page_title: Translation operator (quantum mechanics)
section_title: Time evolution and translational invariance
hierarchical_section_title: Translation operator (quantum mechanics) / Time evolution and translational invariance
context_page_description: 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, 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.
context_section_description: In the passive transformation picture, translational invariance requires,
  It follows that
  where  is the unitary time evolution operator. When the Hamiltonian is time independent, 
  If the Hamiltonian is time dependent, the above commutation relation is satisfied if  or  commutes with  for all t.
caption: Translational Invariance: Time evolution of the wave functions.
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