nLab
frequency

Contents

Idea

Frequency is a number of occurences of a periodic event in time.

For waves: number of wave crests at a given space point per time.

Definition

For pp \in \mathbb{N} and p,1 p+1\mathbb{R}^{p,1} \simeq_{\mathbb{R}} \mathbb{R}^{p+1} the Cartesian space p+1\mathbb{R}^{p+1} regarded as Minkowski spacetime, then the frequency of a plane wave

xe 2πik μx μ x \mapsto e^{ - 2 \pi i k_\mu x^\mu}

with respect to the chosen coordinate system is the 0-component of the given wave vector kk:

νk 0. \nu \coloneqq k_0 \,.

The combination

ω2πν=2πk 0 \omega \coloneqq 2 \pi \nu = 2\pi k_0

is called 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

Last revised on November 7, 2017 at 08:46:56. See the history of this page for a list of all contributions to it.