For example, measuring position is considered to be a measurement of the position operator.

The same applies for the Position operator in the momentum basis:

However, unlike the Hamiltonian, the position operator lacks proper eigenfunctions.

One can show that the Fourier transform of the momentum in quantum mechanics is the position operator.

This is because the position operator is unbounded, and has to be chosen from its domain of definition.

To show this, suppose is an eigenstate of the position operator with eigenvalue .

The position operator, Q, is then defined by:

The eigenfunctions of the position operator, represented in position basis, are dirac delta functions.

Now, because both and are time-independent, the above equation can easily be integrated twice to find the explicit time-dependence of the position operator.

Schrödinger represented the momentum p by a differential operator - which does not commute with the position operator x.

In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle.

In quantum theory, also operators with non-discrete spectrum are in use, such as the position operator in quantum mechanics.

Citations have listed problems including illegal emergency stop switches, inoperative car position operators and inaccessible emergency exits, and carried fines totaling about $2,100.

This can be carried over to quantum mechanics, by reinterpreting r as the quantum position operator and p as the quantum momentum operator.

Orbital angular momentum L is mathematically defined as the cross product of a wave function's position operator (r) and momentum operator (p):

To obtain this operator, we must commute the Hamiltonian operator with the canonical position operators , i.e., we must calculate .

The Newton-Wigner position operators x, x, x, are the premier notion of position in relativistic quantum mechanics of a single particle.

Newton-Wigner localization (named after Theodore Duddell Newton and Eugene Wigner) is a scheme for obtaining a position operator for massive relativistic quantum particles.

Classically, the Fourier coefficients give the intensity of the emitted radiation, so in quantum mechanics the magnitude of the matrix elements of the position operator were the intensity of radiation in the bright-line spectrum.

In addition to some assumptions underlying the interpretation of certain values in the quantum mechanical formulation, one of the fundamental cornerstones to the entire theory lies in the commutator relationship between the position operator and the momentum operator :

If one chooses the eigenfunctions of the position operator as a set of basis functions, one speaks of a state as a wave function ψ(r) in position space (our ordinary notion of space in terms of length).

If we could know both where a particle was and also what it was doing, it would then have to be in a state which was simultaneously an eigenstate of the position operator x and also an eigenstate of the momentum operator p.

The connection between this representation and the more usual wavefunction representation is given by the eigenstate of the position operator representing a particle at position x, which is denoted by an element in the Hilbert space, and which satisfies .

The Stone-von Neumann theorem generalizes Stone's theorem to a pair of self-adjoint operators, Q, P satisfying the canonical commutation relation, and shows that these are all unitarily equivalent to the position operator and momentum operator on L(R).

The Heisenberg group is a central extension (mathematics) of such a commutative Lie group/algebra: the symplectic form defines the commutation, analogously to the canonical commutation relations (CCR), and a Darboux basis corresponds to canonical coordinates - in physics terms, to momentum operators and position operators.