Matter behaves like a wave
In 1924 Louis de Broglie suggested that the wave–particle duality of light also applies to matter. Every moving particle has a wavelength:
λ = h ÷ p
where h = 6.626 × 10⁻³⁴ J·s is Planck’s constant and p is the particle’s momentum. Three years later, Davisson and Germer confirmed it by bouncing electrons off a nickel crystal and seeing a diffraction pattern, exactly as X-rays would produce.
How the calculator works
You choose a particle (electron, proton, neutron or alpha particle) and its kinetic energy in electronvolts. The calculator converts the energy to joules and finds the momentum with the relativistic formula:
pc = √(E² + 2E × mc²)
then divides Planck’s constant by p. It also gives the particle’s speed, v = pc² ÷ (E + mc²), and that speed as a share of the speed of light. Using the relativistic formula means the answer stays correct for fast electrons, where the classical p = √(2mE) starts to drift.
Worked example
An electron with 100 eV of kinetic energy:
| Result | Value |
|---|---|
| Momentum | 5.403 × 10⁻²⁴ kg·m/s |
| Wavelength | 0.1226 nm (122.6 pm) |
| Speed | about 5.93 million m/s |
| Share of c | 1.98% (non-relativistic) |
That wavelength is the same size as the spacing between atoms, which is why low-energy electron diffraction (LEED) is used to study crystal surfaces.
Some reference values
| Particle and energy | Wavelength |
|---|---|
| Electron, 1 eV | 1.23 nm |
| Electron, 10 keV | 0.0122 nm (12.2 pm) |
| Electron, 100 keV (electron microscope) | 3.70 pm |
| Thermal neutron, 0.025 eV | 0.181 nm |
| Proton, 1 MeV | 28.6 fm |
At 100 keV the electron already travels at 55% of the speed of light. The classical formula would give 3.88 pm, about 5% too long, which matters for electron microscope resolution.
Why it matters
The short wavelength of fast electrons is what lets electron microscopes see detail far smaller than light microscopes can, since resolution is limited to roughly half the wavelength. Thermal neutrons, with wavelengths near 0.18 nm, are used to map the positions of atoms in materials, and the same principle underlies quantum effects in electronics and chemistry.
For everyday objects the wavelength is absurdly small. A 145 g baseball at 40 m/s has λ ≈ 1.1 × 10⁻³⁴ m, so its wave nature can never be observed.