De Broglie Wavelength Calculator — matter waves from λ = h/p
In 1924 Louis de Broglie proposed that the wave–particle duality already accepted for light applies to matter as well: every moving particle carries a wave whose length is the Planck constant divided by the particle’s momentum. This de Broglie wavelength calculator evaluates λ = h / p from whichever quantity you actually have — a mass and velocity, a kinetic energy, an accelerating voltage, a momentum, a temperature, or a target wavelength solved backwards.
Why momentum arrives in six different disguises
The relation itself is trivial. What trips students up is that momentum is rarely handed over directly. A diffraction lab quotes an accelerating voltage; a beamline quotes electronvolts; a neutron source quotes a temperature; a textbook problem quotes a speed. Each needs a different algebraic path to the same answer:
- Mass and velocity —
p = m·v, orp = γ·m·vonce the particle passes a tenth of light speed. - Kinetic energy —
p = √(2·m·K)classically, orp = √(K² + 2·K·m·c²) / crelativistically. - Accelerating voltage —
K = q·V, then the energy route. Neutral particles cannot take this path at all. - Temperature —
p = √(3·m·k_B·T)on the root-mean-square convention, or√(2·m·k_B·T)for the most-probable speed. - Reverse solve —
p = h / λ, then back out the velocity, energy or voltage that would produce it.
A worked example, to four significant figures
Take the standard textbook electron travelling at 2.2 × 10⁶ m/s, roughly the speed of an electron in the first Bohr orbit of hydrogen. With the CODATA rest mass 9.1093837015 × 10⁻³¹ kg, the momentum is p = mv = 2.0041 × 10⁻²⁴ kg·m/s, and dividing the exact Planck constant by it gives λ = 3.3063 × 10⁻¹⁰ m — that is 0.33065 nm, 330.63 pm, or 3.3063 Å. That wavelength sits within a factor of three of an atomic diameter, which is precisely why electrons bound in atoms have to be described as waves rather than as orbiting balls.
h and the electron mass to three or four digits before dividing, and land on 0.3305 nm. This calculator divides the exact SI value of h by an unrounded momentum, so its fourth significant figure is trustworthy. If you are checking homework against a printed answer, expect that last digit to differ.Where relativity starts to bite
The classical p = mv is a low-speed approximation. At a tenth of light speed it is already about half a percent short; by 100 kV — an ordinary transmission electron microscope — the electron is moving at 55 % of c and the classical answer of 3.8783 pm overstates the true 3.7014 pm by 4.8 %. The tool switches automatically at 0.1 c and always reports both branches, so you can see the size of the correction rather than take it on trust. The relativistic energy path is deliberately written as √(K² + 2Kmc²) rather than as a difference of large total energies, which would lose precision through catastrophic cancellation at low kinetic energy.
From wavelength to a measurable angle
A wavelength only becomes evidence when something diffracts it. Crystal planes separated by a distance d reflect constructively when 2d·sin θ = nλ, so the Bragg diffraction helper converts the computed wavelength straight into the angle a detector would see. Our 3.3063 Å electron off planes spaced 2.15 Å gives θ = asin(3.3063 / 4.30) = 50.26°. When nλ exceeds 2d no such reflection exists, and the tool says so rather than returning a silent NaN.
The neutron and the baseball
Two results anchor the whole topic. A thermal neutron at 300 K lands at 1.4524 Å on the rms convention and 1.7789 Å on the most-probable one — either way, the spacing of crystal planes, which is exactly why reactors moderate neutrons before sending them at a sample. A baseball of 0.145 kg at 40 m/s, by contrast, comes out at 1.14 × 10⁻³⁴ m: twenty orders of magnitude below a proton. The wave is genuinely there; nothing in the universe is fine enough to reveal it. Mass, not any special quantum-ness, is what separates the two cases.
λ = h/p holds for photons too, but the routes through m do not — a photon has no rest mass, so p = mv and p = √(2mK) both collapse. For light, use E = hc/λ in a photon energy calculator instead.Reading the results
Every panel is derived from one unrounded momentum in SI units, so the wavelength, the wave number k = 2π/λ, the matter-wave frequency, the velocity, the Lorentz factor and the Bragg angle can never disagree with one another. The logarithmic scale bar shows where the answer falls against a proton, a nucleus, an atom, DNA, a virus and visible light, and the observability note turns that position into plain English. Results export as TXT or CSV at whatever precision you set, and the share link encodes every input in the URL.