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Reduced Compton wavelength

The reduced Compton wavelength λ ¯ {\displaystyle \lambda \!\!\!{\bar {}}} (barred lambda) of a particle is defined as its Compton wavelength divided by 2π:

λ ¯ = λ 2 π = ℏ m c , {\displaystyle \lambda \!\!\!{\bar {}}={\frac {\lambda }{2\pi }}={\frac {\hbar }{mc}},}

where ħ 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: ∇ 2 ψ − 1 c 2 ∂ 2 ∂ t 2 ψ = ( m c ℏ ) 2 ψ . {\displaystyle \mathbf {\nabla } ^{2}\psi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\psi =\left({\frac {mc}{\hbar }}\right)^{2}\psi .}

It appears in the Dirac equation (the following is an explicitly covariant form employing the Einstein summation convention): − i γ μ ∂ μ ψ + ( m c ℏ ) ψ = 0. {\displaystyle -i\gamma ^{\mu }\partial _{\mu }\psi +\left({\frac {mc}{\hbar }}\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: i ℏ ∂ ∂ t ψ = − ℏ 2 2 m ∇ 2 ψ − 1 4 π ϵ 0 Z e 2 r ψ . {\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi -{\frac {1}{4\pi \epsilon _{0}}}{\frac {Ze^{2}}{r}}\psi .}

Dividing through by ħc and rewriting in terms of the fine-structure constant, one obtains: i c ∂ ∂ t ψ = − λ ¯ 2 ∇ 2 ψ − α Z r ψ . {\displaystyle {\frac {i}{c}}{\frac {\partial }{\partial t}}\psi =-{\frac {\lambda \!\!\!{\bar {}}}{2}}\nabla ^{2}\psi -{\frac {\alpha Z}{r}}\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: 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. The Compton wavelength for this particle is the wavelength of a photon of the same energy. For photons of frequency f, energy is given by E = h f = h c λ = m c 2 , {\displaystyle E=hf={\frac {hc}{\lambda }}=mc^{2},} which yields the Compton wavelength formula if solved for λ.

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 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 Δ x Δ p ≥ ℏ 2 , {\displaystyle \Delta x\,\Delta p\geq {\frac {\hbar }{2}},} so the uncertainty in the particle's momentum satisfies Δ p ≥ ℏ 2 Δ x . {\displaystyle \Delta p\geq {\frac {\hbar }{2\Delta x}}.}

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: Δ x ≥ 1 2 ( ℏ m c ) . {\displaystyle \Delta x\geq {\frac {1}{2}}\left({\frac {\hbar }{mc}}\right).}

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

Relationship to other constants

Typical atomic lengths, wave numbers, and areas in physics can be related to the reduced Compton wavelength for the electron (⁠ λ ¯ e ≡ λ e 2 π ≃ 386   f m {\displaystyle \textstyle \lambda \!\!\!{\bar {}}_{\text{e}}\equiv {\tfrac {\lambda _{\text{e}}}{2\pi }}\simeq \mathrm {386~fm} } ⁠) and the electromagnetic fine-structure constant (⁠ α ≃ 1 137 {\displaystyle \alpha \simeq {\tfrac {1}{137}}} ⁠).

The classical electron radius is about 3 times larger than the proton radius, and is written: r e = α λ ¯ e ≃ 2.82   fm {\displaystyle r_{\text{e}}=\alpha \lambda \!\!\!{\bar {}}_{\text{e}}\simeq 2.82~{\textrm {fm}}}

The Bohr radius is related to the Compton wavelength by: a 0 = λ ¯ e α ≃ 5.29 × 10 4   fm {\displaystyle a_{0}={\frac {\lambda \!\!\!{\bar {}}_{\text{e}}}{\alpha }}\simeq 5.29\times 10^{4}~{\textrm {fm}}}

The angular wavenumber associated with the Rydberg constant, being (approximately) that of a photon that has barely the energy to ionize a hydrogen atom, is written: 1 2 π R ∞ = 2 λ ¯ e α 2 ≃ 14.5   nm {\displaystyle {\frac {1}{2\pi R_{\infty }}}=2{\frac {\lambda \!\!\!{\bar {}}_{\text{e}}}{\alpha ^{2}}}\simeq 14.5~{\textrm {nm}}}

This yields the sequence: r e = α λ ¯ e = α 2 a 0 = 1 2 α 3 1 2 π R ∞ . {\displaystyle r_{\text{e}}=\alpha \lambda \!\!\!{\bar {}}_{\text{e}}=\alpha ^{2}a_{0}={\frac {1}{2}}\alpha ^{3}{\frac {1}{2\pi R_{\infty }}}.}

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 σ e = 8 π 3 r e 2 ≃ 66.5   f m 2 , {\displaystyle \sigma _{\mathrm {e} }={\frac {8\pi }{3}}r_{\text{e}}{}^{2}\simeq \mathrm {66.5~fm^{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.⁴ In this case, the Compton wavelength is equal to the square root of the quantum metric, a metric describing the quantum space: ⁠ g k k = λ C {\displaystyle {\sqrt {g_{kk}}}=\lambda _{\mathrm {C} }} ⁠.

See also

• de Broglie wavelength
• Planck–Einstein relation

External links

• Length Scales in Physics: the Compton Wavelength

References

1. "2022 CODATA Value: Compton wavelength". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 2024-05-18. https://physics.nist.gov/cgi-bin/cuu/Value?ecomwl ↩

2. Greiner, W., Relativistic Quantum Mechanics: Wave Equations (Berlin/Heidelberg: Springer, 1990), pp. 18–22. /wiki/Walter_Greiner ↩

3. 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. /wiki/International_Journal_of_Modern_Physics#International_Journal_of_Modern_Physics_A ↩

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. 104 (13): 134312. arXiv:2106.12383. Bibcode:2021PhRvB.104m4312L. doi:10.1103/PhysRevB.104.134312. ISSN 2469-9950. S2CID 235606464. https://link.aps.org/doi/10.1103/PhysRevB.104.134312 ↩