\sqrt{\hslash /(2m\omega )}$. Normal ordering the displacement operator gives the displayed matrix element. A sudden momentum kick therefore creates a Poisson distribution of oscillator occupations with mean ${\eta }^2$. Its mean added motional energy is ${\eta }^2\hslash \omega ={\hslash }^2{k}^2/(2m)$, although most events retain $n=0$ when $\eta \ll 1$.

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.