= Harmonic-oscillator recoil matrix element
{title2=$\langle n|e^{-ikx}|0\rangle=e^{-\eta^2/2}(-i\eta)^n/\sqrt{n!}$}
For normalized oscillator states and $x=\sqrt{\hbar/(2m\omega)}(a+a^\dagger)$, define $\eta=k\sqrt{\hbar/(2m\omega)}$. 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 $\eta^2\hbar\omega=\hbar^2k^2/(2m)$, although most events retain $n=0$ when $\eta\ll1$.
Back to article page
