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. It is a special case of the shift operator from functional analysis.

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 ${\displaystyle \mathbf {r} +\mathbf {x} }$.

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 ${\displaystyle y}$ direction has a simple relationship to the ${\displaystyle y}$-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.