When the Compton wavelength is divided by, one obtains the "reduced" Compton wavelength , i.e. the Compton wavelength for radian instead of radians: 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: It appears in the Dirac equation : 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. 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 mass has a rest energy of. The non-reduced Compton wavelength for this particle is the wavelength of a photon of the same energy. For photons of frequency, energy is given by which yields the non-reduced or standard 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. 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 so the uncertainty in the particle's momentum satisfies 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. 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 : Thus the uncertainty in position must be greater than half of the reduced Compton wavelength. 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 and the electromagnetic fine structure constant . The Bohr radius is related to the Compton wavelength by: The classical electron radius is about 3 times larger than the proton radius, and is written: The Rydberg constant, having dimensions of linear wavenumber, is written: 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 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. 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:
