# nLab Compton wavelength

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# Contents

## Definition

The Compton wavelength is a physical unit that governs the dispersion relation/energy-momentum relation of massive fields.

For a particle/field of mass $m$, its Compton wavelength is the length

$\ell_m \coloneqq \frac{2\pi \hbar}{m c}$

where $c$ denotes the speed of light and $2\pi\hbar$ denotes Planck's constant. Correspondingly $\frac{\hbar}{m c}$ is also called the “reduced Compton wavelength”.

The inverse of the Compton wavelength appears as the mass term notably in the Klein-Gordon equation of the scalar field or the Dirac equation of the Dirac field.

## Examples

• The Compton wavelength corresponding to the mass of the electron is about $\ell_{m_e} ~ 386$ fm.

• Another length scale parameterized by a mass $m$ is the Schwarzschild radius $r_m \coloneqq 2 m G/c^2$, where $G$ is the gravitational constant. Solving the equation

$\array{ & \ell_m &=& r_m \\ \Leftrightarrow & 2\pi\hbar / m c &=& 2 m G / c^2 }$

for $m$ yields the Planck mass

$m_{P} \coloneqq \tfrac{1}{\sqrt{\pi}} m_{\ell = r} = \sqrt{\frac{\hbar c}{G}} \,.$

The corresponding Compton wavelength $\ell_{m_{P}}$ is given by the Planck length $\ell_P$

$\ell_{P} \coloneqq \tfrac{1}{2\pi} \ell_{m_P} = \sqrt{ \frac{\hbar G}{c^3} } \,$