As its name suggests, the _displacement operator_ shifts the position of a quantum state in [phase space](Phase%20space%20distributions%20(quantum%20mechanics).md) without changing its shape in phase space. When applied to a [the vacuum state](The%20vacuum%20state.md) it generates a [[coherent state]].
The displacement operator, $\hat{D}(\alpha)$ is written in terms of a complex numbers $\alpha$ and the [creation and annihilation operators](creation%20and%20annihilation%20operators.md)s and appears as
$\hat{D}(\alpha)=e^{\alpha\hat{a}^{\dagger}-\alpha*\hat{a}}$
# Properties
This operator has the following properties
# Coherent states
# Derivation
## From the coherent state definition
We may view the displacement operator from the point of view of needing to derive an operator that creates a [coherent state](coherent%20state.md) when applied to a [the vacuum state](The%20vacuum%20state.md). Thus given the definition of a  where 
Since 
we may rewrite the coherent state as
$|\alpha\rangle=e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n(\hat{a}^{\dagger})^n}{n!}|n\rangle=e^{-|\alpha|^2/2}e^{\alpha a^\dagger}|0\rangle$
where we found the [power series representation](Complex%20matrix%20exponentials.md#Power%20series%20representation) of a complex matrix exponential and converted it to an exponential term.
Note that
and thus we may rewrite the coherent state
$e^{-|\alpha|^2/2}e^{\alpha a^{\dagger}}e^{\alpha^* \hat{a}}|0\rangle$
where we extract the displacement operator by noting that by the [Glauber formula](Glauber%20formula.md)
$e^{-|\alpha|^2/2}e^{\alpha a^{\dagger}}e^{\alpha^* \hat{a}}=e^{\alpha\hat{a}^{\dagger}-\alpha*\hat{a}}=\hat{D}(\alpha)$
#QuantumMechanics/QuantumDynamics/QuantumHarmonicOscillators
#QuantumMechanics/QuantumOptics