What the De Broglie Frequency Calculator does
In 1924, Louis de Broglie proposed that every moving particle has an associated matter wave — an idea confirmed for electrons by the Davisson–Germer diffraction experiment in 1927 and honored with the 1929 Nobel Prize in Physics. This calculator takes a particle's mass and speed and computes the de Broglie wavelength λ = h/(mv) together with the corresponding wave frequency f = v/λ = mv²/h, using the non-relativistic momentum p = mv. It also reports the momentum, the wave period T = 1/f, and the speed as a fraction of the speed of light so you can judge whether the non-relativistic formula is appropriate.
How to use it
- Choose a particle. The electron, proton, and neutron presets fill in the CODATA mass automatically; choose Custom mass to enter any mass in kilograms (scientific notation such as
9.11e-31works). - Enter the speed in meters per second. It must be greater than zero and less than the speed of light, c = 299,792,458 m/s.
- Calculate to get the de Broglie frequency in hertz, the wavelength in meters, the momentum, and the wave period.
Because the calculation uses p = mv rather than the relativistic p = γmv, results are accurate for speeds well below the speed of light. Below about 10% of c the momentum error stays under roughly 0.5%; above that, the calculator shows a warning.
The formula explained
De Broglie's relation ties a particle's wavelength to its momentum through the Planck constant h = 6.62607015×10⁻³⁴ J·s (an exact value since the 2019 SI redefinition):
$$\lambda = \frac{h}{p} = \frac{h}{mv}$$The frequency reported here is the frequency of a wave of wavelength λ travelling at the particle's speed v:
$$f = \frac{v}{\lambda} = \frac{m v^2}{h}$$A note on conventions. Textbooks differ on what "de Broglie frequency" means. This calculator uses f = v/λ, the frequency obtained directly from the computed wavelength and the particle's speed. Another common convention applies the Planck–Einstein relation f = E/h: with the kinetic energy E = ½mv² it gives exactly half the value computed here, while with the total relativistic energy E = γmc² it gives a much larger value whose associated phase velocity is c²/v. None of these is "wrong" — they answer different questions — but when comparing results, always check which energy or velocity the frequency refers to.
Worked example
Take an electron (m = 9.1093837×10⁻³¹ kg) moving at v = 1×10⁶ m/s.
- Momentum: p = mv = 9.1093837×10⁻³¹ × 10⁶ = 9.1094×10⁻²⁵ kg·m/s.
- Wavelength: λ = h/p = 6.62607015×10⁻³⁴ / 9.1094×10⁻²⁵ ≈ 7.274×10⁻¹⁰ m, about 0.727 nm — comparable to atomic spacing, which is exactly why electron beams diffract off crystals.
- Frequency: f = v/λ = 10⁶ / 7.274×10⁻¹⁰ ≈ 1.375×10¹⁵ Hz.
- Validity: v/c ≈ 0.0033, far below light speed, so the non-relativistic formula is accurate.
Frequently asked questions
Which frequency convention does this calculator use? It computes f = v/λ = mv²/h, the frequency of a wave with the particle's de Broglie wavelength moving at the particle's speed. The alternative Planck–Einstein convention f = E/h gives mv²/(2h) if E is the kinetic energy (exactly half this calculator's value) or a much larger number if E is the total relativistic energy, so always confirm the convention before comparing sources.
When is the non-relativistic formula accurate? The formula λ = h/(mv) uses classical momentum p = mv, which is a good approximation while v stays below about 10% of the speed of light — at v = 0.1c the relativistic factor γ is only about 1.005, so the momentum error is roughly 0.5%. At higher speeds you should use the relativistic momentum p = γmv, and this calculator flags such inputs with a warning.
Why don't everyday objects show wave behavior? Because their momentum is enormous compared with the Planck constant. A 0.145 kg baseball thrown at 40 m/s has a de Broglie wavelength of about 1.1×10⁻³⁴ m — around 19 orders of magnitude smaller than a proton — so its wave nature is utterly unobservable, while an electron's wavelength is comparable to atomic dimensions.
