Matter Waves: the de Broglie Wavelength

If waves can be particles...

Light spreads out and interferes like a wave. Yet in the photoelectric effect it arrives as photons, like particles. In 1924 a young French physicist, Louis de Broglie, turned the idea round. If waves can act like particles, perhaps particles can act like waves.

The de Broglie wavelength

De Broglie said that every moving particle has a wavelength

where $p = mv$ is its momentum and $h = 6.63 \times 10^{-34}\ \text{J s}$ is Planck's constant.

An example. In hydrogen's ground state, the electron moves at about $2.19 \times 10^{6}\ \text{m/s}$. Its wavelength is

Keep this number in mind: it comes back at the end of the lesson.

Electrons speeded up by a voltage

An electron that starts from rest and passes through a voltage $V$ gains the kinetic energy $eV$. Since $KE = \dfrac{p^2}{2m}$, its momentum is $p = \sqrt{2meV}$, and

with $V$ in volts.

An example. In 1927, Davisson and Germer fired electrons speeded up by $54\ \text{V}$ at a nickel crystal. The formula gives $\lambda = \dfrac{1.227}{\sqrt{54}}\ \text{nm} = 0.167\ \text{nm}$. The electrons came off the crystal in a diffraction pattern, just as X-rays do, and the pattern gave $0.165\ \text{nm}$. Electrons really do behave as waves.

You should find that a higher voltage gives a shorter wavelength, and that a few tens to a few hundred volts put the wavelength right among the spacings of atoms.

Why does a tennis ball not diffract?

A tennis ball of $0.06\ \text{kg}$ served at $30\ \text{m/s}$ has

This is far smaller than an atom, or even a nucleus. No gap or grating is that small, so the wave nature of everyday objects never shows. It matters only for tiny masses, like electrons, protons and neutrons.

Waves round Bohr's orbits

Bohr's rule was $mvr = \dfrac{nh}{2\pi}$. With $\lambda = \dfrac{h}{mv}$, it becomes

The length of each allowed orbit holds a whole number of electron wavelengths, so the wave joins itself smoothly all the way round. In the ground state, $2\pi r_1 = 2\pi \times 0.0529\ \text{nm} = 0.332\ \text{nm}$: exactly one wavelength, the number we found at the start.

A use: the electron microscope

A microscope cannot show details smaller than about the wavelength it uses. Light has wavelengths of $400\ \text{nm}$ to $700\ \text{nm}$. Electrons speeded up by thousands of volts have wavelengths of a few thousandths of a nanometre, so an electron microscope can show viruses and even rows of atoms.

Try it in Python

The program finds the wavelength of electrons speeded up by five different voltages.

Practise with new numbers

Remember these

• Every moving particle has a de Broglie wavelength $\lambda = \dfrac{h}{p} = \dfrac{h}{mv}$.
• An electron speeded up by $V$ volts: $\lambda = \dfrac{1.227}{\sqrt{V}}\ \text{nm}$.
• Crystals diffract electrons, because the wavelength is about the spacing of the atoms (Davisson and Germer).
• Everyday objects have wavelengths far too small to notice.
• In Bohr's atom, $2\pi r = n\lambda$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. An electron is speeded up from rest by $36\ \text{V}$. What is its de Broglie wavelength?

2. Why do we never see a running person diffract through a doorway?

3. An electron and a proton have the same de Broglie wavelength. Which one moves faster?

4. The speed of an electron doubles. What happens to its de Broglie wavelength?