nLab frequency

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

$$x \mapsto e^{ - 2 \pi i k_\mu x^\mu}$$

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

$$\nu \coloneqq k_0 \,.$$

The combination

$$\omega \coloneqq 2 \pi \nu = 2\pi k_0$$

is called the angular frequency.

plane waves on Minkowski spacetime

$$\array{ \mathbb{R}^{p,1} &\overset{\psi_k}{\longrightarrow}& \mathbb{C} \\ x &\mapsto& \exp\left( \, i k_\mu x^\mu \, \right) \\ (\vec x, x^0) &\mapsto& \exp\left( \, i \vec k \cdot \vec x + i k_0 x^0 \, \right) \\ (\vec x, c t) &\mapsto& \exp\left( \, i \vec k \cdot \vec x - i \omega t \, \right) }$$

symbol	name
$c$	speed of light
$\hbar$	Planck's constant
$\,$	$\,$
$m$	mass
$\frac{\hbar}{m c}$	Compton wavelength
$\,$	$\,$
$k$, $\vec k$	wave vector
$\lambda = 2\pi/{\vert \vec k \vert}$	wave length
${\vert \vec k \vert} = 2\pi/\lambda$	wave number
$\omega \coloneqq k^0 c = -k_0 c = 2\pi \nu$	angular frequency
$\nu = \omega / 2 \pi$	frequency
$p = \hbar k$, $\vec p = \hbar \vec k$	momentum
$E = \hbar \omega$	energy
$\omega(\vec k) = c \sqrt{ \vec k^2 + \left(\frac{m c}{\hbar}\right)^2 }$	Klein-Gordon dispersion relation
$E(\vec p) = \sqrt{ c^2 \vec p^2 + (m c^2)^2 }$	energy-momentum relation

Related concepts

• wavelength

• wave vector

• wave equation, Klein-Gordon equation