Displacement operator

The displacement operator is a generalized version of the translation operator, but with a complex displacement.

The exponent $\alpha a\adj - \alpha\conj a\adj$ is anti-hermitian, and the exponential of an anti-hermitian operator is unitary. The displacement operator can be used to define a generalized coherent state $\coh \alpha = D(\alpha) \ket 0$.

Like translations, displacements form a commutative group.