de Broglie Wavelength

Calculate the matter-wave wavelength of any particle from its speed, kinetic energy or accelerating voltage. Uses relativistic momentum, so it stays correct for fast electrons in microscopes and accelerators.

QuantumWave–ParticleRelativity

Inputs

Input examples

Particle

Charge q — only used for the voltage

Motion given as

Click Solve next to the quantity you want to find: it moves into the result below. Then fill in the others. Example: solve the voltage that gives electrons a wavelength of 0.05 nm.

Mass m

Velocity v

m/s

Wavelength λ

What is a Matter Wave?

In 1924 Louis de Broglie proposed that every moving object has a wavelength λ = h/p. Three years later Davisson and Germer confirmed it: electrons fired at a nickel crystal produced a diffraction pattern, exactly like X-rays.

Wave effects only show up when λ is comparable to the size of the obstacle. For electrons and neutrons that matches the spacing of atoms (~0.1 nm), which is why electron microscopes and neutron diffraction can image atomic structure. For everyday objects λ is absurdly small.

p ↑wavelength shorter

λ ~ 0.1 nmdiffracts off crystals

v → cuse γmv, not mv

baseballλ ~ 10⁻³⁴ m: invisible

Live Graph

λ vs v

relativistic λ = h/γmv

classical h/mv

atomic spacing 0.1 nm

Result

Enter the values above

Typical Wavelengths

Particle	Condition	λ
Electron	100 eV	0.123 nm
Electron (TEM)	200 keV	2.51 pm
Thermal neutron	0.025 eV	0.181 nm
C₆₀ molecule	200 m/s	2.8 pm
Baseball	40 m/s	1.1×10⁻³⁴ m
h = 6.62607015×10⁻³⁴ J·s · 1 eV = 1.602176634×10⁻¹⁹ J

Three Ways to Compute λ

From Velocity

For everyday speeds γ ≈ 1 and this is simply h/mv. Above ~10% of c the factor γ matters.

Example:

From Kinetic Energy

The approximation on the right holds when E_k is much smaller than the rest energy mc² (511 keV for an electron).

Example:

From Accelerating Voltage

A charged particle accelerated through V volts gains |q|V of energy. The shortcut on the right is for non-relativistic electrons.

Example:

Sources & Verification

References

R. A. Serway, C. J. Moses & C. A. Moyer, Modern Physics, 3rd ed., Brooks/Cole, 2005 — Ch. 2 (Relativity II: relativistic momentum) and Ch. 5 (Matter Waves).
E. Tiesinga, P. J. Mohr, D. B. Newell & B. N. Taylor, “CODATA recommended values of the fundamental physical constants: 2018”, Reviews of Modern Physics 93, 025010 (2021) — values of the constants used.