Matter waves.

Step 1 of 5

One electron at a time

We saw that light, a wave, arrives as photons, like particles. Is the reverse true as well? The Scene has an electron gun, a foil with two slits and a screen that keeps a dot wherever an electron lands. The electrons leave one by one: at any moment there is only one electron on the bench. So far ten have been sent, and each arrived whole, as one dot.

In the waves Unit we saw what water waves or light do when they pass two slits: they interfere, and bright and dark stripes appear on the screen. Predict what the screen will show after six thousand electrons.

Step 2 of 5

The electron’s speed

The gun accelerates each electron from rest through a potential difference $V$⁠. The electric field does work $eV$⁠ on it, which becomes kinetic energy:

$eV = \tfrac12 m v^2$⁠

Gun voltage is locked at 100 V. An electron’s charge is $e = 1.6 \times 10^{-19}\ \mathrm{C}$$e =$$1.6 \times 10^{-19}\ \mathrm{C}$⁠ and its mass $m = 9.11 \times 10^{-31}\ \mathrm{kg}$$m =$$9.11 \times 10^{-31}\ \mathrm{kg}$⁠.

Step 3 of 5

The de Broglie wavelength

In 1924 Louis de Broglie proposed that every moving particle has a wave, whose wavelength is set by its momentum, $p = mv$⁠:

$\lambda = \frac{h}{p}$$\lambda =$$\frac{h}{p}$⁠

This is the same Planck’s constant as in a photon’s energy. From $eV = \tfrac12 m v^2$⁠ comes $p = mv = \sqrt{2 m e V}$$p =$$mv =$$\sqrt{2 m e V}$⁠, so $\lambda = \dfrac{h}{\sqrt{2meV}}$$\lambda =$$\dfrac{h}{\sqrt{2meV}}$⁠. At 100 V that is about 0.12 nm, the size of an atom. The gun’s power supply shows the wavelength at each voltage.

The wavelength sets the spacing of the stripes: a shorter wave gives closer stripes. Try Gun voltage at 100 V and at 600 V and compare the stripes.

Step 4 of 5

Watching the slits

If each electron arrives as one dot, surely it went through one slit only. You can check: a lamp by the slits lights up each electron that passes and shows which slit it went through. Predict what will happen to the pattern when Watch the slits is switched on.

Step 5 of 5

A wave of chance

Build up the pattern has laid the wave across the bench: part of it goes through each slit, and beyond them the two parts interfere. The blue curve on the screen is the chance that an electron lands at each place. An electron is neither a tiny ball nor a water wave: it always arrives as a particle, in one place, and the wave sets where that is likely to be. This is the double nature of matter, wave–particle duality, and light has the same double nature.