nLab
wavelength

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Idea

The distance between two crests? of a wave.

Definition

For nn \in \mathbb{N} and

xe 2πikx \vec x \mapsto e^{2 \pi i \vec k \cdot \vec x}

a plane wave with wave vector k n\vec k \in \mathbb{R}^n, then its wavelength is the inverse of the norm of kk:

λ1/|k|. \lambda \coloneqq 1/{\vert k\vert} \,.

The quotient k/|k|S( n)k/{\vert k \vert} \in S(\mathbb{R}^n) is the direction of the plane wave.

The product 2π/λ2\pi/\lambda is also called the wave number.

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

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Examples

Last revised on November 7, 2017 at 08:47:11. See the history of this page for a list of all contributions to it.