Position and momentum spaceIn physics and geometry, there are two intertwined vector spaces.Position space (also real space or coordinate space) is the set of all position vectors r of an object in space(usually 3D). The position vector defines a point in space. If the position vector varies with time it will trace outa path or surface, such as the trajectory of a particle.Momentum space or k-space is the set of all wavevectors k, associated with particles - free and bound. Theterms "momentum" (symbol p, also a vector) and "wavevector" are used interchangeably due to the De Broglierelation p = hk, meaning they are equivalent up to proportionality, although this is not true in a crystal, seebelow.This is an example of Pontryagin duality.The position vector r has dimensions of length, the k-vector has dimensions of reciprocal length, so k is thefrequency analogue of r, just as angular frequency ω is the inverse quantity and frequency analogue of time t.Physical phenomena can be described using either the positions of particles, or their momenta, bothformulations equivalently provide the same information about the system in consideration. Usually r is moreintuitive and simpler than k, though the converse is also true, such as in solid-state physics.

Contents1 Position and momentum spaces in quantum mechanics2 Relation between space and reciprocal space2.1 Functions and operators in position space2.2 Functions and operators in momentum space3 Unitary equivalence between position and momentum operator4 Reciprocal space and crystals5 See also6 References

Position and momentum spaces in quantum mechanicsFurther information: Momentum operatorIn quantum physics, a particle is described by a quantum state. This quantum state can be represented as asuperposition (i.e. a linear combination as a weighted sum) of basis states. In principle one is free to choose theset of basis states, as long as they span the space. If one chooses the eigenfunctions of the position operator as aset 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). The familiar Schrödinger equation in terms of the position r is an example of quantummechanics in the position representation.[2]By choosing the eigenfunctions of a different operator as a set of basis functions, one can arrive at a number ofdifferent representations of the same state. If one picks the eigenfunctions of the momentum operator as a set of1 of 4 01-Feb-14 1:27 PM[1]Position and momentum space - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Position_and_momentum_spacebasis functions, the resulting wave function f(k) is said to be the wave function in momentum space.

Relation between space and reciprocal spaceThe momentum representation of a wave function is very closely related to the Fourier transform and theconcept of frequency domain. Since a quantum mechanical particle has a frequency proportional to themomentum (de Broglie's equation given above), describing the particle as a sum of its momentum components isequivalent to describing it as a sum of frequency components (i.e. a Fourier transform).[3] This becomes clearwhen we ask ourselves how we can transform from one representation to another.Functions and operators in position spaceSuppose we have a three-dimensional wave function in position space ψ(r), then we can write this functions as aweighted sum of orthogonal basis functions ψor, in the continuous case, as an integralIt is clear that if we specify the set of functions ψj(r):(r), say as the set of eigenfunctions of the momentumoperator, the function f(k) holds all the information necessary to reconstruct ψ(r) and is therefore an alternativedescription for the state ψ.In quantum mechanics, the momentum operator is given byj(see matrix calculus for the denominator notation) with appropriate domain. The eigenfunctionsand eigenvalues hk. Soand we see that the momentum representation is related to the position representation by a Fourier transform.