This de Broglie wavelength calculator finds the matter wavelength λ = h/(mv) of any moving particle — from its mass and speed, or, for electrons, directly from accelerating voltage/energy.
Every Moving Mass Is a Wave — Mostly a Hopelessly Small One
The light-side identity is our Photon Energy Calculator (E = hf pairs with λ = h/p). Part of our Physics Calculators collection.
Worked Example (Verified)
- Electron accelerated through 100 V (eV mode, default): v = √(2E/m) = 5.93 × 10⁶ m/s → λ = h/(mv) = 1.23 × 10⁻¹⁰ m = 0.123 nm — X-ray territory, from a battery-scale voltage
- The bench formula behind it: λ(nm) ≈ 1.227/√V for electrons through V volts — at 100 V, 0.1227 nm, matching the full calculation
- Davisson & Germer (1927) measured exactly this: 54 eV electrons diffracted off nickel at λ ≈ 0.167 nm — de Broglie’s thesis idea (1924) turned into a Nobel-winning diffraction pattern
Matter-Wave Discipline
- λ grows as momentum shrinks: slow, light particles are the waviest things in physics (cold neutrons at λ ~ 0.1–1 nm are crystallography tools).
- Relativistic correction: at high energies use p = γmv (or λ = h/√(2mE(1 + E/2mc²))); TEM electrons at 300 keV are ~15% off without it.
- Thermal de Broglie wavelength governs when gases turn quantum (Bose-Einstein condensation when λ rivals particle spacing).
- Nobody diffracts baseballs: λ ~ 10⁻³⁴ m vs any aperture — the wave nature is real and utterly unobservable at macro scale.
Related Tools
- Photon Energy Calculator — E = hf for light
- Photoelectric Effect Calculator — wave-particle evidence #1
- Physics Calculators — the full collection
Frequently Asked Questions
How do you calculate the de Broglie wavelength?
λ = h/p = h/(mv) — Planck’s constant over momentum. For an electron at 5.93 × 10⁶ m/s: λ = 6.626 × 10⁻³⁴/(9.11 × 10⁻³¹ × 5.93 × 10⁶) = 1.23 × 10⁻¹⁰ m. Or, for electrons, directly from energy: λ = h/√(2mE). This calculator does both.
What is the wavelength of a 100 eV electron?
0.123 nm (1.23 Å) — comparable to atomic spacings, which is why 100 eV electrons diffract from crystal surfaces (LEED). The quick formula λ(nm) ≈ 1.227/√(E in eV) gives the same result.
Why don’t everyday objects show wave behavior?
Their wavelengths are absurdly small: a walking human’s λ ≈ 10⁻³⁶ m, dozens of orders below any slit or grating that could reveal diffraction. Quantum wave behavior scales as 1/momentum — macroscopic momentum buries it completely.
What is the thermal de Broglie wavelength?
λ = h/√(2πmkT) for a gas particle at temperature T — the average matter wavelength from thermal motion. When it grows to the inter-particle spacing (very cold, very dense), quantum statistics take over: Bose-Einstein condensates and Fermi degeneracy live past that line.
When do I need the relativistic formula?
When kinetic energy is a non-trivial fraction of rest energy (electron mc² = 511 keV): above ~10 keV the non-relativistic λ errs by > 1%. Use p = √(2mE(1 + E/(2mc²))). At everyday and LEED energies (≤ 1 keV), this calculator’s classical form is fine.
Is this de Broglie wavelength calculator free?
Yes — free, instant, no sign-up.