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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 ...
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. [1]
In a basis of Hilbert space consisting of momentum eigenstates expressed in the momentum representation, the action of the operator is simply multiplication by p, i.e. it is a multiplication operator, just as the position operator is a multiplication operator in the position representation.
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 ...
It is for this reason that the momentum operator is referred to as the generator of translation. [2] A nice way to double-check that these relations are correct is to do a Taylor expansion of the translation operator acting on a position-space wavefunction.
These definitions use the position basis (i.e. for a wavefunction in position space), but momentum space is possible. ... and X is the position operator on ...
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.
In particular, the time-dependence of the position operator is given by = [,] =. where x k (t) is the position operator at time t. The above equation shows that the operator α k can be interpreted as the k-th component of a "velocity operator".