About the De Broglie Wavelength Calculator
This de Broglie wavelength calculator finds the quantum wavelength λ = h/p associated with a moving particle. Pick an electron, proton, neutron, alpha particle or enter any mass, then give its speed, its kinetic energy in electronvolts, or the voltage it has been accelerated through. You get the wavelength in the most readable unit, its value in picometres and nanometres, the momentum, the speed (and as a fraction of light speed) and the kinetic energy.
It is designed for students of modern physics and chemistry, and for anyone working with electron microscopes, electron or neutron diffraction, or crystallography who needs to compare a particle’s wavelength with atomic spacings (around 100–300 pm). Try a baseball to see why wave behaviour never shows up for everyday objects.
Momentum is computed relativistically, so results stay correct for fast electrons in microscopes, where the simple non-relativistic formula is off by several per cent; the non-relativistic value is shown for comparison. Masses are CODATA 2022 values.
How to use the de broglie wavelength calculator
- 1Choose the particle, or enter a custom mass in kilograms.
- 2Choose whether you know its kinetic energy, accelerating voltage or speed.
- 3Enter that value.
- 4Read the de Broglie wavelength and its value in pm and nm.
- 5Compare with the non-relativistic value to see whether relativity matters.
Formula and method
Louis de Broglie proposed that every moving particle has a wavelength equal to Planck’s constant divided by its momentum, λ = h/p. The calculator finds the momentum from whatever you know: from speed it uses the relativistic p = γmv with γ = 1/√(1 − v²/c²); from kinetic energy it uses the energy–momentum relation p = √(K² + 2Kmc²)/c; and for a voltage it first sets K = qV, the energy gained by a charge q falling through potential difference V.
At low energies these reduce to the familiar λ = h/(mv) = h/√(2mK). For a 100 eV electron that gives about 0.123 nm, the right scale to probe atomic structure, which is why electron diffraction (Davisson–Germer) confirmed the theory. Planck’s constant h = 6.62607015 × 10⁻³⁴ J·s is exact in the SI.
- λ
- De Broglie wavelength (m)
- h
- Planck constant, 6.62607015 × 10⁻³⁴ J·s
- p
- Momentum (kg·m/s)
- m, v
- Rest mass (kg) and speed (m/s)
- K
- Kinetic energy (J; 1 eV = 1.602176634 × 10⁻¹⁹ J)
- q, V
- Charge (C) and accelerating voltage (V)
Worked examples
100 eV electron
An electron with 100 eV of kinetic energy has momentum 5.40 × 10⁻²⁴ kg·m/s, so λ = 6.626 × 10⁻³⁴ ÷ 5.40 × 10⁻²⁴ = 1.226 × 10⁻¹⁰ m ≈ 0.123 nm — about the size of an atom. It moves at about 2% of light speed, so relativity barely matters.
Electron in a 50 kV electron microscope
Accelerated through 50 kV, the electron gains 50 keV and reaches 41% of light speed. The relativistic wavelength is 5.36 pm; the non-relativistic formula would give 5.48 pm, about 2.4% too long — a big error in microscopy.
Thermal neutron at 2,200 m/s
p = 1.675 × 10⁻²⁷ × 2200 = 3.685 × 10⁻²⁴ kg·m/s, so λ = 0.180 nm. Neutrons at this standard “thermal” speed (about 0.0253 eV) are ideal for diffraction studies of crystals.
Baseball at 40 m/s
p = 0.145 × 40 = 5.8 kg·m/s, so λ = 6.626 × 10⁻³⁴ ÷ 5.8 = 1.14 × 10⁻³⁴ m — about 10¹⁹ times smaller than a proton, which is why baseballs never diffract.
Frequently asked questions
What is the de Broglie wavelength?+
It is the wavelength associated with a moving particle, λ = h/p, where h is Planck’s constant and p the momentum. It expresses wave–particle duality: electrons, neutrons and even atoms can interfere and diffract like light.
How do you find the wavelength of an electron accelerated through a voltage?+
The electron gains kinetic energy K = eV. For low voltages λ ≈ h/√(2meV), which simplifies to λ ≈ 1.226/√V nm. Above a few kilovolts use the relativistic form p = √(K² + 2Kmc²)/c, as this calculator does.
Why don’t we notice the wave nature of everyday objects?+
Their momentum is enormous compared with Planck’s constant, so their wavelengths are around 10⁻³⁴ m — far smaller than any atom or nucleus. Diffraction needs openings comparable to the wavelength, which cannot exist at that scale.
When should I use the relativistic formula?+
When the kinetic energy is not small compared with the rest energy mc². For electrons (rest energy 511 keV), the simple formula overestimates λ by about 0.25% at 5 keV, 2.4% at 50 keV and 9.3% at 200 keV.
What wavelength do thermal neutrons have?+
Neutrons at room-temperature thermal speeds (about 2,200 m/s, 0.025 eV) have a de Broglie wavelength near 0.18 nm, similar to atomic spacings, which makes them excellent probes for neutron diffraction.