De Broglie wavelength calculator
Pick a particle, enter its speed, energy or voltage, get λ = h/p.
What is the de Broglie wavelength?
Every moving particle behaves as a wave with wavelength λ = h/p, where p is its momentum and h = 6.62607015 × 10⁻³⁴ J·s is Planck’s constant (exact since the 2019 SI). For slow objects p = mv, so λ = h/(mv). Heavier or faster means a shorter wavelength.
Examples from OpenStax
University Physics Volume 3, Example 6.11: a 0.65 kg basketball at 10 m/s has λ = 1.02 × 10⁻³⁴ m — far too small to notice. An electron with 1.0 eV of kinetic energy has λ = 1.23 nm, and one with 108 keV has λ = 3.55 pm. Pick the particle and enter the same numbers to reproduce them.
How do I find the wavelength of an electron from the accelerating voltage?
An electron accelerated through a voltage V gains kinetic energy K = eV, so 54 V gives 54 eV. At low energy p = √(2meV), and λ = h/√(2meV). OpenStax uses this for the Davisson–Germer experiment: 54 V gives λ ≈ 1.67 Å, close to the 1.64 Å they measured from electron diffraction on nickel. For other charged particles K = |q|V — twice as much for an alpha particle.
When is the relativistic formula needed?
When the kinetic energy is not small next to the rest energy mc² — 511 keV for an electron, 938 MeV for a proton. Then p = √(K(K + 2mc²))/c (OpenStax Equation 6.58). This calculator always uses the relativistic momentum and shows the simple non-relativistic answer beside it: for a 108 keV electron the simple formula gives 3.73 pm instead of 3.55 pm. A proton at 0.75c has λ ≈ 1.17 fm (OpenStax Example 6.14 gets 1.16 fm with rounded constants).
What particle masses does it use?
CODATA 2022 values from NIST: electron 9.1093837139 × 10⁻³¹ kg, proton 1.67262192595 × 10⁻²⁷ kg, neutron 1.67492750056 × 10⁻²⁷ kg, alpha particle 6.6446573450 × 10⁻²⁷ kg and muon 1.883531627 × 10⁻²⁸ kg. For anything else choose “Other object” and enter its mass.
How is the wavelength of a thermal neutron found?
A neutron moving at 2,200 m/s, a typical speed for thermal neutrons, has p = mv and a wavelength of λ = h ÷ (mv) = 1.798 × 10⁻¹⁰ m, about 1.80 Å. That is comparable to the spacing between atoms in a crystal, which is why neutrons and electrons can be diffracted by crystals.
Why do we not see the wave nature of everyday objects?
Because h is so small. Even a speck of mass 10⁻¹⁵ kg moving at 1 mm/s has λ = 6.63 × 10⁻¹⁶ m, far below any size we could detect. The wave behavior shows up only when the wavelength is comparable to the size of the openings or spacing the particle meets.
From wavelength to momentum
De Broglie's relation λ = h ÷ p connects a particle's wavelength to its momentum, in the same way that a photon's momentum is h ÷ λ. If you know the speed, p = γmv, which reduces to mv at low speeds; if you know the kinetic energy K, p = √(K(K + 2mc²)) ÷ c, which reduces to √(2mK) at low energy. Both forms are valid at all speeds below c. The ratio of the wavelength to the particle's size or to the spacing of the structure decides whether diffraction effects appear.
Worked example: electrons at 100 eV and 10 keV
For an electron with 100 eV of kinetic energy the calculator gives λ = 0.1226 nm, or 1.226 Å, and the non-relativistic estimate is 0.0049% too long. At 10 keV the wavelength is 12.20 pm. Raising the energy by a factor of 100 shortens the wavelength by a factor of about 10, because λ is proportional to 1 ÷ √K at low energies; at 10 keV the relativistic correction (the simple formula is 0.49% too long) is already visible. Wavelengths of a few picometers, against hundreds of nanometers for visible light, are one reason electron microscopes can show far finer detail than optical ones.
Common mistakes
Entering kinetic energy in electron-volts without converting to joules (1 eV = 1.602176634 × 10⁻¹⁹ J) is the usual one, and the calculator handles the units if you choose them. The second is using the simple formulas for fast particles: above a few percent of the speed of light the non-relativistic wavelength is noticeably too long. Third, the wavelength of a macroscopic object is not physically measurable, so a result like 10⁻³⁴ m is a statement that wave effects are negligible, not a size to be observed.
Sources: OpenStax University Physics, Volume 3, chapter 6 (matter waves, de Broglie relation, Eq. 6.58 for the relativistic momentum); constants from NIST CODATA 2022 (h and the electron-volt are exact). Related: photoelectric effect calculator, momentum calculator, kinetic energy calculator and Rydberg equation calculator.