De Broglie Wavelength Calculator

How it works

• Matter waves: De Broglie’s relation says particles have wavelength λ = h / p with momentum p = m·v.
• Planck’s constant: We use h = 6.62607015×10⁻³⁴ J·s. For non-relativistic speeds, p = m·v is enough.
• Units: Outputs show meters, nanometers (1e-9 m), and picometers (1e-12 m) for easier reading across scales.
• Kinetic energy (classical): We also show KE = ½ m v² as a helpful check (approximate for low speeds).

Assumes non-relativistic speeds (v ≪ c). For relativistic particles, momentum is higher and λ is slightly shorter than shown.

Why de Broglie wavelengths matter (and how to read the numbers)

The de Broglie hypothesis connects classical momentum to wave behavior through λ = h / p, with p = m·v for slow particles. That simple division answers two questions at once: will a particle diffract through a grating, and how “wave-like” will it behave? If λ is on the order of the obstacle spacing—crystal planes, slits, nanopores—interference shows up. If λ is vastly smaller, the wave nature is effectively invisible and the particle behaves classically. An electron with kinetic energy of a few hundred eV has a wavelength near inter-atomic spacings, which is why electron diffraction and electron microscopes work. A baseball’s wavelength, even at fast speeds, is so tiny that no realistic grating could reveal interference.

Wavelengths are often given in meters, but that gets unwieldy fast. Nanometers (nm) match atomic spacing (~0.1–0.3 nm), while picometers (pm) suit high-energy electrons and X-ray regimes. In this tool you’ll see all three, plus momentum in SI units and a classical kinetic-energy estimate KE = ½mv² as a sanity check. If your output shows “—”, the input is missing or zero; if you see astronomically small numbers (like 10⁻³⁴ m), that tells you you’re firmly in the classical limit where diffraction experiments are impractical.

The comparison box uses the speed you entered and plugs it into two fixed masses: an electron (9.109×10⁻³¹ kg) and a 0.145 kg baseball. That instantly shows the mass effect alone: at the same speed, λ scales inversely with mass. You can also hit the presets for electron, proton, and baseball to populate realistic masses. Keep in mind that this calculator assumes non-relativistic speeds. Once speeds approach a meaningful fraction of c, you should replace p = mv with p = γmv, where γ = 1/√(1 - v²/c²). That will shave λ even shorter. For most homework problems below ~10% of light speed, the classical version is close enough and far simpler to work with.

A quick way to estimate experiment feasibility: compare λ to feature size. If λ ≈ lattice spacing, you can expect diffraction. If λ is 1000× smaller, treat the particle as point-like. Likewise, slowing particles increases λ; that’s why cold-neutron and cold-atom experiments use slowed beams or trapped atoms to make interference visible. Play with the velocity input—drop it by factors of ten—to see how quickly λ expands for light particles and how stubbornly tiny it stays for heavy ones.