Comprimento de Onda de De Broglie

De Broglie wavelength

In 1924 Louis de Broglie made the striking claim that every particle with mass carries a wavelength of its own, given by λ = h/p = h/(m·v), where h = 6.626 × 10⁻³⁴ J·s is Planck's constant and p is the momentum. Three years later Davisson and Germer (1927) backed him up by watching electrons diffract off a nickel crystal. Take Example 1: an electron pushed through 100 V ends up with p ≈ 5.4 × 10⁻²⁴ kg·m/s, which works out to λ ≈ 0.12 nm, roughly the size of an atom, so it diffracts inside crystals. Now Example 2: a 0.5 kg ball moving at 30 m/s has p = 15 kg·m/s and a wavelength of λ ≈ 4.4 × 10⁻³⁵ m, well below the Planck length, with no wave behavior anyone could ever detect. That gap is exactly why everyday objects look perfectly classical. Their wavelength is just absurdly small.

Applications

You meet this idea in transmission electron microscopy (TEM), where the tiny electron wavelength buys atomic-scale resolution around 0.05 nm, far past what any optical microscope reaches. It also drives electron and neutron diffraction for pinning down crystal and protein structures, sits behind scanning tunneling microscopy, and is a staple of quantum mechanics teaching for showing wave–particle duality. And in particle physics it tells you how deeply an accelerator can probe: more energy means smaller λ, which means finer resolution.

FAQ

Why don't I see the wavelength of a soccer ball? Its λ is around 10⁻³⁵ m, smaller than the Planck length itself. Nothing can measure waves at that scale. Quantum effects only become observable once you drop to very small masses, like electrons, atoms, or large molecules up to roughly 10⁻²⁴ kg.

Does the formula hold for photons? It does. Photons have no rest mass, yet they still carry momentum p = E/c = h/λ. That is the same relation, and historically it was the starting point.

What is the relativistic version? At high speeds you swap in the relativistic momentum p = γmv, with γ = 1/√(1−v²/c²). The wavelength formula λ = h/p still holds.