Hong Life CalculatorsUnderstanding the de Broglie wavelength calculator — concept and principle
Every moving particle behaves partly like a wave, and the de Broglie wavelength tells you how long that wave is. This calculator takes a mass and a speed and returns the wavelength in nanometres. It is used in chemistry and physics courses, and to understand why electrons and neutrons can be diffracted by crystals and why electron microscopes resolve far finer detail than light microscopes.
How it is calculated
λ = h / p = h / (m × v)
h = 6.62607015 × 10⁻³⁴ J·s is the Planck constant (exact in the SI). m is the mass in kilograms, v the speed in metres per second, and p = mv the momentum. The result in metres is multiplied by 10⁹ to show nanometres. The default mass 9.109 × 10⁻³¹ kg is the electron mass. The formula is non-relativistic: it uses p = mv, which is accurate when v is well below the speed of light.
The principle
In 1924 Louis de Broglie proposed that the relation p = h/λ, already known for photons, holds for matter too. A particle with larger momentum has a shorter wavelength. For everyday objects the wavelength is absurdly small, so wave effects are invisible; for electrons, whose mass is tiny, it is comparable to the spacing of atoms in a crystal. Davisson and Germer, and independently G. P. Thomson, confirmed this in 1927 by observing electron diffraction. The same idea underlies quantised energy levels in atoms: only orbits that fit whole numbers of wavelengths are stable in the simple picture.
Worked example
- Inputs: m = 9.109 × 10⁻³¹ kg (an electron), v = 1 × 10⁶ m/s.
- Momentum p = 9.109 × 10⁻³¹ × 10⁶ = 9.109 × 10⁻²⁵ kg·m/s.
- λ = 6.62607015 × 10⁻³⁴ ÷ 9.109 × 10⁻²⁵ = 7.2742 × 10⁻¹⁰ m = 0.7274 nm.
That electron has a kinetic energy of ½mv² ≈ 4.55 × 10⁻¹⁹ J, about 2.84 eV.
| Mass (kg) | Wavelength (nm) |
|---|---|
| 2.28e-31 | 2.909680602 |
| 4.55e-31 | 1.454840301 |
| 6.83e-31 | 0.9698935339 |
| 9.11e-31 | 0.7274201504 |
| 1.14e-30 | 0.5819386758 |
| 1.37e-30 | 0.4849467669 |
| 1.82e-30 | 0.3637100752 |
How to read the result
Wavelengths around 0.1–1 nm match interatomic distances, so such particles diffract from crystals. Electrons accelerated through 100 V have about 0.123 nm. A 0.15 kg ball at 40 m/s has a wavelength near 10⁻³⁴ m, far too small to detect, which is why classical mechanics works for large objects.
Common mistakes
- Entering mass in grams or atomic mass units instead of kilograms.
- Using the formula at speeds above about 10% of light speed without relativistic momentum.
- Confusing the de Broglie wavelength with the size of the particle.
Frequently asked questions
How do I get the wavelength from kinetic energy?
Use p = √(2mE) for non-relativistic particles, then λ = h/p.
Why do neutrons work for crystal studies?
Thermal neutrons have wavelengths around 0.18 nm and interact with nuclei, so they complement X-rays.
Does a heavier particle at the same speed have a longer wavelength?
No. Greater mass means greater momentum, so the wavelength is shorter.
Sources: L. de Broglie, Recherches sur la théorie des quanta, doctoral thesis, Paris (1924) · C. Davisson and L. H. Germer, Physical Review 30, 705 (1927) · BIPM, The International System of Units (SI Brochure), 9th edition