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
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.
