nLab
wave vector

Contents

Idea

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

Definition

Let nn \in \mathbb{N} and write n\mathbb{R}^n the Cartesian space of dimension nn. Thinking of n\mathbb{R}^n as a vector space, then each point in it is a vector x n\vec x \in \mathbb{R}^n and hence a smooth function f: nf \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^: n\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 xe 2πixk\vec x \mapsto e^{2\pi i \vec x \cdot \vec k} as

f(x)=k nf^(k)e 2πikxdk. 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 n\vec k \in \mathbb{R}^n determines

  1. the wavelength λ1/|k|\lambda \coloneqq 1/{\vert \vec k\vert} (the inverse of the norm of k\vec k);

  2. the direction k|k|S( n)\frac{\vec k}{{\vert \vec k\vert }} \in S(\mathbb{R}^n) (the corresponding unit vector in the unit sphere)

of the “plane wavexe 2πixk\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 n 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

νk 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

p,1 ψ k x exp(ik μx μ) (x,x 0) exp(ikx+ik 0x 0) (x,ct) exp(ikxiωt) \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) }
symbolname
ccspeed of light

\hbarPlanck's constant

\,\,

mmmass

mc\frac{\hbar}{m c}Compton wavelength

\,\,

kk, k\vec kwave vector

λ=2π/|k|\lambda = 2\pi/{\vert \vec k \vert}wave length

|k|=2π/λ{\vert \vec k \vert} = 2\pi/\lambdawave number

ωk 0c=k 0c=2πν\omega \coloneqq k^0 c = -k_0 c = 2\pi \nuangular frequency

ν=ω/2π\nu = \omega / 2 \pifrequency

p=kp = \hbar k, p=k\vec p = \hbar \vec kmomentum

E=ωE = \hbar \omegaenergy

ω(k)=ck 2+(mc) 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

</table>

References

See also

Last revised on August 2, 2018 at 03:11:17. See the history of this page for a list of all contributions to it.