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1 Equivalent photon wavelength
2 Significance
3 References
4 External links

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, $\lambda \$ , of a particle is given by

$\lambda = \frac{h}{m c} = 2 \pi \frac{\hbar}{m c} \$

where

$h \$ is the Planck constant, $m \$ is the particle's rest mass, $c \$ is the speed of light.

The significance of this formula is shown in the derivation of the Compton shift formula.

The CODATA 2006 value for the Compton wavelength of the electron is 2.4263102175×10^-12 meters with a standard uncertainty of 0.0000000033×10^-12 m.[1] Other particles have different Compton wavelengths.

Equivalent photon wavelength

A particle of rest mass $m \$ has the rest energy $E = m c^2 \$ . The Compton wavelength for this particle is the wavelength of a photon of this same energy $E \$ since for photons of frequency $f \$ , energy is given by

$E = h f = \frac{h c}{\lambda} = m c^2 \$

which yields the above formula if solved for $\lambda \$ .

Significance

Limitation on measurement

The 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 $mc^2 \$ , 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. This renders moot the question of the original particle's location.

This argument also shows that the 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 $\Delta x \$ . Then the uncertainty relation for position and momentum says that

$\Delta x\,\Delta p\ge \hbar/2$

so the uncertainty in the particle's momentum satisfies

$\Delta p \ge \frac{\hbar}{2\Delta x}$

Using the relativistic relation between momentum and energy $\mathbf{p} = \gamma m_0\mathbf{v}$ , when $Δ p$ exceeds $m c$ then the uncertainty in energy is greater than $mc^2 \$ , which is enough energy to create another particle of the same type. So, with a little algebra, we see there is a fundamental limitation

$\Delta x \ge \frac{\hbar}{2mc}$

So, at least to within an order of magnitude, the uncertainty in position must be greater than the Compton wavelength $h/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.

Fermion cross-section of interactions

For fermions, the 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

$\frac{8\pi}{3}\alpha^2\lambda_e^2$ ,

where $\alpha \$ is the fine-structure constant and $\lambda_e \$ is the Compton wavelength of the electron. 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.

Relationship to Planck units

The Compton wavelength of the electron is one of a trio of related units of length, the other two being the Bohr radius $a 0$ and the classical electron radius $r e$ . The Compton wavelength is built from the electron mass $m e$ , Planck's constant $h$ and the speed of light $c$ . The Bohr radius is built from $m e$ , $h$ and the electron charge $e$ . The classical electron radius is built from $m e$ , $c$ and $e$ . Any one of these three lengths can be written in terms of any other using the fine structure constant $α$ :

$r_e = {\alpha \lambda_e \over 2\pi} = \alpha^2 a_0$

The Planck mass is special because ignoring factors of $2π$ and the like, the Compton wavelength for this mass is equal to its Schwarzschild radius. This special distance is called the Planck length. 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.

References

^ CODATA 2006 value for Compton wavelength for the electron from NIST

External links

Length Scales in Physics: the Compton Wavelength

Categories: Atomic physics | Foundational quantum physics

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