What is the de Broglie wavelength formula?

The de Broglie wavelength is given by lambda = h / p, where h is Planck's constant (6.62607015 x 10^-34 J s) and p is the momentum of the particle (p = m x v for non-relativistic speeds). A smaller wavelength means a more wave-like particle is harder to observe; larger, faster objects have wavelengths far too small to detect, which is why everyday objects do not show observable wave behaviour.

Why do electrons have a measurable de Broglie wavelength but a thrown ball does not?

Wavelength is inversely proportional to momentum (mass times velocity). An electron has an extremely small mass (about 9.109 x 10^-31 kg), so even at everyday laboratory speeds its momentum is tiny and its wavelength (typically sub-nanometre) is large enough to cause observable diffraction, as seen in electron microscopy and electron diffraction experiments. A thrown ball has a mass around a million trillion trillion times larger, so its momentum is enormous and its de Broglie wavelength works out to roughly 10^-34 metres, far too small to ever detect.

Does the de Broglie wavelength formula work for photons?

The relationship lambda = h / p applies to photons too, but photon momentum is calculated differently since photons have zero rest mass. For photons, momentum is p = E / c (energy divided by the speed of light), which combined with E = hc / lambda gives the same lambda = h / p result. This calculator uses p = m x v, which is the standard non-relativistic form for particles with mass such as electrons, protons and neutrons. For a light-specific calculation, use a dedicated photon energy calculator.

De Broglie Wavelength Calculator

This calculator works out the de Broglie wavelength of a moving particle, the wave-like property every piece of matter carries alongside its familiar particle behaviour. You choose a particle type, electron, proton, neutron or alpha particle, or enter a custom mass, then enter a velocity in metres per second, with preset speeds available for common cases such as a thermal electron, a Bohr-model electron or a thermal neutron. Using the relation lambda equals h divided by mass times velocity, the calculator returns the wavelength in nanometres and picometres, the particle's momentum in kilogram metres per second, and its kinetic energy in electron volts. A breakdown panel shows the mass, velocity and momentum used, plus a check on what fraction of the speed of light your velocity represents, confirming whether the non-relativistic approximation still applies. A comparison table also shows how the wavelength changes across a range of velocities for the same particle, so you can see wavelength shrink as speed rises. This suits physics students working through wave-particle duality problems, and anyone curious why electrons show measurable diffraction, the basis of electron microscopy, while thrown balls never do. The formula used is the standard non-relativistic one, accurate for speeds well below light speed; above roughly ten percent of light speed, a relativistic calculation gives a shorter, more accurate wavelength than shown here.

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Reviewed July 2026 Standard de Broglie relation, CODATA 2018 physical constants.

1. Particle

Particle type

2. Velocity

Velocity (m/s, must be well below 3 x 10⁸ m/s)

Common presets

De Broglie Wavelength Result

Wavelength
-
Nanometres (nm)

Wavelength

Picometres (pm)

Momentum

kg m/s

Kinetic Energy

Electron volts (eV)

Calculation Breakdown

Particle-

Mass (m)-

Velocity (v)-

Momentum (p = m x v)-

Planck's constant (h)6.62607015 x 10⁻³⁴ J s

Wavelength (lambda = h / p)-

Kinetic Energy and Speed Check

Kinetic energy (KE = 1/2 m v²)-

Kinetic energy-

Speed of light (c)299,792,458 m/s

Velocity as fraction of c-

Non-relativistic approximation valid?-

Wavelength at Different Velocities (Same Particle)

Velocity (m/s)	Momentum (kg m/s)	Wavelength (nm)	Wavelength (pm)

Summary: Enter a particle and velocity above.

What Is the De Broglie Wavelength?

In 1924, French physicist Louis de Broglie proposed that all matter has wave-like properties, not just light. He suggested that any moving particle has an associated wavelength, now called the de Broglie wavelength, given by lambda = h / p, where h is Planck's constant and p is the particle's momentum. This idea extended wave-particle duality (already established for light) to matter, and was confirmed experimentally in 1927 when electron diffraction patterns were observed, exactly as de Broglie's equation predicted.

The De Broglie Formula

The wavelength is calculated as:

lambda = h / p = h / (m x v)

lambda is the de Broglie wavelength, in metres
h is Planck's constant, 6.62607015 x 10⁻³⁴ joule seconds
p is the particle's momentum, in kilogram metres per second (kg m/s)
m is the particle's rest mass, in kilograms
v is the particle's velocity, in metres per second

This calculator uses the non-relativistic form p = m x v, which is accurate for velocities well below the speed of light (roughly under 10% of c, or about 3 x 10⁷ m/s). At higher speeds, momentum must be calculated relativistically, and this simple form under-estimates momentum, giving a wavelength that is too long.

Typical Particle Masses

Particle	Rest Mass (kg)	Notes
Electron	9.1093837015 x 10⁻³¹	Lightest common charged particle; shows large, easily measured wavelengths
Proton	1.67262192369 x 10⁻²⁷	About 1,836 times heavier than an electron
Neutron	1.67492749804 x 10⁻²⁷	Slightly heavier than a proton; used in neutron diffraction studies
Alpha particle (helium-4 nucleus)	6.6446573357 x 10⁻²⁷	Two protons and two neutrons bound together

Why This Matters: Electron Microscopy and Diffraction

Because electrons have a much smaller mass than atoms or everyday objects, they can be accelerated to speeds where their de Broglie wavelength is comparable to atomic spacings (roughly 0.01 to 1 nanometre). This is the basis of electron microscopy, which achieves far higher resolution than light microscopy because the electron wavelength is thousands of times shorter than visible light wavelengths (400 to 700 nm). Electron diffraction, first demonstrated by Davisson and Germer in 1927, provided the first direct experimental proof of de Broglie's hypothesis.

Worked Example

An electron (mass 9.10938 x 10⁻³¹ kg) travelling at 1,000,000 m/s has momentum p = m x v = 9.10938 x 10⁻³¹ x 1,000,000 = 9.10938 x 10⁻²⁵ kg m/s. Its de Broglie wavelength is lambda = h / p = 6.62607015 x 10⁻³⁴ / 9.10938 x 10⁻²⁵ ≈ 7.274 x 10⁻¹⁰ m, or about 0.727 nm (727.4 pm). Its kinetic energy is 1/2 x m x v² ≈ 4.555 x 10⁻¹⁹ J, or about 2.84 eV.

Limitations of This Calculator

This calculator uses the classical (non-relativistic) momentum formula p = m x v. For particles travelling at a significant fraction of the speed of light (typically above about 10% of c), relativistic momentum must be used instead, and the wavelength calculated here will be slightly longer than the true relativistic value. For most textbook problems involving electrons, protons and neutrons at laboratory speeds, the non-relativistic approximation is accurate to within a fraction of a percent.

Related Calculators

Science and Engineering Calculators: the full hub of physics, chemistry and engineering tools.
Wavelength Calculator: convert between frequency and wavelength for waves.
Photon Energy Calculator: find the energy of a photon from its wavelength or frequency.
Momentum Calculator: calculate momentum from mass and velocity.
Kinetic Energy Calculator: calculate kinetic energy from mass and velocity.

Sources: CODATA 2018 recommended values for physical constants (physics.nist.gov/cuu/Constants). De Broglie, L. (1924), original hypothesis on wave-particle duality. Davisson, C. and Germer, L. (1927), experimental confirmation of electron diffraction, Physical Review.

This calculator uses the standard non-relativistic de Broglie relation (lambda = h / (m x v)), suitable for typical physics coursework involving electrons, protons, neutrons and similar particles at speeds well below the speed of light. For particles approaching relativistic speeds, a relativistic momentum calculation is required for an accurate result.