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wave vector in nLab

Contents

Context

Harmonic analysis

Idea

A wave vector is a vector that encodes wavelength and direction of a plane wave.

Definition

Let n∈ℕn \in \mathbb{N} and write ℝ n\mathbb{R}^n the Cartesian space of dimension nn. Thinking of ℝ n\mathbb{R}^n as a vector space, then each point in it is a vector x→∈ℝ n\vec x \in \mathbb{R}^n and hence a smooth function f:ℝ n→ℂf \colon \mathbb{R}^n \to \mathbb{C} may be thought of as a function of these “position vectors”.

If ff is a function with rapidly decreasing partial derivatives, then its Fourier transform f^:ℝ n→ℂ\hat f \;\colon\; \mathbb{R}^n \to \mathbb{C} exists. By the Fourier inversion theorem, this function is such that it expresses ff as a superposition of “plane wave” functions x→↦e 2πix→⋅k→\vec x \mapsto e^{2\pi i \vec x \cdot \vec k} as

f(x→)=∫k→∈ℝ nf^(k)e 2πik→⋅x→dk→. f(\vec x) \;=\; \underset{\vec k \in \mathbb{R}^n}{\int} \hat f(k) \, e^{2 \pi i \vec k \cdot \vec x} \, d \vec k \,.

Here the vector k→∈ℝ n\vec k \in \mathbb{R}^n determines

the wavelength λ≔1/|k→|\lambda \coloneqq 1/{\vert \vec k\vert} (the inverse of the norm of k→\vec k);
the direction k→|k→|∈S(ℝ n)\frac{\vec k}{{\vert \vec k\vert }} \in S(\mathbb{R}^n) (the corresponding unit vector in the unit sphere)

of the “plane wave” x→↦e 2πix→⋅k→\vec x \mapsto e^{2 \pi i \vec x \cdot \vec k}.

The product 2π|k|2 \pi {\vert k \vert} is also called the wave number and 2πk2 \pi k then the wave number vector. Beware that elsewhere the wave number vector is denoted “kk”, which makes the “wave vector” become k/2πk / 2 \pi. (See e.g. Wikipedia, “Physics definition” as opposed to “Crystallography definition”.)

If here ℝ n≃ℝ p,1\mathbb{R}^n \simeq \mathbb{R}^{p,1} is identified with Minkowski spacetime with canonical coordinates denoted (x 0,x 1,⋯,x p)(x^0, x^1, \cdots, x^p), then the 0-component of the wave vector

ν≔k 0 \nu \coloneqq k_0

is called the frequency of the corresponding plane wave (in the chosen coordinate system); this ω=2πvu\omega = 2 \pi \vu is the angular frequency.

plane waves on Minkowski spacetime

ℝ p,1 ⟶ψ k ℂ x ↦ exp(ik μx μ) (x→,x 0) ↦ exp(ik→⋅x→+ik 0x 0) (x→,ct) ↦ exp(ik→⋅x→−iωt) \array{ \mathbb{R}^{p,1} &\overset{\psi_k}{\longrightarrow}& \mathbb{C} \\ x &\mapsto& \exp\left( \, i k_\mu x^\mu \, \right) \\ (\vec x, x^0) &\mapsto& \exp\left( \, i \vec k \cdot \vec x + i k_0 x^0 \, \right) \\ (\vec x, c t) &\mapsto& \exp\left( \, i \vec k \cdot \vec x - i \omega t \, \right) }

symbol	name
cc	speed of light
ℏ\hbar	Planck's constant
\,	\,
mm	mass
ℏmc\frac{\hbar}{m c}	Compton wavelength
\,	\,
kk, k→\vec k	wave vector
λ=2π/\|k→\|\lambda = 2\pi/{\vert \vec k \vert}	wave length
\|k→\|=2π/λ{\vert \vec k \vert} = 2\pi/\lambda	wave number
ω≔k 0c=−k 0c=2πν\omega \coloneqq k^0 c = -k_0 c = 2\pi \nu	angular frequency
ν=ω/2π\nu = \omega / 2 \pi	frequency
p=ℏkp = \hbar k, p→=ℏk→\vec p = \hbar \vec k	momentum
E=ℏωE = \hbar \omega	energy
ω(k→)=ck→ 2+(mcℏ) 2\omega(\vec k) = c \sqrt{ \vec k^2 + \left(\frac{m c}{\hbar}\right)^2 }	Klein-Gordon dispersion relation
E(p→)=c 2p→ 2+(mc 2) 2E(\vec p) = \sqrt{ c^2 \vec p^2 + (m c^2)^2 }	energy-momentum relation