Position and momentum spaces

In <a href="/facts/Physics/a7aH2y08">physics</a> and <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, there are two closely related <a href="/facts/Vector_space/M1pxMLx2">vector spaces</a>, usually <a href="/facts/Three-dimensional_space/KAqKWUbS">three-dimensional</a> but in general of any finite dimension.
Position space (also real space or <a href="/facts/Coordinate_system/Mtd0q4SP">coordinate</a> space) is the set of all <a href="/facts/Position_vector/Tpk5d9f2">position vectors</a> r in <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>, and has <a href="/facts/Quantity_dimension/t6UskoCD">dimensions</a> of <a href="/facts/Length/EwFyAqGH">length</a>; a position vector defines a point in space. (If the position vector of a <a href="/facts/Point_particle/4MPToN8W">point particle</a> varies with time, it will trace out a path, the <a href="/facts/Trajectory/STNN33FK">trajectory</a> of a particle.) Momentum space is the set of all <a href="/facts/Momentum_vector/TTkGv5eY">momentum vectors</a> p a physical system can have; the momentum vector of a particle corresponds to its motion, with dimension of mass ⋅ length ⋅ time−1.
Mathematically, the duality between position and momentum is an example of <a href="/facts/Pontryagin_duality/fbD3d8kz">Pontryagin duality</a>. In particular, if a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> is given in position space, f(r), then its <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a> obtains the function in momentum space, φ(p). Conversely, the inverse Fourier transform of a momentum space function is a position space function.
These quantities and ideas transcend all of classical and quantum physics, and a physical system can be described using either the positions of the constituent particles, or their momenta, both formulations equivalently provide the same information about the system in consideration. Another quantity is useful to define in the context of <a href="/facts/Wave/fU0HH6St">waves</a>. The <a href="/facts/Wave_vector/Jk8wUEft">wave vector</a> k (or simply "k-vector") has dimensions of <a href="/facts/Reciprocal_length/mlrDdsFD">reciprocal length</a>, making it an analogue of <a href="/facts/Angular_frequency/VqLm3L3R">angular frequency</a> ω which has dimensions of reciprocal <a href="/facts/Time/HceJKUGB">time</a>. The set of all wave vectors is k-space. Usually, the position vector r is more intuitive and simpler than the wave vector k, though the converse can also be true, such as in <a href="/facts/Solid-state_physics/Cmc6hvyR">solid-state physics</a>.
<a href="/facts/Quantum_mechanics/BRc3Mzgr">Quantum mechanics</a> provides two fundamental examples of the duality between position and momentum, the <a href="/facts/Uncertainty_principle/UUSoSp4T">Heisenberg uncertainty principle</a> ΔxΔp ≥ ħ/2 stating that position and momentum cannot be simultaneously known to arbitrary precision, and the <a href="/facts/De_Broglie_relation/Rpszb8f1">de Broglie relation</a> p = ħk which states the momentum and wavevector of a <a href="/facts/Free_particle/Fjbx90yW">free particle</a> are proportional to each other. In this context, when it is unambiguous, the terms "<a href="/facts/Momentum/TTkGv5eY">momentum</a>" and "wavevector" are used interchangeably. However, the de Broglie relation is not true in a crystal.

Position and momentum spaces open-in-new

Position and momentum spaces