De Broglie Wavelength Calculator
Every moving particle has a matter wavelength set by its momentum, a cornerstone of quantum mechanics. Pick a particle (or enter a custom mass), then solve from its velocity, kinetic energy or momentum. Switch units freely and turn on the relativistic correction for fast electron and accelerator beams. The result comes back in metres, nanometres, ångströms and picometres, with the momentum, energy and Lorentz factor alongside.
Formula
Worked example
An electron (m = 9.109×10⁻³¹ kg) at v = 1×10⁶ m/s, classically: p = m·v = 9.109×10⁻²⁵ kg·m/s, so λ = h ÷ p = 6.626×10⁻³⁴ ÷ 9.109×10⁻²⁵ = 7.27×10⁻¹⁰ m ≈ 0.727 nm ≈ 7.27 Å.
How the de Broglie wavelength is calculated
In 1924 Louis de Broglie proposed that every particle of matter behaves like a wave, with a wavelength fixed by its momentum: λ = h ÷ p, where h is the Planck constant (6.626×10⁻³⁴ joule-seconds). The only quantity that matters is the momentum p. This calculator lets you reach p three ways. From a velocity it uses p = m·v; from a kinetic energy it uses p = √(2·m·K); and if you already know the momentum you can enter it directly, which is handy for photons and other cases where rest mass is not the natural input. Pick a particle preset to load its rest mass, or choose Custom mass to type your own.
Velocity, kinetic energy and momentum modes
The three solve-from modes describe the same physics from different starting points. Velocity mode is the textbook form and accepts metres per second, kilometres per hour, or a fraction of the speed of light. Kinetic energy mode is the practical choice for beams: electron microscopes and accelerators are specified by their energy, often in electronvolts, kiloelectronvolts or megaelectronvolts, and the non-relativistic relation p = √(2·m·K) converts that energy to momentum. Momentum mode skips the mass entirely and applies λ = h ÷ p straight away. Whichever mode you use, the calculator also reports the wavelength in nanometres, ångströms and picometres so you can compare it directly with atomic spacings and X-ray wavelengths.
The relativistic correction
The simple formula p = m·v is only accurate while the particle moves well below the speed of light, roughly under 10% of c. Above that, special relativity makes the true momentum larger than m·v, so the real wavelength is shorter than the classical estimate. Turn on the relativistic correction and the calculator uses p = γ·m·v in velocity mode, where the Lorentz factor γ = 1 ÷ √(1 - (v/c)²), and p = √(E_total² - (mc²)²) ÷ c in energy mode. This matters in practice: a 100 keV electron in a transmission electron microscope is already moving at more than half the speed of light, so its relativistic wavelength (about 3.7 pm) is noticeably shorter than the classical figure. The reported Lorentz factor and speed-as-a-fraction-of-c tell you at a glance whether the correction is needed.
Where matter waves matter
The de Broglie relation is the experimental backbone of quantum mechanics. The Davisson-Germer experiment confirmed it by diffracting electrons off a nickel crystal, and the electron microscope exploits the very short wavelength of fast electrons to resolve detail far finer than visible light ever could. Neutron diffraction, used to map atomic positions in materials, relies on the same principle, and matter-wave interference has even been demonstrated with large molecules such as C60 fullerene. Because Planck’s constant is so tiny, the wavelength of any everyday object is fantastically small: a 0.145 kg baseball at 40 m/s has a wavelength near 10⁻³⁴ m, which is why its wave nature is utterly undetectable.
De Broglie wavelengths of common particles
| Particle | Mass (kg) | Speed or energy | Wavelength | Observable? |
|---|---|---|---|---|
| Electron | 9.1×10⁻³¹ | 1×10⁶ m/s | ~0.73 nm | High |
| Electron (100 keV, rel.) | 9.1×10⁻³¹ | 100 keV | ~3.7 pm | High |
| Proton | 1.67×10⁻²⁷ | 1×10⁴ m/s | ~4.0×10⁻¹¹ m | High |
| Thermal neutron | 1.67×10⁻²⁷ | 2.2×10³ m/s | ~1.8×10⁻¹⁰ m | High |
| C60 fullerene | 1.2×10⁻²⁴ | 200 m/s | ~2.8 pm | Med |
| Baseball | 0.145 | 40 m/s | ~1.1×10⁻³⁴ m | Low |
Approximate wavelengths at the indicated speeds or energies (classical unless noted).
Frequently asked questions
Should I solve from velocity, kinetic energy or momentum?
Use whichever quantity you actually know. Velocity mode applies λ = h ÷ (m·v) and is the textbook form. Kinetic energy mode is best for beams specified in electronvolts (electron microscopes, accelerators) and uses p = √(2·m·K). Momentum mode applies λ = h ÷ p directly and needs no mass, which is convenient for photons or when momentum is given. All three describe the same wavelength.
When do I need the relativistic correction?
Turn it on whenever the particle moves above roughly 10% of the speed of light, or whenever the kinetic energy is a sizeable fraction of the rest energy mc². The correction uses the Lorentz factor γ so the momentum is γ·m·v rather than m·v, which makes the wavelength shorter and more accurate. A 100 keV electron, for example, is already past half the speed of light, so the relativistic result (about 3.7 pm) differs noticeably from the classical one.
Why is the wavelength so incredibly small?
The de Broglie wavelength is Planck’s constant divided by momentum, and Planck’s constant is only 6.626×10⁻³⁴ J·s. Unless the momentum is also extremely small, which requires a very light particle moving slowly, the result is minuscule. That is why matter waves are noticeable only for electrons, neutrons and similar subatomic particles, never for everyday objects.
What units does the calculator report?
The wavelength is shown in metres (scientific notation) and also converted to nanometres, ångströms (1 Å = 0.1 nm) and picometres, the units physicists use at atomic and X-ray scales. Alongside it you get the momentum in kg·m/s, the kinetic energy in electronvolts, the speed as a fraction of the speed of light, and the Lorentz factor.