Momentum operator

The quantum observable associated with momentum. In position space it acts as a derivative operator and generates spatial translations.

Canonical commutation relation: The commutator relation between position and momentum, conventionally written as [x̂, p̂] = iℏ. This relation fixes the momentum operator’s derivative form in the position representation.
Position operator: The quantum observable for position, represented by an operator whose eigenvalues are position coordinates. Its commutation relation with momentum constrains both observables’ representations.
Heisenberg uncertainty principle: A family of bounds limiting the precision of jointly measured quantum observables, including position and momentum. The position–momentum commutator gives the standard uncertainty bound.
Mechanical momentum: The momentum associated with a particle’s motion, equal to mass times velocity in nonrelativistic mechanics. In an electromagnetic field it differs from canonical momentum by a vector-potential term.
Spatial translation operator: A quantum operator that shifts states or wavefunctions by a specified spatial displacement. Exponentiating momentum produces the operator for continuous spatial translations.
Wavefunction: A complex-valued function that encodes a quantum state and determines probabilities for measurement outcomes. The position-space momentum operator acts directly on the wavefunction.
Free particle: An idealized particle subject to no external potential or force. Its momentum operator helps define the kinetic-energy term in the Hamiltonian.
Canonical momentum: The momentum variable conjugate to a coordinate in a system’s Hamiltonian formulation. The usual translation-generator operator quantizes this canonical quantity.
Momentum eigenstate: A state that is an eigenvector of a momentum observable, with a definite momentum eigenvalue. Applying the differential operator to a plane wave yields its momentum eigenvalue.
Hilbert space: A complete inner-product vector space used to represent quantum states and observables. The momentum operator is an observable acting on states in this mathematical setting.

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