What the de Broglie wavelength calculator does
This page turns mass and speed into a wavelength. The formula is lambda = h / (m v). Here lambda is the de Broglie wavelength in meters, h is the Planck constant (6.62607015 x 10^-34 J s), m is the mass in kilograms, and v is the speed in meters per second. The result is a matter wave, the same kind of wave that makes electron diffraction work.
You can pick a preset for an electron, a proton, or a neutron. Those masses come from CODATA 2018. You can also type any mass in kilograms. The calculator uses the non-relativistic formula. Above about 0.1 c, where c is the speed of light (299,792,458 m/s), the momentum is larger than m x v, so the real wavelength is shorter than this tool reports. The calculator shows a warning at that point. It does not compute the relativistic value.
Momentum and why mass matters
Momentum is p = m v. A heavy object moving at the same speed as a light one carries more momentum. Since lambda = h / p, more momentum means a shorter wavelength. That single idea explains almost every result on this page.
Planck's constant is tiny. A 0.145 kg baseball thrown at 40 m/s has a wavelength near 1.1 x 10^-34 m. That is far smaller than an atom, so no experiment can see it. Everyday objects have matter waves, but they are unobservable in practice.
Electrons are different. An electron has a mass of 9.1093837015 x 10^-31 kg. At 1,000,000 m/s its wavelength is about 0.727 nm. That is close to the spacing between atoms in a crystal. When such electrons pass through a thin crystal, the atomic planes act like a diffraction grating. This is why electron diffraction works and why electron microscopes can resolve details far smaller than light microscopes.
Worked example: electron at 1,000,000 m/s
Start with the formula: lambda = h / (m v).
- h = 6.626 x 10^-34 J s
- m = 9.109 x 10^-31 kg
- v = 1,000,000 m/s = 1 x 10^6 m/s
- m v = 9.109 x 10^-31 x 1 x 10^6 = 9.109 x 10^-25 kg m/s
- lambda = 6.626 x 10^-34 / 9.109 x 10^-25 = 7.27 x 10^-10 m
That is 0.727 nm. For comparison, a green laser pointer at 532 nm has a photon wavelength about 730 times longer. The electron wavelength sits in the X-ray range, which is why electron beams can probe atomic structure.
Presets and when to use them
The preset masses are:
| Particle | Mass (kg) |
|---|---|
| Electron | 9.1093837015 x 10^-31 |
| Proton | 1.67262192369 x 10^-27 |
| Neutron | 1.67492749804 x 10^-27 |
Use the electron preset for electron diffraction, electron microscopy, and low-energy electron experiments. Use the proton or neutron preset for beam physics and neutron scattering. For anything else, enter the mass in kilograms. If you only know the mass in atomic mass units, multiply by 1.66053906660 x 10^-27 kg per u before you type it in.
Speed must be in meters per second. If you have kinetic energy instead, you can find v from E = 1/2 m v^2 for slow particles. Rearranged, v = sqrt(2E/m). Keep the energy in joules. If your energy is in electronvolts, multiply by 1.602176634 x 10^-19 J per eV first.
Non-relativistic limit and the warning
The formula lambda = h / (m v) uses the classical momentum. At low speeds this is accurate to many decimal places. As speed rises, the relativistic momentum p = gamma m v grows faster than m v, where gamma is the Lorentz factor. The true wavelength becomes shorter than the non-relativistic result.
Above about 0.1 c, the difference passes one percent and keeps growing. For an electron, 0.1 c is about 3.0 x 10^7 m/s. The calculator warns you when your input crosses that line. It does not compute the relativistic value, so treat any result above 0.1 c as an upper bound. For those cases, use p = gamma m v in the same lambda = h / p formula.
Below 0.1 c, no correction is needed. The electron example at 1,000,000 m/s is only 0.0033 c, so the classical formula is fine there.
Where matter waves show up
Electron diffraction is the classic case. A beam of electrons with wavelengths near atomic spacing scatters off a crystal and forms a pattern of bright and dark spots. The pattern matches the crystal structure. Neutron diffraction uses the same idea with neutrons, which penetrate deeper into many materials.
Matter waves also matter in quantum mechanics courses. The Bohr model, the particle in a box, and the uncertainty principle all use lambda = h / p. If you are studying any of those topics, this calculator gives you a quick way to check your numbers.
For photons, the related formula is lambda = c / f, and photon energy is E = h f. Those live on the photon energy calculator. For radio waves and antennas, see the antenna length calculator. Every formula used here is listed at how we calculate.