Matter waves

If light waves can act like particles, can particles act like waves? de Broglie said yes. Every moving particle has a wavelength:

λ = h / p = h / (mv)

Heavier or faster particles have shorter wavelengths. For a photon, the formula gives back λ = c/ν, so it fits light too.

Everyday objects: a car of 1000 kg at 20 m/s has λ = 6.63 × 10⁻³⁴ / 20 000 ≈ 3 × 10⁻³⁸ m, far smaller than any atom. No opening could ever diffract it, so we never see waves in the motion of everyday things.

Electrons are so light that their wavelengths are about the size of the gaps between atoms in crystals, like X-rays. So electron beams diffract off crystals.

For an electron accelerated from rest through V volts: eV = ½mv², which gives a quick formula:

λ ≈ 1.227 / √V nm

Example: through 100 V, λ = 1.227/10 ≈ 0.12 nm.

Proof: in 1927, Davisson and Germer fired electrons at a nickel crystal and found them scattered strongly in particular directions, exactly as waves of λ = h/p would be. Matter really does behave as a wave.

Use: the electron microscope. Electron wavelengths are thousands of times shorter than light's, so it can resolve far finer detail, down to the shapes of viruses and molecules.

So light and matter both have a dual nature: wave-like in some experiments, particle-like in others.