De Broglie Wavelength Calculator

Physics

The particle

Every particle carries a wave whose length is Planck's constant divided by its momentum. Nothing about the relation is special to electrons — a thrown baseball has one too, about 10⁻³⁴ m, which is why nobody has ever watched a baseball diffract.

Worked example preset

The speed of an electron in the first Bohr orbit of hydrogen, near 0.73 % of light speed. Its wavelength lands within a factor of three of an atomic diameter — which is exactly why electrons in atoms have to be described as waves.

Particle

The lightest particle in ordinary matter, and the reason electron microscopes resolve atoms: at 100 kV its wavelength is about 3.7 pm, some 150 000 times shorter than green light.

Relativistic treatment

Auto applies p = γmv above a tenth of light speed and reports the size of the correction either way.

Rest mass

Fixed by the Electron preset — switch to Custom to edit it.

Mass unit

Kilograms, grams, atomic mass units or MeV/c².

Charge (multiples of e)

The Electron carries -1 e.

What do you know?

p = m·v (γ·m·v when relativistic), λ = h / p — The textbook route. Below a tenth of light speed p = mv is within 0.5 % of the truth; above it the Lorentz factor is applied automatically.

Velocity

Must be greater than zero and below the speed of light.

Velocity unit

Including fractions of c, so 0.5 means half light speed.

De Broglie wavelength — λ = h / p

Classicalp = mvComparable to atomic spacing — diffraction observable

3.30632 Å

The wavelength sits in the same decade as the spacing of crystal planes, so a crystal acts as a diffraction grating. This is the band electron diffraction, neutron scattering and transmission electron microscopy all work in.

Wavelength in metres: 3.30632e-10 m

Momentum p: 2.00406e-24 kg·m/s

Wave number k = 2π/λ: 1.90036e+10 rad/m

Frequency f = K/h: 3.32697e+15 Hz

Velocity

2.20000e+6 m/s

Supplied in velocity mode, derived from the momentum in every other mode.

v / c

0.00733841

The relativistic switch trips at 0.1.

Lorentz factor γ

1.000027

Fixed decimals, because γ − 1 is the interesting part.

Rest energy mc²

510.999 keV

Kinetic energy approaching this is what makes a particle relativistic.

Kinetic energy (J)

2.20447e-18 J

The energy of motion alone.

Kinetic energy (eV)

13.7592 eV

Auto-scaled through eV, keV, MeV and GeV.

Equivalent voltage

13.7592 V

The potential difference that would produce this energy. Blank for neutral particles.

Classical vs relativistic λ

0.00269272 %

How much longer the classical wavelength is than the relativistic one.

The matter wave

The wave that travels with the particle, with one period dimensioned between two crests. Raise the velocity or the energy and the crests crowd together — that is the whole content of λ = h / p.

Where the wavelength lands

A logarithmic ruler from sub-nuclear to sub-millimetre, with the result pinned. Diffraction becomes visible when the wavelength is comparable to the spacing of whatever it passes through.

Comparison object

The crystallography sweet spot — diffraction is observable here.

λ ÷ comparison object

3.30632 ×

A ratio near one is the regime where diffraction is observable.

The same wavelength in every unit

Unit	Wavelength
fm	330632
pm	330.632
Å	3.30632
nm	0.330632
µm	0.000330632
mm	3.30632e-7
m	3.30632e-10

Momentum in every unit

Unit	Momentum
kg·m/s	2.00406e-24
eV/c	3749.92
MeV/c	0.00374992

Relativistic regime

Below a tenth of light speed the classical p = mv is within half a percent of the truth. Above it the Lorentz factor matters, and by 100 kV it is worth nearly five percent of the wavelength.

Classical λ

3.30632 Å

From p = mv, ignoring relativity entirely.

Relativistic λ

3.30623 Å

From p = γmv, the physically correct branch.

Branch used

Classical

Set by the treatment control above.

Wavelength against kinetic energy

Bragg diffraction helper

Crystal planes separated by d reflect constructively when 2d·sin θ = nλ. Supply the spacing and the tool closes the loop from wavelength to a measurable angle.

Lattice spacing d

Leave blank to hide the helper. Silicon (111) is 3.135 Å; nickel (111) is 2.03 Å.

Spacing unit

Crystallography uses ångström almost exclusively.

Diffraction order n

A positive integer. Higher orders need a longer path difference and so a steeper angle.

Bragg angle θ

50.2562°

Measured from the plane surface, as Bragg's law defines it.

sin θ = nλ / 2d

0.768911

Greater than one means no such reflection exists.

Highest order

The largest n that still satisfies nλ ≤ 2d.

Constructive interference at θ = 50.26° from the planes, measured from the plane surface as Bragg's law defines it.

The same condition, other particles

Mass is the only thing that changes across these rows. At equal kinetic energy λ scales as 1/√m, so an electron comes out about 43 times longer than a proton — which is √(m_p/m_e), and nothing more mysterious than that.

Particle	Wavelength	Momentum	Velocity	Relative to longest
Electron	3.30632 Å	2.00406e-24 kg·m/s	2.20000e+6 m/s	1 ×
Proton	180.068 fm	3.67977e-21 kg·m/s	2.20000e+6 m/s	0.000544617 ×
Neutron	179.82 fm	3.68484e-21 kg·m/s	2.20000e+6 m/s	0.000543867 ×
Alpha particle (He²⁺)	45.3274 fm	1.46182e-20 kg·m/s	2.20000e+6 m/s	0.000137093 ×

Step-by-step derivation

Show steps

Step	Substitution
Classical momentum	`p = m·v = 9.109384e-31 × 2200000`
De Broglie relation	`λ = h / p = 6.62607015e-34 / 2.004064e-24 = 3.306316e-10 m`
Bragg angle	`θ = asin(n·λ / 2d) = asin(1 × 3.306316e-10 / (2 × 2.150000e-10)) = 50.256°`
Wave number	`k = 2π / λ = 1.900358e+10 rad/m`
Kinetic energy	`K = 2.204471e-18 J = 13.75922 eV`
Speed and Lorentz factor	`v = 2200000 m/s = 0.00733841 c, γ = 1.000027`

Precision (significant figures)

Clamped to 0–10. It drives every readout on this page and the TXT and CSV exports too.

Headline unit chosen

Picked automatically so results read as 3.3 Å rather than 3.3e-10 m.

Constants used

Symbol	Name	Value	Status
`h`	Planck constant	6.62607015e-34 J·s	Exact by definition
`c`	Speed of light in vacuum	299792458 m/s	Exact by definition
`e`	Elementary charge	1.602176634e-19 C	Exact by definition
`k_B`	Boltzmann constant	1.380649e-23 J/K	Exact by definition
`u`	Unified atomic mass unit	1.6605390666e-27 kg	CODATA 2018
`mₑ`	Electron rest mass	9.1093837015e-31 kg	CODATA 2018
`m_p`	Proton rest mass	1.67262192369e-27 kg	CODATA 2018
`m_n`	Neutron rest mass	1.67492749804e-27 kg	CODATA 2018

De Broglie Wavelength Calculator — matter waves from λ = h/p

In 1924 Louis de Broglie proposed that the wave–particle duality already accepted for light applies to matter as well: every moving particle carries a wave whose length is the Planck constant divided by the particle’s momentum. This de Broglie wavelength calculator evaluates λ = h / p from whichever quantity you actually have — a mass and velocity, a kinetic energy, an accelerating voltage, a momentum, a temperature, or a target wavelength solved backwards.

Why momentum arrives in six different disguises

The relation itself is trivial. What trips students up is that momentum is rarely handed over directly. A diffraction lab quotes an accelerating voltage; a beamline quotes electronvolts; a neutron source quotes a temperature; a textbook problem quotes a speed. Each needs a different algebraic path to the same answer:

Mass and velocity — p = m·v, or p = γ·m·v once the particle passes a tenth of light speed.
Kinetic energy — p = √(2·m·K) classically, or p = √(K² + 2·K·m·c²) / c relativistically.
Accelerating voltage — K = q·V, then the energy route. Neutral particles cannot take this path at all.
Temperature — p = √(3·m·k_B·T) on the root-mean-square convention, or √(2·m·k_B·T) for the most-probable speed.
Reverse solve — p = h / λ, then back out the velocity, energy or voltage that would produce it.

A worked example, to four significant figures

Take the standard textbook electron travelling at 2.2 × 10⁶ m/s, roughly the speed of an electron in the first Bohr orbit of hydrogen. With the CODATA rest mass 9.1093837015 × 10⁻³¹ kg, the momentum is p = mv = 2.0041 × 10⁻²⁴ kg·m/s, and dividing the exact Planck constant by it gives λ = 3.3063 × 10⁻¹⁰ m — that is 0.33065 nm, 330.63 pm, or 3.3063 Å. That wavelength sits within a factor of three of an atomic diameter, which is precisely why electrons bound in atoms have to be described as waves rather than as orbiting balls.

Rounded constants move the fourth digit

Many textbooks round h and the electron mass to three or four digits before dividing, and land on 0.3305 nm. This calculator divides the exact SI value of h by an unrounded momentum, so its fourth significant figure is trustworthy. If you are checking homework against a printed answer, expect that last digit to differ.

Where relativity starts to bite

The classical p = mv is a low-speed approximation. At a tenth of light speed it is already about half a percent short; by 100 kV — an ordinary transmission electron microscope — the electron is moving at 55 % of c and the classical answer of 3.8783 pm overstates the true 3.7014 pm by 4.8 %. The tool switches automatically at 0.1 c and always reports both branches, so you can see the size of the correction rather than take it on trust. The relativistic energy path is deliberately written as √(K² + 2Kmc²) rather than as a difference of large total energies, which would lose precision through catastrophic cancellation at low kinetic energy.

From wavelength to a measurable angle

A wavelength only becomes evidence when something diffracts it. Crystal planes separated by a distance d reflect constructively when 2d·sin θ = nλ, so the Bragg diffraction helper converts the computed wavelength straight into the angle a detector would see. Our 3.3063 Å electron off planes spaced 2.15 Å gives θ = asin(3.3063 / 4.30) = 50.26°. When nλ exceeds 2d no such reflection exists, and the tool says so rather than returning a silent NaN.

The neutron and the baseball

Two results anchor the whole topic. A thermal neutron at 300 K lands at 1.4524 Å on the rms convention and 1.7789 Å on the most-probable one — either way, the spacing of crystal planes, which is exactly why reactors moderate neutrons before sending them at a sample. A baseball of 0.145 kg at 40 m/s, by contrast, comes out at 1.14 × 10⁻³⁴ m: twenty orders of magnitude below a proton. The wave is genuinely there; nothing in the universe is fine enough to reveal it. Mass, not any special quantum-ness, is what separates the two cases.

Massless particles need a different tool

λ = h/p holds for photons too, but the routes through m do not — a photon has no rest mass, so p = mv and p = √(2mK) both collapse. For light, use E = hc/λ in a photon energy calculator instead.

Reading the results

Every panel is derived from one unrounded momentum in SI units, so the wavelength, the wave number k = 2π/λ, the matter-wave frequency, the velocity, the Lorentz factor and the Bragg angle can never disagree with one another. The logarithmic scale bar shows where the answer falls against a proton, a nucleus, an atom, DNA, a virus and visible light, and the observability note turns that position into plain English. Results export as TXT or CSV at whatever precision you set, and the share link encodes every input in the URL.

Is the De Broglie Wavelength Calculator free?

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Yes, you can install the webapp as PWA.

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How does this de Broglie wavelength calculator work?

Whichever quantity you supply — a mass and velocity, a kinetic energy, an accelerating voltage, a momentum, a temperature, or a target wavelength to solve backwards — is converted to SI and reduced to a single unrounded momentum in kg·m/s. The wavelength then follows from λ = h / p, and every other figure on the page (wave number, frequency, velocity, γ, kinetic energy, Bragg angle) is derived from that same momentum, so no two panels can disagree because one of them started from a rounded display value.

Why does an electron at 2.2 × 10⁶ m/s come out at 0.33065 nm and not 0.3305 nm?

Because the calculator uses the exact SI value h = 6.62607015 × 10⁻³⁴ J·s together with the CODATA electron mass 9.1093837015 × 10⁻³¹ kg. Working it through, p = mv = 2.004064 × 10⁻²⁴ kg·m/s and λ = h/p = 3.30632 × 10⁻¹⁰ m — that is 0.33065 nm, 330.63 pm, or 3.3063 Å. Textbooks that round the constants to three or four digits before dividing frequently land on 0.3305 nm, which is off in the fourth significant figure.

When does the calculator switch to the relativistic formula?

On the Auto setting it applies p = γmv once the particle passes a tenth of light speed, and it always reports both branches so you can see the size of the correction. At 100 kV an electron is travelling at 55 % of c, where the relativistic 3.7014 pm is 4.8 % shorter than the classical 3.8783 pm — a difference that matters in electron microscopy. Below about 10 kV the two agree to better than a tenth of a percent.

Why can a neutron not be given an accelerating voltage?

A voltage does work only on charge, and the neutron, the hydrogen atom, helium-4 and C₆₀ are all electrically neutral, so no potential difference can speed them up. Neutron sources moderate their neutrons thermally instead: at 300 K the root-mean-square convention gives 1.4524 Å and the most-probable convention 1.7789 Å, both close to the spacing of crystal planes, which is exactly what makes neutron diffraction work.

Why does a baseball never show any diffraction?

It does have a matter wave — the relation is universal — but a 0.145 kg baseball at 40 m/s carries so much momentum that λ works out to about 1.14 × 10⁻³⁴ m. That is twenty orders of magnitude smaller than a proton, and diffraction only becomes visible when the wavelength is comparable to the obstacle. No aperture that small exists, or could exist, so the wave is unobservable in principle rather than merely in practice.

How accurate are the results?

The physical constants are the exact SI definitions of h, c, e and k_B plus CODATA 2018 rest masses, and all arithmetic is done in double precision on unrounded SI values, so the figures are good to roughly twelve significant digits before display rounding. The relativistic energy route is written as √(K² + 2Kmc²) rather than as a difference of large total energies, which avoids the cancellation error that form would suffer at low kinetic energy.