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Position and momentum spaces. In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension. 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.
e. The phase-space formulation of quantum mechanics places the position and momentum variables on equal footing in phase space. In contrast, the Schrödinger picture uses the position or momentum representations (see also position and momentum space). The two key features of the phase-space formulation are that the quantum state is described by ...
For mechanical systems, the phase space usually consists of all possible values of position and momentum variables. It is the direct product of direct space and reciprocal space. [clarification needed] The concept of phase space was developed in the late 19th century by Ludwig Boltzmann, Henri Poincaré, and Josiah Willard Gibbs. [1]
The position-space and momentum-space wave functions are thus found to be Fourier transforms of each other. [30] They are two representations of the same state; containing the same information, and either one is sufficient to calculate any property of the particle.
Canonical commutation relation. In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example,
The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant. It is a differential equation version of the relativistic energy–momentum relation .
Momentum operator. In quantum mechanics, the momentum operator is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimension, the definition is: where ħ is the reduced Planck constant, i the imaginary ...
v. t. e. Canonical commutation rule for position q and momentum p variables of a particle, 1927. pq − qp = h / (2 πi). Uncertainty principle of Heisenberg, 1927. The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with ...