Translation operator (quantum mechanics)

In quantum mechanics, a translation operator is defined as an operator which shifts particles and fields by a certain amount in a certain direction. More specifically, for any displacement vector ${\displaystyle \mathbf {x} }$, there is a corresponding translation operator ${\displaystyle {\hat {T}}(\mathbf {x} )}$ that shifts particles and fields by the amount ${\displaystyle \mathbf {x} }$. For example, if ${\displaystyle {\hat {T}}(\mathbf {x} )}$ acts on a particle located at position ${\displaystyle \mathbf {r} }$, the result is a particle at position ${\displaystyle \mathbf {(r+x)} }$.

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