The displacement operator is a unitary operator, and therefore obeys ${\displaystyle {\hat {D}}(\alpha ){\hat {D}}^{\dagger }(\alpha )={\hat {D}}^{\dagger }(\alpha ){\hat {D}}(\alpha )={\hat {1}}}$ , where ${\displaystyle {\hat {1}}}$  is the identity operator. Since ${\displaystyle {\hat {D}}^{\dagger }(\alpha )={\hat {D}}(-\alpha )}$ , the hermitian conjugate of the displacement operator can also be interpreted as a displacement of opposite magnitude (${\displaystyle -\alpha }$ ). The effect of applying this operator in a similarity transformation of the ladder operators results in their displacement.

${\displaystyle {\hat {D}}^{\dagger }(\alpha ){\hat {a}}{\hat {D}}(\alpha )={\hat {a}}+\alpha }$ 

${\displaystyle {\hat {D}}(\alpha ){\hat {a}}{\hat {D}}^{\dagger }(\alpha )={\hat {a}}-\alpha }$ 

The product of two displacement operators is another displacement operator whose total displacement, up to a phase factor, is the sum of the two individual displacements. This can be seen by utilizing the Baker–Campbell–Hausdorff formula.

${\displaystyle e^{\alpha {\hat {a}}^{\dagger }-\alpha ^{*}{\hat {a}}}e^{\beta {\hat {a}}^{\dagger }-\beta ^{*}{\hat {a}}}=e^{(\alpha +\beta ){\hat {a}}^{\dagger }-(\beta ^{*}+\alpha ^{*}){\hat {a}}}e^{(\alpha \beta ^{*}-\alpha ^{*}\beta )/2}.}$ 

which shows us that:

${\displaystyle {\hat {D}}(\alpha ){\hat {D}}(\beta )=e^{(\alpha \beta ^{*}-\alpha ^{*}\beta )/2}{\hat {D}}(\alpha +\beta )}$ 

When acting on an eigenket, the phase factor ${\displaystyle e^{(\alpha \beta ^{*}-\alpha ^{*}\beta )/2}}$  appears in each term of the resulting state, which makes it physically irrelevant.^[1]

It further leads to the braiding relation

${\displaystyle {\hat {D}}(\alpha ){\hat {D}}(\beta )=e^{\alpha \beta ^{*}-\alpha ^{*}\beta }{\hat {D}}(\beta ){\hat {D}}(\alpha )}$ 

The Kermack-McCrae identity gives two alternative ways to express the displacement operator:

${\displaystyle {\hat {D}}(\alpha )=e^{-{\frac {1}{2}}|\alpha |^{2}}e^{+\alpha {\hat {a}}^{\dagger }}e^{-\alpha ^{*}{\hat {a}}}}$ 

${\displaystyle {\hat {D}}(\alpha )=e^{+{\frac {1}{2}}|\alpha |^{2}}e^{-\alpha ^{*}{\hat {a}}}e^{+\alpha {\hat {a}}^{\dagger }}}$ 

The displacement operator can also be generalized to multimode displacement. A multimode creation operator can be defined as

${\displaystyle {\hat {A}}_{\psi }^{\dagger }=\int d\mathbf {k} \psi (\mathbf {k} ){\hat {a}}^{\dagger }(\mathbf {k} )}$ ,

where ${\displaystyle \mathbf {k} }$  is the wave vector and its magnitude is related to the frequency ${\displaystyle \omega _{\mathbf {k} }}$  according to ${\displaystyle |\mathbf {k} |=\omega _{\mathbf {k} }/c}$ . Using this definition, we can write the multimode displacement operator as

${\displaystyle {\hat {D}}_{\psi }(\alpha )=\exp \left(\alpha {\hat {A}}_{\psi }^{\dagger }-\alpha ^{\ast }{\hat {A}}_{\psi }\right)}$ ,

and define the multimode coherent state as

${\displaystyle |\alpha _{\psi }\rangle \equiv {\hat {D}}_{\psi }(\alpha )|0\rangle }$ .

• Optical phase space