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In the [[quantum mechanics]] study of [[optical phase space]], the '''displacement operator''' for one mode is the [[shift operator]] in [[quantum optics]],
:<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^*</math> is the complex conjugate of that displacement, and <math>\hat{a}</math> and <math>\hat{a}^\dagger</math> are the [[creation and annihilation operators|lowering and raising operators]], respectively.▼
▲where <math>\alpha</math> is the amount of displacement in [[optical phase space]], <math>\alpha^*</math> is the complex conjugate of that displacement, and <math>\hat{a}</math> and <math>\hat{a}^\dagger</math> are the [[creation and annihilation operators|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 <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>|\alpha\rangle</math> is a [[coherent state]], which is the [[eigenstates]] of the annihilation (lowering) operator.
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