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Concept
Compton wavelength
Summary
The Compton wavelength is a quantum mechanical property of a particle, defined as the wavelength of a photon whose energy is the same as the rest energy of that particle (see mass–energy equivalence). It was introduced by Arthur Compton in 1923 in his explanation of the scattering of photons by electrons (a process known as Compton scattering).
The standard Compton wavelength λ of a particle is given by
while its frequency f is given by
where h is the Planck constant, m is the particle's proper mass, and c is the speed of light. The significance of this formula is shown in the derivation of the Compton shift formula.
The CODATA 2018 value for the Compton wavelength of the electron is 2.42631023867e-12m. Other particles have different Compton wavelengths.
The reduced Compton wavelength ƛ (barred lambda, denoted below by ) is defined as the Compton wavelength divided by 2π:
where ħ is the reduced Planck constant.
The inverse reduced Compton wavelength is a natural representation for mass on the quantum scale, and as such, it appears in many of the fundamental equations of quantum mechanics. The reduced Compton wavelength appears in the relativistic Klein–Gordon equation for a free particle:
It appears in the Dirac equation (the following is an explicitly covariant form employing the Einstein summation convention):
The reduced Compton wavelength is also present in Schrödinger's equation, although this is not readily apparent in traditional representations of the equation. The following is the traditional representation of Schrödinger's equation for an electron in a hydrogen-like atom:
Dividing through by and rewriting in terms of the fine-structure constant, one obtains:
The reduced Compton wavelength is a natural representation of mass on the quantum scale and is used in equations that pertain to inertial mass, such as the Klein–Gordon and Schrödinger's equations.
Equations that pertain to the wavelengths of photons interacting with mass use the non-reduced Compton wavelength. A particle of mass m has a rest energy of E = mc2.
Official source
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