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Compton wavelength

## Summary

The Compton wavelength is a quantum mechanical property of a particle. The Compton wavelength of a particle is equal to the wavelength of a photon whose energy is the same as the mass of that particle (see mass–energy equivalence). It was introduced by Arthur Compton 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,

${\displaystyle \lambda ={\frac {h}{mc}}\ }$

while its frequency is given by,

${\displaystyle f={\frac {mc^{2}}{h}}}$

where h is the Planck constant, m is the particle's rest mass, and c is the speed of light. The significance of this formula is shown in the derivation of the Compton shift formula. It is equivalent to the de Broglie wavelength with ${\displaystyle v=c}$.

The CODATA 2018 value for the Compton wavelength of the electron is 2.42631023867(73)×10−12 m.[1] Other particles have different Compton wavelengths.

## Reduced Compton wavelength

When the Compton wavelength is divided by 2π, one obtains the "reduced" Compton wavelength ƛ (barred lambda), i.e. the Compton wavelength for 1 radian instead of 2π radians:

ƛ = λ/2π = ħ/mc,

where ħ is the "reduced" Planck constant.

## Role in equations for massive particles

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:

${\displaystyle \mathbf {\nabla } ^{2}\psi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\psi =\left({\frac {mc}{\hbar }}\right)^{2}\psi .}$

It appears in the Dirac equation (the following is an explicitly covariant form employing the Einstein summation convention):

${\displaystyle -i\gamma ^{\mu }\partial _{\mu }\psi +\left({\frac {mc}{\hbar }}\right)\psi =0.}$

The reduced Compton wavelength also appears in Schrödinger's equation, although its presence is obscured in traditional representations of the equation. The following is the traditional representation of Schrödinger's equation for an electron in a hydrogen-like atom:

${\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi -{\frac {1}{4\pi \epsilon _{0}}}{\frac {Ze^{2}}{r}}\psi .}$

Dividing through by ${\displaystyle \hbar c}$, and rewriting in terms of the fine structure constant, one obtains:

${\displaystyle {\frac {i}{c}}{\frac {\partial }{\partial t}}\psi =-{\frac {1}{2}}\left({\frac {\hbar }{mc}}\right)\nabla ^{2}\psi -{\frac {\alpha Z}{r}}\psi .}$

## Distinction between reduced and non-reduced

The reduced Compton wavelength is a natural representation of mass on the quantum scale. Equations that pertain to inertial mass like Klein-Gordon and Schrödinger's, use the reduced Compton wavelength.[2]: 18–22  The non-reduced Compton wavelength is a natural representation for mass that has been converted into energy. Equations that pertain to the conversion of mass into energy, or 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. The non-reduced Compton wavelength for this particle is the wavelength of a photon of the same energy. For photons of frequency f, energy is given by

${\displaystyle E=hf={\frac {hc}{\lambda }}=mc^{2},\ }$

which yields the non-reduced or standard Compton wavelength formula if solved for λ.

## Limitation on measurement

The Compton wavelength expresses a fundamental limitation on measuring the position of a particle, taking into account quantum mechanics and special relativity.[3]

This limitation depends on the mass m of the particle. To see how, note that we can measure the position of a particle by bouncing light off it – but measuring the position accurately requires light of short wavelength. Light with a short wavelength consists of photons of high energy. If the energy of these photons exceeds mc2, when one hits the particle whose position is being measured the collision may yield enough energy to create a new particle of the same type.[citation needed] This renders moot the question of the original particle's location.

This argument also shows that the reduced Compton wavelength is the cutoff below which quantum field theory – which can describe particle creation and annihilation – becomes important. The above argument can be made a bit more precise as follows. Suppose we wish to measure the position of a particle to within an accuracy Δx. Then the uncertainty relation for position and momentum says that

${\displaystyle \Delta x\,\Delta p\geq {\frac {\hbar }{2}},}$

so the uncertainty in the particle's momentum satisfies

${\displaystyle \Delta p\geq {\frac {\hbar }{2\Delta x}}.}$

Using the relativistic relation between momentum and energy E2 = (pc)2 + (mc2)2, when Δp exceeds mc then the uncertainty in energy is greater than mc2, which is enough energy to create another particle of the same type. But we must exclude this greater energy uncertainty. Physically, this is excluded by the creation of one or more additional particles to keep the momentum uncertainty of each particle at or below mc. In particular the minimum uncertainty is when the scattered photon has limit energy equal to the incident observing energy. It follows that there is a fundamental minimum for Δx:

${\displaystyle \Delta x\geq {\frac {1}{2}}\left({\frac {\hbar }{mc}}\right).}$

Thus the uncertainty in position must be greater than half of the reduced Compton wavelength ħ/mc.

The Compton wavelength can be contrasted with the de Broglie wavelength, which depends on the momentum of a particle and determines the cutoff between particle and wave behavior in quantum mechanics. Notably, de Broglie's derivation of the de Broglie wavelength is based on the assumption that an observed particle is associated with a periodic phenomenon of the particle's Compton frequency.

## Relationship to other constants

Typical atomic lengths, wave numbers, and areas in physics can be related to the reduced Compton wavelength for the electron (${\textstyle {\bar {\lambda }}_{\text{e}}\equiv {\tfrac {\lambda _{\text{e}}}{2\pi }}\simeq 386~{\textrm {fm}}}$) and the electromagnetic fine structure constant (${\textstyle \alpha \simeq {\tfrac {1}{137}}}$).

The Bohr radius is related to the Compton wavelength by:

${\displaystyle a_{0}={\frac {1}{\alpha }}\left({\frac {\lambda _{\text{e}}}{2\pi }}\right)={\frac {{\bar {\lambda }}_{\text{e}}}{\alpha }}\simeq 137\times {\bar {\lambda }}_{\text{e}}\simeq 5.29\times 10^{4}~{\textrm {fm}}}$

${\displaystyle r_{\text{e}}=\alpha \left({\frac {\lambda _{\text{e}}}{2\pi }}\right)=\alpha {\bar {\lambda }}_{\text{e}}\simeq {\frac {{\bar {\lambda }}_{\text{e}}}{137}}\simeq 2.82~{\textrm {fm}}}$

The Rydberg constant, having dimensions of linear wavenumber, is written:

${\displaystyle {\frac {1}{R_{\infty }}}={\frac {2\lambda _{\text{e}}}{\alpha ^{2}}}\simeq 91.1~{\textrm {nm}}}$
${\displaystyle {\frac {1}{2\pi R_{\infty }}}={\frac {2}{\alpha ^{2}}}\left({\frac {\lambda _{\text{e}}}{2\pi }}\right)=2{\frac {{\bar {\lambda }}_{\text{e}}}{\alpha ^{2}}}\simeq 14.5~{\textrm {nm}}}$

This yields the sequence:

${\displaystyle r_{\text{e}}=\alpha {\bar {\lambda }}_{\text{e}}=\alpha ^{2}a_{0}=\alpha ^{3}{\frac {1}{4\pi R_{\infty }}}}$.

For fermions, the reduced Compton wavelength sets the cross-section of interactions. For example, the cross-section for Thomson scattering of a photon from an electron is equal to[clarification needed]

${\displaystyle \sigma _{T}={\frac {8\pi }{3}}\alpha ^{2}{\bar {\lambda }}_{\text{e}}^{2}\simeq 66.5~{\textrm {fm}}^{2},}$

which is roughly the same as the cross-sectional area of an iron-56 nucleus. For gauge bosons, the Compton wavelength sets the effective range of the Yukawa interaction: since the photon has no mass, electromagnetism has infinite range.

The Planck mass is the order of mass for which the Compton wavelength and the Schwarzschild radius ${\displaystyle r_{\rm {S}}=2GM/c^{2}}$ are the same, when their value is close to the Planck length (${\displaystyle l_{\rm {P}}}$). The Schwarzschild radius is proportional to the mass, whereas the Compton wavelength is proportional to the inverse of the mass. The Planck mass and length are defined by:

${\displaystyle m_{\rm {P}}={\sqrt {\hbar c/G}}}$
${\displaystyle l_{\rm {P}}={\sqrt {\hbar G/c^{3}}}.}$