Compton wavelength

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 fundamental equations of quantum mechanics. The reduced Compton wavelength appears in the relativistic Klein–Gordon equation for a free particle: $${\displaystyle {\mathrm{\nabla }}^2\psi -\frac{1}{c^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}\psi ={\left(\frac{mc}{\hslash }\right)}^2\psi .}$$

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

The reduced Compton wavelength is also present in the Schrödinger equation for an electron in a hydrogen-like atom, although this is not readily apparent in traditional representations of the equation. The following is the traditional representation of the Schrödinger equation: $${\displaystyle i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}\psi =-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2\psi -\frac{1}{4\pi {ϵ}_0}\frac{Z{e}^2}{r}\psi .}$$

Dividing through by ħc and rewriting in terms of the fine-structure constant, one obtains: $${\displaystyle \frac{i}{c}\frac{\mathrm{\partial }}{\mathrm{\partial }t}\psi =-\frac{\lambda \overline{}}{2}{\mathrm{\nabla }}^2\psi -\frac{\alpha Z}{r}\psi .}$$

Table of values

Relationship to other constants

Typical atomic lengths, wave numbers, and areas in physics can be related to the reduced Compton wavelength for the electron (⁠${\displaystyle \lambda {\overline{}}_{\text{e}}\equiv \frac{{\lambda }_{\text{e}}}{2\pi }\simeq 386\mathrm{f}\mathrm{m}}$⁠) and the electromagnetic fine-structure constant (⁠${\displaystyle \alpha \simeq \frac{1}{137}}$⁠).

The classical electron radius is about 3 times larger than the proton radius, and is written: $${\displaystyle r_{\text{e}}=\alpha \lambda {\overline{}}_{\text{e}}\simeq 2.82\text{fm}}$$

The Bohr radius is related to the Compton wavelength by: $${\displaystyle a_0=\frac{\lambda {\overline{}}_{\text{e}}}{\alpha }\simeq 5.29\times {10}^4\text{fm}}$$

The angular wavenumber of a photon with one hartree (the atomic unit of energy ⁠${\displaystyle E_{\text{h}}=4\pi \hslash c{R}_{\mathrm{\infty }}}$⁠, where ⁠${\displaystyle R_{\mathrm{\infty }}}$⁠ is the Rydberg constant), being (approximately) the negative potential energy of the electron in the hydrogen atom, and twice the energy needed to ionize it, is: $${\displaystyle \frac{\hslash c}{E_{\text{h}}}=\frac{1}{4\pi R_{\mathrm{\infty }}}=\frac{\lambda {\overline{}}_{\text{e}}}{{\alpha }^2}\simeq 7.25\text{nm}}$$

This yields the sequence: $${\displaystyle {\alpha }^{-1}{r}_{\text{e}}=\lambda {\overline{}}_{\text{e}}=\alpha a_0={\alpha }^2\hslash c/E_{\text{h}}.}$$

For fermions, the classical (electromagnetic) radius sets the cross-section of electromagnetic interactions of a particle. For example, the cross-section for Thomson scattering of a photon from an electron is equal to $${\displaystyle {\sigma }_{\mathrm{e}}=\frac{8\pi }{3}{r}_{\text{e}}{}^2\simeq 66.5\mathrm{f}{\mathrm{m}}^2,}$$ which is roughly the same as the cross-sectional area of an iron-56 nucleus.

Geometrical interpretation

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: ⁠${\displaystyle \sqrt{g_{kk}}={\lambda }_{\mathrm{C}}}$⁠.