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In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle. When the position operator is considered with a wide enough domain (e.g. the space of tempered distributions ), its eigenvalues are the possible position vectors of the particle.
This operator occurs in relativistic quantum field theory, such as the Dirac equation and other relativistic wave equations, since energy and momentum combine into the 4-momentum vector above, momentum and energy operators correspond to space and time derivatives, and they need to be first order partial derivatives for Lorentz covariance.
The Ehrenfest theorem, named after Austrian theoretical physicist Paul Ehrenfest, relates the time derivative of the expectation values of the position and momentum operators x and p to the expectation value of the force = ′ on a massive particle moving in a scalar potential (), [1]
between the position operator x and momentum operator p x in the x direction of a point particle in one dimension, where [x, p x] = x p x − p x x is the commutator of x and p x , i is the imaginary unit, and ℏ is the reduced Planck constant h/2π, and is the unit operator. In general, position and momentum are vectors of operators and their ...
Position space (also real space or coordinate space) is the set of all position vectors r in Euclidean space, and has dimensions of length; a position vector defines a point in space. (If the position vector of a point particle varies with time, it will trace out a path, the trajectory of a particle.) Momentum space is the set of all momentum ...
The Newton–Wigner position operators x 1, x 2, x 3, are the premier notion of position in relativistic quantum mechanics of a single particle. They enjoy the same commutation relations with the 3 space momentum operators and transform under rotations in the same way as the x , y , z in ordinary QM .
FILE - A solider wears a U.S. Space Force uniform during a ceremony for U.S. Air Force airmen transitioning to U.S. Space Force guardian designations at Travis Air Force Base, Calif., Feb. 12, 2021.
Informally stated, with certain technical assumptions, every representation of the Heisenberg group H 2n + 1 is equivalent to the position operators and momentum operators on R n. Alternatively, that they are all equivalent to the Weyl algebra (or CCR algebra ) on a symplectic space of dimension 2 n .