nLab
dispersion relation

Context

Harmonic analysis

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Physics

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    • Axiomatizations

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Contents

Idea

In harmonic analysisa dispersion relation is a relation between the frequency and the wavelength of plane waves. Typically this relation expresses the frequency ν(λ)\nu(\lambda) as a function of the wavelength.

In special relativity the frequency of a plane wave is proportional to its energy, and its wave vector is proportional to its momentum, so that now a dispersion relation becomes an energy-momentum relation.

on

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
cc
\hbar
\,\,
mm
mc\frac{\hbar}{m c}
\,\,
kk, k\vec k
λ=2π/|k|\lambda = 2\pi/{\vert \vec k \vert}
|k|=2π/λ{\vert \vec k \vert} = 2\pi/\lambda
ωk 0c=k 0c=2πν\omega \coloneqq k^0 c = -k_0 c = 2\pi \nu
ν=ω/2π\nu = \omega / 2 \pi
p=kp = \hbar k, p=k\vec p = \hbar \vec k
E=ωE = \hbar \omega
ω(k)=ck 2+(mc) 2\omega(\vec k) = c \sqrt{ \vec k^2 + \left(\frac{m c}{\hbar}\right)^2 }
E(p)=c 2p 2+(mc 2) 2E(\vec p) = \sqrt{ c^2 \vec p^2 + (m c^2)^2 }

References

Last revised on November 8, 2017 at 14:10:07. See the history of this page for a list of all contributions to it.