In quantum mechanics, the displacement operator is an important concept in the context of quantum harmonic oscillators and coherent states. The displacement operator, often denoted as \( D(\alpha) \), is used to shift the state of a quantum system in phase space. ### Definition The displacement operator is defined as: \[ D(\alpha) = e^{\alpha a^\dagger - \alpha^* a} \] where: - \( \alpha \) is a complex number.
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The displacement operator translates oscillator phase space. It is unitary and obeys and , a coherent state. The Baker--Campbell--Hausdorff formula with a scalar commutator givesThe conjugation on ensures that the exponent is anti-Hermitian for arbitrary complex displacement. For real , this becomes the spatial translation .