= Displacement operator
{title2=$D(\alpha)=e^{\alpha a^\dagger-\alpha^*a}$}
{wiki}
The displacement operator translates oscillator <phase space>. It is <unitary> and obeys $D(\alpha)^\dagger aD(\alpha)=a+\alpha$ and $D(\alpha)|0\rangle=|\alpha\rangle$, a <coherent state>. The <Baker--Campbell--Hausdorff formula> with a scalar <commutator> gives
$$
D(\alpha)=e^{-|\alpha|^2/2}e^{\alpha a^\dagger}e^{-\alpha^*a},\qquad
D(\alpha)D(\beta)=e^{(\alpha\beta^*-\alpha^*\beta)/2}D(\alpha+\beta).
$$
The conjugation on $\alpha^*$ ensures that the exponent is anti-Hermitian for arbitrary complex displacement. For real $\alpha=\sqrt{m\omega/(2\hbar)}x_0$, this becomes the spatial translation $e^{-ix_0p/\hbar}$.
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