Explain wave-particle duality, apply the de Broglie relation, and describe the experimental evidence such as electron diffraction

What this dot point is asking

SEAB wants you to explain wave-particle duality for both light and matter, to apply the de Broglie relation $\lambda = h/p$, and to describe the experimental evidence, especially electron diffraction. Duality is the central conceptual idea of quantum physics: the same entity shows wave or particle behaviour depending on the experiment.

The answer

The dual nature of light

Light shows wave behaviour in interference and diffraction (the double slit, the grating) and particle behaviour in the photoelectric effect (photons of energy $hf$). Neither model alone is complete: light is described as a wave when it propagates and as particles when it exchanges energy with matter. This is wave-particle duality.

The de Broglie hypothesis

In 1924 de Broglie proposed that if light can behave as particles, then matter can behave as waves. Any particle of momentum $p$ has an associated wavelength:

This is the de Broglie wavelength. For everyday objects $\lambda$ is far too small to detect (a moving ball has a wavelength around $10^{-34}\ \text{m}$), but for tiny, fast particles such as electrons it becomes comparable to atomic spacings and the wave behaviour shows up.

Evidence: electron diffraction

The decisive evidence is electron diffraction. When a beam of electrons passes through a thin crystal or graphite film, it produces a diffraction pattern of rings, exactly as X-rays do. Diffraction is a wave phenomenon, so this proves electrons have a wave nature. The ring spacing matches the de Broglie wavelength calculated from the electrons' momentum, confirming the relation quantitatively.

When does each behaviour appear?

Wave behaviour dominates when the de Broglie wavelength is comparable to the size of the obstacle or aperture (so an electron diffracts off atomic-scale crystals but not a macroscopic slit). Particle behaviour appears in localised interactions such as collisions and energy exchange. The two descriptions are complementary: a single experiment reveals one or the other, never both at once.

Examples in context

Example 1. The electron microscope. Electrons accelerated to high speed have a de Broglie wavelength thousands of times shorter than visible light. Because resolution improves with shorter wavelength, an electron microscope resolves far finer detail than a light microscope, a direct technological payoff of the wave nature of matter.

Example 2. Neutron diffraction. Slow neutrons have a de Broglie wavelength comparable to atomic spacings, so they diffract off crystals and reveal atomic structure, including the positions of light atoms that X-rays miss. This is the de Broglie relation put to work for materials science.

Try this

Q1. State the de Broglie relation and explain what it tells us about matter. [2 marks]

• Cue. $\lambda = h/p$; all matter has an associated wavelength and can show wave behaviour.

Q2. Find the de Broglie wavelength of a proton (mass $1.67 \times 10^{-27}\ \text{kg}$) moving at $2.0 \times 10^4\ \text{m s}^{-1}$ ($h = 6.63 \times 10^{-34}\ \text{J s}$). [2 marks]

• Cue. $p = 1.67 \times 10^{-27} \times 2.0 \times 10^4 = 3.34 \times 10^{-23}$; $\lambda = h/p = 2.0 \times 10^{-11}\ \text{m}$.

Q3. Explain why a moving car does not show observable wave behaviour. [2 marks]

• Cue. Its large momentum gives a de Broglie wavelength around $10^{-38}\ \text{m}$, vastly smaller than any aperture, so no diffraction is detectable.