Displacement operator

\[ D(\alpha) = e^{\alpha a^{\dagger} - \alpha^* a} \] \[ \alpha a^{\dagger} - \alpha^* a = \mathrm{i} (\mathrm{Im}[\alpha] x - \mathrm{Re}[\alpha]p) \] \[ e^{\mathrm{i} a p } \sim 1 - \mathrm{i} a p = 1 - a \frac{\partial}{\partial x} \] \[ e^{\mathrm{i} a p } f(x)\sim f(x) - a \frac{\partial}{\partial x} f(x) = f(x - a) \]

Reference