Momentum space

The momentum space or k-space associated with a particle is a vector space in which every point {k_x,k_y,k_z} corresponds to a possible value of the momentum vector $\vec{k}$. Representing a problem in terms of the momenta of the particles involved, rather than in terms of their positions, can greatly simplify some problems in physics.

Relation to quantum mechanics

Further information: Momentum operator

In quantum physics, a particle is described by a quantum state. This quantum state can be represented as a superposition (weighted sum) of basis states. In principle one is free to choose the set of basis states, as long as they span state space. If one chooses the eigenfunctions of the position operator as a set of basis functions, one speaks of a state as a wave function $\psi(\vec{x})$ in position space (normal space as we know it). The familiar Schrödinger equation in terms of the position $\vec{x}$ is an example of quantum mechanics in the position representation. By choosing the eigenfunctions of a different operator as a set of basis functions, one can arrive at a number of different representations of the same state. If one picks the eigenfunctions of the momentum operator as a set of basis functions, the resulting wave function $\phi(\vec{k})$ is said to be the wave function in momentum space.

Relation to frequency domain

The momentum representation of a wave function is very closely related to the Fourier transform and the concept of frequency domain. Since a quantum mechanical particle has a frequency proportional to the momentum, describing the particle as a sum of its momentum components is equivalent to describing it as a sum of frequency components (i.e. a Fourier transform). This becomes clear when we ask ourselves how we can transform from one representation to another. Suppose we have a one dimensional wave function in position space ψ(x), then we can write this functions as a weighted sum of orthogonal basis functions ψ_k(x)

ψ(x) =	∑	ϕkψk(x)
	k

or, in the continuous case, as an integral

ψ(x) =	∫	ϕ(k)ψk(x)
	k

It is clear that if we specify the set of functions ψ_k(x), say as the set of eigenfunctions of the momentum operator, the function φ(k) holds all the information necessary to reconstruct ψ(x) and is therefore an alternative description for the state ψ. In quantum mechanics, the momentum operator is given by

$\hat p = -i \hbar\frac{d}{d x}$

with eigenfunctions

$\frac{1}{\sqrt{2\pi}} e^{i k x}$

and eigenvalues $\hbar k$ then

$\psi(x)=\frac{1}{\sqrt{2\pi}} \int_k \phi(k) e^{i k x} dk$

and we see that the momentum representation is related to the position representation by a Fourier transform.

Conversely, in momentum space the position operator is given by

$\hat x = i \hbar\frac{d}{d p}$

and a similar decomposition can be made of φ(k) in terms of this operators eigenfunctions.