Momentum space - Citizendia

The Momentum 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 $\vec{k}$ . In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product 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 (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion.

Relation to quantum mechanics

In quantum physics, a particle is described by a quantum state. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons 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 wave function $\psi(\vec{x})$ in position space (normal space as we know it). In Mathematics, an eigenfunction of a Linear operator, A, defined on some Function space is any non-zero function f in A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system The familiar Schrödinger equation in terms of the position $\vec{x}$ is an example of quantum mechanics in the position representation. In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system 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 $\phi(\vec{k})$ 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. This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and Frequency domain is a term used to describe the analysis of Mathematical functions or signals with respect to frequency 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 sum of orthogonal basis functions $φ k (x)$

ψ(x) =	∑	φ_kψ_k(x)
	k

or, in the continuous case, as an integral

ψ(x) =	∫	φ(k)ψ_k(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

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

and eigenvalues $\hbar k$ then

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

and we see that the momentum representation is related to the position representation by a Fourier transform. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes

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