In the quantum mechanics study of optical phase space, the displacement operator for one mode is the shift operator in quantum optics,

where ${\displaystyle \alpha }$ is the amount of displacement in optical phase space, ${\displaystyle \alpha ^{*}}$ is the complex conjugate of that displacement, and ${\displaystyle {\hat {a}}}$ and ${\displaystyle {\hat {a}}^{\dagger }}$ are the lowering and raising operators, respectively.

The name of this operator is derived from its ability to displace a localized state in phase space by a magnitude ${\displaystyle \alpha }$. It may also act on the vacuum state by displacing it into a coherent state. Specifically, ${\displaystyle {\hat {D}}(\alpha )|0\rangle =|\alpha \rangle }$ where ${\displaystyle |\alpha \rangle }$ is a coherent state, which is an eigenstate of the annihilation (lowering) operator. This operator was introduced independently by Richard Feynman and Roy J. Glauber in 1951.

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.

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.

which shows us that:

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.

It further leads to the braiding relation