Physics › ⚛️ Modern & Quantum

Wave–Particle Duality

A single electron, fired alone through two slits, still "knows" about both of them — and that changes what "particle" means.

AP Physics 2 levelUni Year 1
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The big idea: Every particle of matter and every photon of light behaves as both a wave and a particle — never fully one or the other. The wave, described by de Broglie's λ = h/p, doesn't compete with the particle picture; it governs the odds of where that particle will be detected, one localized event at a time.
🎯 By the end, you'll be able to
  • Explain what the single-particle double-slit experiment reveals about the nature of matter and light
  • Calculate the de Broglie wavelength of a particle from its mass and velocity
  • Explain why wave-particle duality is invisible for everyday objects but essential for electrons
  • Interpret an interference pattern as a probability distribution built up one particle at a time
📎 You should already know
  • Waves & Interference
  • Momentum (p = mv)
  • The Photoelectric Effect

Two centuries of arguing about light

For a long time, physicists fought over whether light was a wave or a stream of particles. Newton bet on particles; Young's double-slit experiment in 1801 showed clean interference fringes — alternating bands of light and dark — which seemed to settle things firmly in favor of waves. Then came the photoelectric effect, which only made sense if light arrived in discrete packets of energy: photons. Light, it turned out, was both. In 1924, a graduate student named Louis de Broglie asked an audacious question: if light can act like particles, can particles like electrons act like waves? The answer, confirmed by experiment, changed physics for good.

The experiment that still feels impossible

Here is the version that matters most: send electrons through a double slit one at a time, slow enough that only one electron is ever in flight. Each electron hits the detector screen as a single, localized dot — exactly like a particle should. There is no way to predict exactly where any individual dot will land. But let the experiment run for thousands of electrons, and the dots do not pile up randomly. They accumulate into the same interference pattern of bright and dark bands you would get from overlapping water waves. Each electron, alone, behaved as if it took both paths through the slits and interfered with itself.

🔑 Every particle carries a wave

Wave–particle duality means matter and light are not exclusively "wave" or "particle" — they are quantum objects whose behavior is governed by a wave of probability, but which are always detected as discrete, localized events. Light shows this (photons); so does matter (electrons, atoms, even large molecules).

\[ \lambda = \dfrac{h}{p} \]
The de Broglie wavelength: h is Planck's constant (6.626×10⁻³⁴ J·s), p is the particle's momentum.
\[ \lambda = \dfrac{h}{mv} \]
For a non-relativistic particle, p = mv, so the wavelength depends on both mass and speed — heavier or faster means a shorter wavelength.
🎮 Interactive: Single Particles, One at a Time LIVE
Fire particles through a double slit one by one and watch the interference pattern emerge dot by dot. Each landing spot is unpredictable, but the pattern the dots build together is not.

What the pattern is really telling you

The interference pattern is a map of probability. At each point on the screen, the wave associated with the particle can arrive in phase with itself after passing both slits (bright fringe, high probability) or out of phase (dark fringe, near-zero probability). Cover one slit, and the interference term disappears — you get a simple particle-like pile-up instead, no fringes. That single change, from two open slits to one, is the cleanest evidence that the wave nature is doing real physical work in the outcome, not just describing our ignorance of a hidden path.

📝 Worked example: An electron is accelerated from rest through a potential difference of 100 V. Find its de Broglie wavelength.
  1. Find the kinetic energy gained: \(KE = eV = (1.602\times10^{-19}\,\text{C})(100\,\text{V}) = 1.602\times10^{-17}\,\text{J}\)
  2. Solve for speed from \(KE = \frac{1}{2}mv^2\): \(v = \sqrt{\dfrac{2KE}{m}} = \sqrt{\dfrac{2(1.602\times10^{-17})}{9.109\times10^{-31}}} \approx 5.93\times10^{6}\,\text{m/s}\)
  3. Find momentum: \(p = mv = (9.109\times10^{-31})(5.93\times10^{6}) \approx 5.40\times10^{-24}\,\text{kg·m/s}\)
  4. Apply de Broglie's relation: \(\lambda = \dfrac{h}{p} = \dfrac{6.626\times10^{-34}}{5.40\times10^{-24}} \approx 1.23\times10^{-10}\,\text{m}\)
✓ λ ≈ 1.23×10⁻¹⁰ m (about 123 pm) — comparable to typical crystal-lattice spacing, which is why electrons diffract visibly off crystal lattices.
📝 Worked example: A 0.145 kg baseball is thrown at 40 m/s (about 90 mph). Find its de Broglie wavelength.
  1. Find momentum: \(p = mv = (0.145\,\text{kg})(40\,\text{m/s}) = 5.80\,\text{kg·m/s}\)
  2. Apply de Broglie's relation: \(\lambda = \dfrac{h}{p} = \dfrac{6.626\times10^{-34}}{5.80} \approx 1.14\times10^{-34}\,\text{m}\)
✓ λ ≈ 1.14×10⁻³⁴ m — about 10³³ times smaller than the ball itself, and utterly undetectable. That is why baseballs never show diffraction or interference.
⚠️ Common trap: "it's a wave until you look, then it becomes a particle"

It's tempting to picture a particle physically morphing between wave and particle forms. Cleaner way to think about it: every quantum object always has a wave of probability associated with it (governed by \(\lambda = h/p\)), and every measurement always registers a localized, particle-like event. The wave does not "become" the particle — it sets the odds of where the particle-like detection will happen. Also note: you do not need a beam of many particles at once for interference. Fire electrons one at a time, minutes apart, and the fringes still build up — each electron interferes with itself, not with the others.

Check your understanding

1. What did the double-slit experiment reveal when electrons were sent through one at a time?
Each electron is detected as a single localized dot — a particle-like event. But accumulated over many electrons, the dots form the same fringe pattern as a wave, showing that a single electron's probability of landing somewhere is governed by a wave that effectively samples both slits.
2. Calculate the de Broglie wavelength of an electron moving at 2.00×10⁶ m/s. (mₑ = 9.109×10⁻³¹ kg, h = 6.626×10⁻³⁴ J·s)
p = mv = (9.109×10⁻³¹)(2.00×10⁶) ≈ 1.82×10⁻²⁴ kg·m/s. Then λ = h/p = (6.626×10⁻³⁴)/(1.82×10⁻²⁴) ≈ 3.64×10⁻¹⁰ m.
3. Why don't thrown baseballs show any observable diffraction or interference?
Every object technically has a de Broglie wavelength, but for a baseball it works out to around 10⁻³⁴ m — far too small compared to any real slit or obstacle to ever produce a visible diffraction or interference effect.
4. If a particle's momentum is doubled, what happens to its de Broglie wavelength?
Since λ = h/p, wavelength and momentum are inversely proportional. Doubling p halves λ.
✅ Key takeaways
  • Wave–particle duality means every particle of matter and light carries an associated wave, but is always detected as a localized, particle-like event.
  • The de Broglie wavelength λ = h/p (or h/mv) applies to everything with momentum — electrons, atoms, even baseballs — not just light.
  • In the single-particle double-slit experiment, individual particles land unpredictably, but their accumulated pattern shows interference: each particle interferes with itself, not with others.
  • Wave effects only show up when λ is comparable to the size of the relevant slits or obstacles, which is why electrons diffract but everyday objects never do.