In quantum mechanics, a translation operator is defined as an operator which shifts particles and fields by a certain amount in a certain direction. It is a special case of the shift operator from functional analysis.
More specifically, for any displacement vector , there is a corresponding translation operator that shifts particles and fields by the amount .
For example, if acts on a particle located at position , the result is a particle at position .
Translation operators are unitary.
Translation operators are closely related to the momentum operator; for example, a translation operator that moves by an infinitesimal amount in the direction has a simple relationship to the -component of the momentum operator. Because of this relationship, conservation of momentum holds when the translation operators commute with the Hamiltonian, i.e. when laws of physics are translation-invariant. This is an example of Noether's theorem.
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