Compton wavelength

The Compton wavelength is a quantum mechanical property of a particle. It was introduced by Arthur Compton in his explanation of the scattering of photons by electrons (a process known as Compton scattering). The Compton wavelength of a particle is equivalent to the wavelength of a photon whose energy is the same as the rest-mass energy of the particle.

The Compton wavelength, λ, of a particle is given by

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

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.

The CODATA 2010 value for the Compton wavelength of the electron is Template:Val. Other particles have different Compton wavelengths.

Significance

Reduced Compton wavelength

When the Compton wavelength is divided by ${2\pi }$, one obtains a smaller or “reduced” Compton wavelength:

${\frac {\lambda }{2\pi }}={\frac {\hbar }{mc}}\$

The 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:

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

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

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

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:

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

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

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

Relationship between the reduced and non-reduced Compton wavelength

The reduced Compton wavelength is a natural representation for mass on the quantum scale. Equations that pertain to mass in the form of mass, like Klein-Gordon and Schrödinger's, use the reduced Compton wavelength. 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 rest 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

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

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

Limitation on measurement

The reduced Compton wavelength can be thought of as a fundamental limitation on measuring the position of a particle, taking quantum mechanics and special relativity into account. This depends on the mass m of the particle. To see this, 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 have enough energy to create a new particle of the same type.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}} }} 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.

We can make the above argument 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

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

so the uncertainty in the particle's momentum satisfies

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

Using the relativistic relation between momentum and energy p = γm₀v, 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. It follows that there is a fundamental limitation on Δx:

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

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.

Relationship to other constants

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

The Bohr radius is related to the Compton wavelength by:

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

The classical electron radius is about 3 times larger than the proton radius, and is written:

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

The Rydberg constant is written:

$R_{\infty }={\frac {\alpha ^{2}}{2\lambda _{e}}}$

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

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

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 rest mass, electromagnetism has infinite range.

Typical lengths and areas in gravitational physics can be related to the Compton wavelength and the gravitational coupling constant ($\alpha _{G}$ which is the gravitational analog of the fine structure constant):

The Planck mass is special because the reduced Compton wavelength for this mass is equal to half of the Schwarzschild radius. This special distance is called the Planck length ($\ell _{P}$). This is a simple case of dimensional analysis: the Schwarzschild radius is proportional to the mass, whereas the Compton wavelength is proportional to the inverse of the mass. The Planck length is written:

$\ell _{P}=\lambda _{e}\,{\frac {\sqrt {\alpha _{G}}}{2\pi }}$

External links

• Length Scales in Physics: the Compton Wavelength