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. 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. The translation operator moves particles and fields by the amount . Therefore, if a particle is in an eigenstate of the position operator (i.e., precisely located at the position ), then after acts on it, the particle is at the position : 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 , and acts on the particle, the new position-space wavefunction is defined by This relation is easier to remember as which can be read as: "The value of the new wavefunction at the new point equals the value of the old wavefunction at the old point". Here is an example showing that these two descriptions are equivalent. The state corresponds to the wavefunction (where is the Dirac delta function), while the state corresponds to the wavefunction These indeed satisfy Shift operator and Momentum operator In introductory physics, momentum is usually defined as mass times velocity. However, there is a more fundamental way to define momentum, in terms of translation operators.