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 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
Reduced Compton wavelength
The reduced Compton wavelength ƛ (barred lambda) is defined as the Compton wavelength is divided by 2π:
where ħ is the reduced Planck constant.
Further the relativistic Compton Wavelength is given by
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:
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:
Dividing through by and rewriting in terms of the fine-structure constant, one obtains:
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.: 18–22
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. Compton wavelength for this particle is the wavelength of a photon of the same energy. For photons of frequency f, energy is given by
Limitation on measurement
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. 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
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:
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 () and the electromagnetic fine-structure constant ().
The Bohr radius is related to the Compton wavelength by:
This yields the sequence:
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]
The Planck mass is the order of mass for which the Compton wavelength and the Schwarzschild radius are the same, when their value is close to the Planck length (). 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:
A geometrical origin of the Compton wavelength has been demonstrated using semiclassical equations describing the motion of a wavepacket. In this case, the Compton wavelength is equal to the square root of the quantum metric, a metric describing the quantum space:
- CODATA 2018 value for Compton wavelength for the electron from NIST.
- Greiner, W., Relativistic Quantum Mechanics: Wave Equations (Berlin/Heidelberg: Springer, 1990), pp. 18–22.
- Garay, Luis J. (1995). "Quantum Gravity And Minimum Length". International Journal of Modern Physics A. 10 (2): 145–65. arXiv:gr-qc/9403008. Bibcode:1995IJMPA..10..145G. doi:10.1142/S0217751X95000085. S2CID 119520606.
- 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. 104 (13): 134312. arXiv:2106.12383. doi:10.1103/PhysRevB.104.134312. ISSN 2469-9950. S2CID 235606464.