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 x {\displaystyle \mathbf {x} } , there is a corresponding translation operator T ^ ( x ) {\displaystyle {\hat {T}}(\mathbf {x} )} that shifts particles and fields by the amount x {\displaystyle \mathbf {x} } . For example, if T ^ ( x ) {\displaystyle {\hat {T}}(\mathbf {x} )} acts on a particle located at position r {\displaystyle \mathbf {r} } , the result is a particle at position r + x {\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 y {\displaystyle y} direction has a simple relationship to the y {\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.
Action on position eigenkets and wavefunctions The translation operator T ^ ( x ) {\displaystyle {\hat {T}}(\mathbf {x} )} moves particles and fields by the amount x {\displaystyle \mathbf {x} } . Therefore, if a particle is in an eigenstate | r ⟩ {\displaystyle |\mathbf {r} \rangle } of the position operator (i.e., precisely located at the position r {\displaystyle \mathbf {r} } ), then after T ^ ( x ) = ∫ d r | r + x ⟩ ⟨ r | {\displaystyle {\hat {T}}(\mathbf {x} )=\int \!d\mathbf {r} ~|\mathbf {r+x} \rangle \langle \mathbf {r} |} acts on it, the particle is at the position r + x {\displaystyle \mathbf {r} +\mathbf {x} } :
T ^ ( x ) | r ⟩ = | r + x ⟩ . {\displaystyle {\hat {T}}(\mathbf {x} )|\mathbf {r} \rangle =|\mathbf {r} +\mathbf {x} \rangle .}
An alternative (and equivalent) way to describe what the translation operator determines is based on position-space wavefunctions. If a particle has a position-space wavefunction ψ ( r ) {\displaystyle \psi (\mathbf {r} )} , and T ^ ( x ) {\displaystyle {\hat {T}}(\mathbf {x} )} acts on the particle, the new position-space wavefunction is ψ ′ ( r ) = T ^ ( x ) ψ ( r ) {\displaystyle \psi '(\mathbf {r} )={\hat {T}}(\mathbf {x} )\psi (\mathbf {r} )} defined by
ψ ′ ( r ) = ψ ( r − x ) . {\displaystyle \psi '(\mathbf {r} )=\psi (\mathbf {r} -\mathbf {x} ).}
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