Momentum space

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Momentum space

The Momentum space associated with a particle is a vector space in which every point corresponds to a possible value of the momentum

Momentum

In classical mechanics, momentum is the product of the mass and velocity of an object . For more accurate measures of momentum, see the section Momentum#Modern definitions of momentum on this page....
vector . 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

Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
.

Relation to quantum mechanics
In quantum physics, a particle is described by a quantum state.

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The Momentum space associated with a particle is a vector space in which every point corresponds to a possible value of the momentum

Momentum

Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
.

Relation to quantum mechanics

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 eigenfunction

Eigenfunction

In mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor....
s of the position operator

Position operator

In quantum mechanics, the position operator corresponds to the position observable of a particle. Consider, for example, the case of a spinless particle moving on a line....
as a set of basis functions, one speaks of a state as wave function in position space (normal space as we know it). The familiar Schrödinger equation

Schrödinger equation

In physics, especially quantum mechanics, the Schr?dinger equation is an equation that describes how the quantum state of a physical system changes in time....
in terms of the position is an example of quantum mechanics in the position representation. One can however choose 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 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

Fourier transform

In mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions....
and the concept of frequency domain

Frequency domain

In electronics and control systems engineering, frequency domain is a term used to describe the analysis of mathematical functions or Signal with respect to frequency, rather than time....
. 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 , then we can write this functions as a weighted sum of orthogonal basis functions or, in the continuous case, as an integral It is clear that if we specify the set of functions , say as the set of eigenfunctions of the momentum operator, the function holds all the information necessary to reconstruct and is therefore an alternative description for the state . In quantum mechanics, the momentum operator is given by with eigenfunctions and eigenvalues then 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 and a similar decomposition can be made of in terms of this operators eigenfunctions.