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:<math>\hat{D}(\alpha)=\exp \left ( \alpha \hat{a}^\dagger - \alpha^\ast \hat{a} \right ) </math>,
where <math>\alpha</math> is the amount of displacement in [[optical phase space]], <math>\alpha^\ast</math> is the complex conjugate of that displacement, and <math>\hat{a} </math> and <math>\hat{a}^\dagger</math> are the [[lowering and raising operators|creation and annihilation operators]], respectively.
The name of this operator is derived from its ability to displace a localized state in phase space by a magnitude <math>\alpha</math>. It may also act on the vacuum state by displacing it into a [[coherent state]]. Specifically,
<math>\hat{D}(\alpha)|0\rangle=|\alpha\rangle</math> where <math>
Displaced states are [[eigenfunctions]] of the annihilation (lowering) operator.
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