Compton Wavelength Calculator – Quantum Wavelength of Particles
Compute the Compton wavelength λ = h/(mc) for electrons, protons, neutrons, or any custom particle mass using fundamental quantum constants.
Select a particle type (electron, proton, neutron) or enter a custom particle mass in kilograms. The calculator returns the Compton wavelength and the reduced Compton wavelength.
Compton Wavelength Calculator – Quantum Wavelength of Particles
Compute the Compton wavelength λ = h/(mc) for electrons, protons, neutrons, or any custom particle mass using fundamental quantum constants.
Result
Compton Wavelength λ = 2.42631 pm
Reduced Compton Wavelength ƛ = 386.159 fm
λ = 2.42631e-12 m
λ = h/(m₀c), ƛ = ℏ/(m₀c) = λ/(2π); h = 6.626 × 10⁻³⁴ J·s, c = 2.998 × 10⁸ m/s
About the Compton Wavelength Calculator
The Compton wavelength is one of the most fundamental length scales in quantum physics. For a particle of rest mass m₀, it is defined as λ = h/(m₀c), where h is Planck's constant (6.62607 × 10⁻³⁴ J·s) and c is the speed of light (2.99792 × 10⁸ m/s). The reduced Compton wavelength is ƛ = λ/(2π) = ℏ/(m₀c), where ℏ is the reduced Planck constant. This length scale represents the quantum mechanical size of a particle — the scale at which quantum field effects become important and pair creation becomes energetically possible.
For the electron, the Compton wavelength is approximately 2.42631 × 10⁻¹² m = 2.42631 pm (picometres). This is about 137 times larger than the classical electron radius and approximately 20 times smaller than the Bohr radius (the characteristic size of a hydrogen atom). For the proton, the Compton wavelength is about 1.32141 × 10⁻¹⁵ m = 1.32141 fm (femtometres), close to the measured proton charge radius. For the neutron, it is approximately 1.31959 × 10⁻¹⁵ m = 1.31959 fm, very close to the proton value since their masses are nearly equal.
The Compton wavelength was named after Arthur H. Compton following his 1923 discovery of the Compton effect — the inelastic scattering of X-rays by free electrons. The wavelength shift Δλ = λ_c(1 − cosθ) observed in this scattering process directly reveals the electron's Compton wavelength. Compton's Nobel Prize-winning work established that electromagnetic radiation behaves as a stream of discrete photons with definite energy and momentum, providing pivotal evidence for quantum mechanics.
In quantum field theory, the Compton wavelength has profound significance. Below the reduced Compton wavelength of a particle, quantum field effects dominate over ordinary quantum mechanics — in particular, the energy required to confine a particle to this scale is comparable to its rest mass energy m₀c², at which point particle-antiparticle pair production becomes possible. This makes the Compton wavelength a natural boundary between relativistic quantum mechanics and non-relativistic quantum mechanics.
The Compton wavelength appears throughout modern physics: in the hydrogen atom energy levels, in the fine structure constant (α = r_e/ƛ_e where r_e is the classical electron radius), in nuclear physics for setting the scale of nuclear forces, and in cosmology when discussing quantum gravity effects. For composite particles such as atomic nuclei, the Compton wavelength can be calculated using their total rest mass, though the interpretation differs from that of fundamental point particles.
Compton Wavelength Examples
Compton wavelengths for fundamental particles and comparison with other quantum length scales.
| Particle / Mass | Compton Wavelength | Physical Significance |
|---|---|---|
| Electron (m = 9.109 × 10⁻³¹ kg) | λ = 2.4263 pm | Sets the quantum scale for electron–photon interactions; 137× larger than classical electron radius. |
| Proton (m = 1.673 × 10⁻²⁷ kg) | λ = 1.3214 fm | Comparable to the measured proton charge radius (~0.87 fm); scale of strong nuclear force effects. |
| Neutron (m = 1.675 × 10⁻²⁷ kg) | λ = 1.3196 fm | Nearly identical to the proton wavelength because the proton and neutron masses differ by less than 0.14%. |
| Custom: m = 1.00 × 10⁻²⁷ kg | λ ≈ 2.210 fm | Demonstrates how the Compton wavelength scales inversely with mass — heavier particles have shorter wavelengths. |
How to Use the Compton Wavelength Calculator
- Select the particle type — Electron, Proton, or Neutron — for standard fundamental particles. The calculator uses the 2018 CODATA recommended values for their masses.
- To calculate the Compton wavelength for any other particle, select Custom Mass and enter the rest mass in kilograms (kg). Use scientific notation, for example 1.67e-27.
- Click Calculate. The result displays both the Compton wavelength λ = h/(m₀c) and the reduced Compton wavelength ƛ = ℏ/(m₀c) in appropriate units (pm for electrons, fm for nucleons).
- Compare the result to other quantum length scales: the Bohr radius (52.9 pm) is about 22× the electron Compton wavelength; nuclear radii are on the order of a few fm.
- Use the example buttons to load common particles instantly for reference and comparison.
Compton Wavelength FAQ
What is the Compton wavelength?
The Compton wavelength of a particle is λ = h/(m₀c), where h is Planck's constant, m₀ is the particle's rest mass, and c is the speed of light. It represents the quantum mechanical length scale characteristic of that particle. For the electron, λ = 2.42631 pm. The Compton wavelength was first identified in Arthur Compton's 1923 study of X-ray scattering, where it appeared as the characteristic wavelength shift per unit of (1 − cosθ) in the scattering formula.
What is the difference between the Compton wavelength and the reduced Compton wavelength?
The Compton wavelength is λ = h/(m₀c) and the reduced Compton wavelength is ƛ = ℏ/(m₀c) = λ/(2π), where ℏ = h/(2π) is the reduced Planck constant. The reduced version appears more naturally in quantum field theory equations and is sometimes called the 'Compton radius'. For the electron, ƛ_e = 0.38616 pm. Both are fundamental constants of quantum mechanics; which one to use depends on whether the formula uses h or ℏ.
How does the Compton wavelength relate to the de Broglie wavelength?
The de Broglie wavelength λ_dB = h/p depends on the particle's momentum p, while the Compton wavelength λ_C = h/(m₀c) depends only on the rest mass. For a particle moving at speed v, the de Broglie wavelength equals the Compton wavelength when the particle's momentum equals m₀c, which occurs at relativistic speeds (v ≈ c/√2). At non-relativistic speeds, the de Broglie wavelength is much longer than the Compton wavelength.
Why does the Compton wavelength matter in quantum field theory?
In quantum field theory, the reduced Compton wavelength ƛ sets the length scale below which a particle cannot be localized without pair production. If you try to confine a particle to a region smaller than ƛ, the required energy exceeds the rest mass energy m₀c², making spontaneous particle-antiparticle creation possible. This makes the Compton wavelength a fundamental boundary between single-particle quantum mechanics and full quantum field theory where particle number is not conserved.
What is the Compton wavelength of a proton compared to nuclear scales?
The proton Compton wavelength is approximately 1.321 fm (femtometres = 10⁻¹⁵ m), which is comparable to the proton's measured charge radius of about 0.87 fm. The range of the strong nuclear force (mediated by pion exchange) is roughly 1.4 fm — close to the pion's Compton wavelength of about 1.4 fm. This is not coincidental: the Compton wavelength of the exchange particle sets the range of the associated force through the Yukawa potential.
Can the Compton wavelength be measured experimentally?
Yes. The electron Compton wavelength was first measured by Compton himself through X-ray scattering experiments in 1923, confirming the formula Δλ = λ_c(1 − cosθ). Modern precision measurements use Penning trap experiments and x-ray spectroscopy to determine it with extraordinary accuracy. The 2018 CODATA value is λ_e = 2.42631023867 × 10⁻¹² m with a relative uncertainty of 3.0 × 10⁻¹⁰, and it can also be derived from the fine structure constant and Rydberg constant.