A _coherent state_ (for the single-mode case) is mathematically expressed as the following [Quantum superposition](Quantum%20superposition.md) of a number of [fock state](fock%20state.md)s approaching infinity: $|\alpha\rangle=e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}|n\rangle$ ^e84aa9 which alternatively may be expressed in terms of the [displacement operator](displacement%20operator.md) applied to a [the vacuum state](The%20vacuum%20state.md), resulting in the following expression $|\alpha\rangle = \hat{D}(\alpha)|0\rangle.$ These states are notable because they most closely resemble the [classical harmonic oscillator](coherent%20state.md#Resemblance%20to%20the%20classical%20harmonic%20oscillator) in their behavior. # Completeness relation These states form an [[overcomplete basis]], thus they are _not_ [orthonormal eigenstates.](State%20vector.md#Decomposing%20state%20vectors%20into%20sets%20of%20orthonormal%20eigenstates) # As the eigenstate of the annihilation operator The coherent state is itself the [eigenstate](State%20vector.md#Decomposing%20state%20vectors%20into%20sets%20of%20orthonormal%20eigenstates) of the [annihilation operator](creation%20and%20annihilation%20operators.md) thus we may state that $\hat{a}|\alpha\rangle=\alpha|\alpha\rangle$ ^cc774f # Resemblance to the classical harmonic oscillator # Multimode coherent states # Schrödinger's cat states  #QuantumMechanics/StationaryStateQuantumSystems/QuantumHarmonicOscillators #QuantumMechanics/QuantumStateRepresentations/StateVectors #QuantumMechanics/QuantumHarmonicOscillators