Compton wavelength explained

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).

m is given by$$\backslash lambda\; =\; \backslash frac,$$where is the Planck constant and is the speed of light.The corresponding frequency is given by$$f\; =\; \backslash frac,$$ and the angular frequency is given by$$\backslash omega\; =\; \backslash frac.$$

The CODATA 2018 value for the Compton wavelength of the electron is .^[1] Other particles have different Compton wavelengths.

Reduced Compton wavelength

The reduced Compton wavelength (barred lambda, denoted below by

\bar λ ) is defined as the Compton wavelength divided by :\bar λ = (λ)/(2 \pi) = (\hbar)/(m c) , where 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:$$\backslash mathbf^2\backslash psi-\backslash frac\backslash frac\backslash psi\; =\; \backslash left(\backslash frac\; \backslash right)^2\; \backslash psi.$$

It appears in the Dirac equation (the following is an explicitly covariant form employing the Einstein summation convention):$$-i\; \backslash gamma^\backslash mu\; \backslash partial\_\backslash mu\; \backslash psi\; +\; \backslash left(\backslash frac\; \backslash right)\; \backslash psi\; =\; 0.$$

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:$$i\backslash hbar\backslash frac\backslash psi=-\backslash frac\backslash nabla^2\backslash psi\; -\backslash frac\; \backslash frac\; \backslash psi.$$

Dividing through by

\hbar c and rewriting in terms of the fine-structure constant, one obtains:$$\backslash frac\backslash frac\backslash psi=-\backslash frac\; \backslash nabla^2\backslash psi\; -\; \backslash frac\; \backslash psi.$$

Distinction between reduced and non-reduced

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.^[2]

Equations that pertain to the wavelengths of photons interacting with mass use the non-reduced Compton wavelength. A particle of mass has a rest energy of . The Compton wavelength for this particle is the wavelength of a photon of the same energy. For photons of frequency, energy is given by$$E\; =\; h\; f\; =\; \backslash frac\; =\; m\; c^2,$$which yields the Compton wavelength formula if solved for .

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 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, 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. 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 . Then the uncertainty relation for position and momentum says that$$\backslash Delta\; x\backslash ,\backslash Delta\; p\backslash ge\; \backslash frac,$$so the uncertainty in the particle's momentum satisfies$$\backslash Delta\; p\; \backslash ge\; \backslash frac.$$

Using the relativistic relation between momentum and energy, when exceeds then the uncertainty in energy is greater than, 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 . 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 :$$\backslash Delta\; x\; \backslash ge\; \backslash frac\; \backslash left(\backslash frac\; \backslash right).$$

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

Relationship to other constants

The Bohr radius is related to the Compton wavelength by:$$a\_0\; =\; \backslash frac\backslash left(\backslash frac\backslash right)\; =\; \backslash frac\; \backslash simeq\; 137\backslash times\backslash bar\_\backslash text\backslash simeq\; 5.29\backslash times\; 10^4~\backslash textrm$$

The classical electron radius is about 3 times larger than the proton radius, and is written:$$r\_\backslash text\; =\; \backslash alpha\backslash left(\backslash frac\backslash right)\; =\; \backslash alpha\backslash bar\_\backslash text\; \backslash simeq\backslash frac\backslash simeq\; 2.82~\backslash textrm$$

The Rydberg constant, having dimensions of linear wavenumber, is written:$$\backslash frac=\backslash frac\; \backslash simeq\; 91.1~\backslash textrm$$$$\backslash frac\; =\; \backslash frac\backslash left(\backslash frac\backslash right)\; =\; 2\; \backslash frac\; \backslash simeq\; 14.5~\backslash textrm$$

This yields the sequence:$$r\_\; =\; \backslash alpha\; \backslash bar\_\; =\; \backslash alpha^2\; a\_0\; =\; \backslash alpha^3\; \backslash frac.$$

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$$\backslash sigma\_\backslash mathrm\; =\; \backslash frac\backslash alpha^2\backslash bar\_\backslash text^2\; \backslash simeq\; 66.5~\backslash textrm^2,$$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.

r _{\rm} = 2 G M /c ^{2} are the same, when their value is close to the Planck length (l _{\rm} ). 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:

$$m\_\; =\; \backslash sqrt$$$$l\_\; =\; \backslash sqrt.$$

Geometrical interpretation

A geometrical origin of the Compton wavelength has been demonstrated using semiclassical equations describing the motion of a wavepacket.^[4] In this case, the Compton wavelength is equal to the square root of the quantum metric, a metric describing the quantum space:

\sqrt{g _{kk} }=\lambda_\mathrm.

See also

• de Broglie wavelength
• Planck–Einstein relation

External links

• Length Scales in Physics: the Compton Wavelength

Notes and References

1. CODATA 2018 value for Compton wavelength for the electron from NIST.
2. [Walter Greiner|Greiner, W.]
3. Garay . Luis J. . Quantum Gravity And Minimum Length . . 10 . 2 . 1995 . 145–65 . gr-qc/9403008 . 10.1142/S0217751X95000085 . 1995IJMPA..10..145G . 119520606 .
4. Leblanc . C. . Malpuech . G. . Solnyshkov . D. D. . 2021-10-26 . Universal semiclassical equations based on the quantum metric for a two-band system . Physical Review B . en . 104 . 13 . 134312 . 10.1103/PhysRevB.104.134312 . 2106.12383 . 2021PhRvB.104m4312L . 235606464 . 2469-9950.