The de Broglie wavelength

The de Broglie wavelength is the wavelength of a moving particle. Every particle with momentum behaves partly like a wave, with wavelength λ = h/mv, where h is the Planck constant (6.63 × 10⁻³⁴ J s) and mv is the particle's momentum. The faster or heavier the particle, the shorter its wavelength. For electrons the wavelength is about the size of an atom, which is why electrons can be diffracted.

The de Broglie wavelength equation

This is the form in AQA's A-level Physics specification, λ = h/mv, where mv is the momentum. Because λ and p are inversely proportional, doubling a particle's momentum halves its wavelength. Rearranged, the equation gives the momentum or speed needed for a particular wavelength: p = h/λ and v = h/mλ.

How the wavelength changes with speed

Here x is the electron's speed in millions of metres per second and y is its de Broglie wavelength in nanometres. The curve shows inverse proportion: double the speed and the wavelength halves. (At much higher speeds, approaching the speed of light, the simple formula mv no longer gives the momentum, but A-level questions stay well below that.)

Wave-particle duality

Light was long thought of as a wave, because it diffracts and interferes. Then the photoelectric effect showed it also comes in particles, photons, each with energy E = hf. Louis de Broglie suggested the reverse: if waves can act like particles, particles should act like waves. That idea, that light and matter both have wave and particle properties, is called wave-particle duality.

AQA's specification puts the evidence side by side: electron diffraction suggests particles have wave properties, and the photoelectric effect suggests electromagnetic waves have a particle nature. The photon side uses the Planck constant too. Try a photon energy question, then press Mark to check it in the course.

Electron diffraction

Waves diffract noticeably when the gap or obstacle is about the same size as their wavelength. Atoms in a crystal are spaced about 10⁻¹⁰ m apart, about the size of an atom, so a beam of electrons with a similar de Broglie wavelength spreads out when it passes through a thin layer of graphite or metal. It makes a pattern of bright rings on a screen, like light through a tiny hole. Particles alone wouldn't do that; waves do.

How the amount of diffraction depends on momentum

• Faster electrons (a higher accelerating voltage) have more momentum, so a shorter wavelength.
• A shorter wavelength diffracts less, so the rings get smaller.
• Slower electrons have a longer wavelength, diffract more, and give bigger rings.

This is exactly what AQA asks students to explain: how and why the amount of diffraction changes when the momentum of the particle changes. Details of particular diffraction methods are not expected.

Worked examples

These use h = 6.63 × 10⁻³⁴ J s and an electron mass of 9.11 × 10⁻³¹ kg. For the momentum step on its own, the momentum calculator works out p = mv.

Example 1: an electron's wavelength

An electron moves at 2.0 × 10⁶ m s⁻¹. Find its de Broglie wavelength.

The wavelength is 3.6 × 10⁻¹⁰ m, or 0.36 nm, close to the size of an atom. AQA's GCSE Chemistry specification gives an atom's radius as about 1 × 10⁻¹⁰ m.

Example 2: an electron accelerated through a voltage

An electron starts from rest and is accelerated through a potential difference of 250 V. Find its de Broglie wavelength.

The kinetic energy it gains is eV = 1.60 × 10⁻¹⁹ × 250 = 4.00 × 10⁻¹⁷ J. Since kinetic energy is p²/2m, the momentum is:

The wavelength is 7.8 × 10⁻¹¹ m, about 0.08 nm. That is close to the spacing of atoms, so electrons like these are diffracted by a thin crystal.

Example 3: why a ball doesn't diffract

A ball of mass 58 g travels at 50 m s⁻¹. Its momentum is 0.058 × 50 = 2.9 kg m s⁻¹, so:

That is far, far smaller than an atom, let alone any gap the ball could pass through, so no diffraction can ever be seen. Everyday objects have wavelengths too small to notice, which is why wave behaviour only shows up for tiny particles.

Example 4: finding a speed

What speed must a neutron (mass 1.675 × 10⁻²⁷ kg) have for a de Broglie wavelength of 0.20 nm?

About 2.0 × 10³ m s⁻¹. A neutron is about 1,839 times heavier than an electron, so it needs a much lower speed for the same wavelength.

Photon momentum

The same equation works the other way for light. A photon has no mass, but it does have momentum, p = h/λ. A photon of green light with a wavelength of 500 nm has momentum 6.63 × 10⁻³⁴ ÷ 5.00 × 10⁻⁷ = 1.33 × 10⁻²⁷ kg m s⁻¹. Don't use mv for a photon.

Common mistakes

• Using grams. Mass must be in kilograms: 58 g is 0.058 kg.
• Putting kinetic energy in place of momentum. λ = h/p, not h/Eₖ. Convert with p = √(2mEₖ).
• Forgetting to convert eV or volts. An electron accelerated through V volts gains eV joules, with e = 1.60 × 10⁻¹⁹ C.
• Getting the diffraction direction backwards. More momentum means a shorter wavelength and less diffraction.
• Using mv for a photon. For light, p = h/λ.

To practise quantum and particle questions under time pressure, try today's A-level Physics: Year 1 short timed paper. Each paper mixes topics. For the particles themselves, see the mass of an electron.

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