How to Calculate De Broglie Wavelength
In 1924, Louis de Broglie proposed that every moving particle has a wave associated with it, with wavelength λ = h / p, where h is Planck's constant (6.62607015 × 10⁻³⁴ J·s) and p is the particle's momentum. This extended the wave-particle duality already known for light to all matter, and the Davisson-Germer experiment confirmed it in 1927 by diffracting electrons off a nickel crystal. This calculator computes λ from a mass and velocity you supply, along with the resulting momentum and Lorentz factor.
Deriving the formula
De Broglie combined two known relations: Planck's E = hf for photon energy, and the relativistic photon momentum p = E/c. Substituting and using c = fλ gives λ = h/p for light. De Broglie's insight was to apply the same relation to matter by defining a particle's momentum as p = mv (or, at relativistic speeds, p = γmv with the Lorentz factor γ = 1/√(1 − v²/c²)). Rearranging p = h/λ gives the working formula λ = h/(mv) used by this calculator's non-relativistic mode.
Non-relativistic vs. relativistic momentum
For everyday speeds (v well under about 10% of the speed of light), p = mv is accurate to within roughly 0.5% and is the standard textbook formula. It breaks down for fast particles: electrons accelerated through 100 kV in a transmission electron microscope reach about 0.55c, where γ ≈ 1.2 — using p = mv there would understate momentum and overstate the wavelength by a similar margin. Switch to the relativistic model whenever v/c exceeds about 0.1, or whenever you know the particle was accelerated through a large voltage or found in a particle accelerator.
Real-world applications
- Electron microscopy uses the short de Broglie wavelength of accelerated electrons (picometer scale) to resolve details far smaller than visible light (hundreds of nanometers) allows.
- Neutron and electron diffraction are used in crystallography to map atomic structures, since particle wavelengths on the order of interatomic spacing produce interference patterns.
- Quantum mechanics uses λ = h/p to explain why macroscopic objects (baseballs, cars) show no observable wave behavior: their wavelengths are absurdly smaller than any physical measurement.
Frequently Asked Questions
What is the de Broglie wavelength formula?
Why do electrons in an electron microscope need the relativistic formula?
Does a thrown baseball have a de Broglie wavelength?
How accurate is the Planck's constant value used here?
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Common Measurement Conversions for Everyday Use
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