nLab
dispersion relation

Contents

Context

Harmonic analysis

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

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.

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

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.