de Broglie Wavelength Calculator

Every massive particle has wave properties with wavelength λ = h/p — Louis de Broglie’s hypothesis (1924). This tool computes the matter wavelength for electrons, protons, neutrons, or custom massive particles, in 4 modes: from mass & velocity, from kinetic energy, relativistic version, and inverse mode (energy/velocity from measured λ, e.g. electron diffraction results).

Verified examples

Frequently asked questions

What is the de Broglie wavelength?

Louis de Broglie’s hypothesis (1924): every massive particle also behaves as a wave with wavelength λ = h/p, where h is Planck’s constant (6.62607015×10⁻³⁴ J·s, exact) and p the particle momentum. Proven by the Davisson–Germer experiment (1927), which detected diffraction of 54 eV electrons by a nickel crystal (λ ≈ 1.669 Å) — the foundation of quantum-mechanical wave–particle duality.

What is the de Broglie wavelength of a 100 eV electron?

λ = h/√(2·m_e·E) = 1.2264×10⁻¹⁰ m ≈ 0.12264 nm (122.64 pm). This is comparable to X-ray wavelengths, so a 100 eV electron beam can be diffracted by a crystal lattice — the same principle used by electron microscopes.

When should I use the relativistic formula?

When a particle’s speed exceeds 0.1c (10% of the speed of light), the non-relativistic approximation λ = h/(mv) starts to deviate. For electrons this limit is reached at a kinetic energy of about 2.6 keV. Mode 3 of this calculator uses the full relativistic formula λ = hc/√(E_total² − (mc²)²), and Modes 1/2/4 automatically warn when v/c > 0.1.

How do I compute velocity from a measured wavelength?

Use Mode 4 (inverse): from the measured λ, momentum p = h/λ, then velocity v = p/m and kinetic energy E = p²/2m (non-relativistic) — the calculator also shows the relativistic version for honesty. Example: λ = 1.2264×10⁻¹⁰ m for an electron reproduces exactly E = 100 eV.

Do macroscopic objects also have a de Broglie wavelength?

Yes, the formula is universal: a 1 kg ball moving at 1 m/s has λ = h/(mv) = 6.62607015×10⁻³⁴ m — exactly Planck’s constant. Because this is far smaller than any atomic scale (≈10⁻¹⁰ m), the wave effect is impossible to observe; wave–particle duality only matters at microscopic scales.

What is the difference between the de Broglie wavelength and the Compton wavelength?

Both use h, but the physics differs: the de Broglie wavelength (λ = h/p) is the matter wave of a moving MASSIVE particle — an intrinsic property of the particle. The Compton wavelength instead arises from SCATTERING: the shift in a photon’s wavelength when it strikes an electron, Δλ = λ_c(1 − cos θ). To compute photon scattering, use the Compton scattering calculator.