Formula and Method for the Compton Wavelength
The Compton wavelength of a particle is the wavelength of a photon whose energy equals that particle's rest-mass energy. It is defined as λ = h / (m c), where h is Planck's constant, m is the particle's rest mass, and c is the speed of light in a vacuum. Arthur Compton introduced the quantity in 1923 while explaining why X-ray photons shift to a longer wavelength when they scatter off electrons — a discovery that helped confirm light carries momentum in discrete quanta. This calculator returns the Compton wavelength, the reduced Compton wavelength, the rest energy, and the Compton frequency for any particle's rest mass.
How the calculation works
Pick a preset particle (electron, proton, neutron, or muon) or choose "Custom mass" and enter a rest mass in MeV/c², kilograms, or atomic mass units (u). The calculator converts that mass to kilograms, then applies λ = h / (m c) to get the Compton wavelength. Dividing by 2π gives the reduced Compton wavelength ƛ = ħ / (m c), where ħ = h / (2π) is the reduced Planck constant. The same mass fixes the particle's rest energy through Einstein's E = m c², and the Compton frequency f = c / λ = m c² / h — the frequency of a photon carrying exactly that much energy.
Common mistakes
- Confusing λ and ƛ: the reduced Compton wavelength is smaller than the ordinary Compton wavelength by a factor of 2π ≈ 6.283 — mixing them up changes the answer by that factor.
- Mismatched mass units: 1 MeV/c² ≈ 1.7827 × 10⁻³⁰ kg and 1 u ≈ 1.6605 × 10⁻²⁷ kg — picking the wrong unit for a given number produces a wavelength off by many orders of magnitude.
- Mixing up Compton and de Broglie wavelengths: the Compton wavelength depends only on rest mass (λ = h/mc); the de Broglie wavelength depends on momentum (λ = h/p) and describes a moving particle, not a stationary one.
Where the Compton wavelength matters
- Compton scattering: the wavelength shift of an X-ray or gamma-ray photon after scattering off an electron is Δλ = λ_C(1 − cos θ), where λ_C is the electron's Compton wavelength found here — see the Compton Scattering Calculator for the shift at a chosen angle.
- Quantum field theory: the reduced Compton wavelength marks the length scale below which a particle's own quantum fluctuations (pair production) become significant, bounding the validity of single-particle quantum mechanics.
- Nuclear and particle physics: comparing a particle's Compton wavelength to a system's nuclear or de Broglie length scale shows whether relativistic quantum effects matter for that system.
Frequently Asked Questions
What is the Compton wavelength?
The Compton wavelength of a particle is the wavelength of a photon whose energy equals that particle's rest-mass energy: λ = h / (m c), where h is Planck's constant, m is the rest mass, and c is the speed of light. It is a fixed physical property of the particle, not something that changes with its motion.
What is the Compton wavelength of an electron?
About 2.42631 × 10⁻¹² m (2.42631 pm). This is the standard reference value used across physics, and it is the constant λ_C in the Compton scattering formula Δλ = λ_C(1 − cos θ).
What is the difference between the Compton wavelength and the reduced Compton wavelength?
The reduced Compton wavelength ƛ = λ / (2π) = ħ / (m c) is smaller than the ordinary Compton wavelength by a factor of 2π ≈ 6.283. It shows up more often in quantum mechanics and field theory because it removes the 2π that otherwise appears when converting between wavelength and angular quantities.
How is the Compton wavelength used in Compton scattering?
When a photon scatters off a free electron at angle θ, its wavelength increases by Δλ = λ_C(1 − cos θ), where λ_C is the electron's Compton wavelength. This calculator finds that fixed constant for any particle; to find the scattered wavelength itself at a chosen angle, use the Compton Scattering Calculator.
Common Tip Percentages
| Service | Typical Tip Range | Notes |
|---|
| Restaurant (sit-down) | 15–20% | 20%+ for exceptional service |
| Food delivery | 10–20% | Consider extra for long distances |
| Taxi / rideshare | 10–20% | More for assistance with luggage |
| Hotel housekeeping | $2–5/night | Leave daily, not just at checkout |
| Barber / hair salon | 15–20% | On pre-discount total |
| Pizza delivery | 10–15% or $3–5 minimum | Higher in bad weather |
General Household Budget Ratios (50/30/20 Rule)
| Category | % of Net Income | Examples |
|---|
| Needs | 50% | Rent, groceries, utilities, insurance |
| Wants | 30% | Dining out, entertainment, hobbies |
| Savings / Debt | 20% | Emergency fund, retirement, extra payments |
Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |