Translation operator (quantum mechanics)

In <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>, a translation operator is defined as an <a href="/facts/Operator_(physics)/rCJw8UFz">operator</a> which shifts particles and <a href="/facts/Field_(physics)/FJdAKeRA">fields</a> by a certain amount in a certain direction. It is a special case of the <a href="/facts/Shift_operator/6c6lc0ks">shift operator</a> from functional analysis.
More specifically, for any <a href="/facts/Displacement_vector/jkmEfS3C">displacement vector</a> 
 
 
 
 
 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 <a href="/facts/Unitary_operator/rkypfQQ9">unitary</a>.
Translation operators are closely related to the <a href="/facts/Momentum_operator/nPx4UL3k">momentum operator</a>; 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, <a href="/facts/Conservation_of_momentum/TTkGv5eY">conservation of momentum</a> holds when the translation operators commute with the Hamiltonian, i.e. when laws of physics are translation-invariant. This is an example of <a href="/facts/Noether%2527s_theorem/VFjqrEvz">Noether's theorem</a>.

Translation operator (quantum mechanics) open-in-new

Translation operator (quantum mechanics)