F. Revolutions in Modern Physics: Quantum Mechanics and Special Relativity · Quantum MechanicsWave-Particle Duality of Light
F3.1 F1.1 F2.1 F1.2About 65 minutes
Learning goals- I can summarize the experimental evidence for the wave model and for the particle model of light.
- I can describe the single-photon double-slit experiment and explain it with a probability interpretation.
- I can explain pair production and annihilation, and how annihilation is used in PET scans.
- I can explain the idea of complementarity and how wave-particle duality changed scientific thought.
Words to know
wave-particle duality
The idea that light (and matter) shows wave properties in some experiments and particle properties in others.
complementarity
Bohr's principle that the wave and particle descriptions are both needed, but no single experiment shows both fully at once.
probability interpretation
The idea that the intensity of a light wave at a point tells us the probability of finding a photon there.
pair production
The conversion of a high-energy photon into a particle and its antiparticle, such as an electron and a positron.
positron
The antiparticle of the electron: same mass, opposite (positive) charge.
annihilation
A particle and its antiparticle meet and their mass becomes energy, usually two photons.
Wave or particle? Yes.
In the unit The Wave Nature of Light you saw overwhelming evidence that light is a wave: it diffracts around edges, two beams interfere to make bright and dark fringes (Young's double slit, thin films, gratings), and it can be polarized. In this unit you have seen equally strong evidence that light is made of particles: the blackbody spectrum, the photoelectric effect and the Compton effect.
Physicists spent years trying to decide which model was "really" right. The answer turned out to be neither, and both. Light is something new that does not match our everyday ideas of a wave (like a ripple on a pond) or a particle (like a tiny marble). It propagates like a wave and interacts (is emitted or absorbed) like a particle. This is called wave-particle duality.
| Evidence for waves | Evidence for particles (photons) |
|---|
| Interference (double slit, thin films) | Blackbody spectrum (Planck, energy quanta) |
| Diffraction (single slit, gratings) | Photoelectric effect (threshold, no time delay) |
| Polarization | Compton effect (photon momentum) |
| Refraction explained by a change in wave speed | Pair production and annihilation |
One photon at a time
Here is the experiment that shows duality most clearly. Make light so dim that only one photon is in the apparatus at a time, and send it toward a double slit. Put a very sensitive detector (a photon-counting camera) on the screen.
- Each photon arrives at one single spot, like a particle. You never see half a photon spread out across the screen.
- But where it lands seems random. Watch for a while and the spots pile up in bright and dark fringes: the double-slit interference pattern.
- Close one slit and the fringes disappear, even though each photon could only have gone through one slit at a time... or could it?
An early version of this was done by G. I. Taylor in 1909, using light so faint that the longest photographic exposure lasted about three months; the fringes still appeared. Modern photon-counting cameras show the pattern building up dot by dot.
The probability interpretation
How can one photon make an interference pattern? The modern answer, built on ideas of Max Born (1926), is that the wave is a wave of probability:
- The light wave (worked out with ordinary wave theory) passes through both slits and interferes.
- The intensity of the wave at each point (proportional to amplitude squared) tells you the probability that a photon will be detected there.
- When the photon is detected, it is found at one point, as a whole particle.
Bright fringes are places where photons are likely to land; dark fringes are where they almost never land. We cannot predict where any one photon will go, only the pattern that many photons make. This was a shocking change: classical physics was deterministic (know the starting conditions, predict the result exactly); quantum physics is fundamentally probabilistic.
Complementarity (F1.1)
Niels Bohr summed up duality as complementarity: the wave and particle pictures are both necessary to describe light fully, but an experiment designed to show one hides the other. If you put detectors at the slits to find out which slit each photon goes through (particle information), the interference pattern disappears (wave behaviour lost). Physicists disagreed sharply about this. Einstein famously objected that God "does not play dice"; Bohr defended the probability interpretation. Their debates, often using clever thought experiments, shaped quantum mechanics for decades.
Worked exampleExample 1: Is there really only one photon at a time?
A dim laser beam (633 nm) has a power of 1.0 × 10⁻¹² W. How many photons per second does it carry, and on average how far apart are they along the beam? Is it reasonable to say only one photon is in a 1 m long apparatus at a time?
- Energy per photon: E = hc/λ = (6.63 × 10⁻³⁴)(3.00 × 10⁸) ÷ (6.33 × 10⁻⁷) = 3.14 × 10⁻¹⁹ J.
- Photons per second: N = P/E = 1.0 × 10⁻¹² ÷ 3.14 × 10⁻¹⁹ = 3.2 × 10⁶ photons/s.
- In one second the beam's front travels 3.00 × 10⁸ m, shared among 3.2 × 10⁶ photons. Average spacing = 3.00 × 10⁸ ÷ 3.2 × 10⁶ = 94 m.
About 3 million photons per second, spaced on average 94 m apart. In a 1 m apparatus there is usually no more than one photon at a time, so each photon must make its share of the pattern on its own.
Pair production and annihilation
Another striking particle-like behaviour: a high-energy gamma photon passing near a nucleus can disappear and create an electron and a positron (the electron's antiparticle). Energy turns into mass according to E = mc² (see Special Relativity). Each particle has rest energy mₑc² = 0.511 MeV, so the photon needs at least 2 × 0.511 = 1.022 MeV. Any extra becomes the particles' kinetic energy. (The nucleus is needed nearby to take up a little momentum; it gets almost no energy.)
The reverse also happens. When a positron meets an electron they annihilate: both vanish and usually two gamma photons of 0.511 MeV each fly off in opposite directions, conserving momentum.
PET scans (positron emission tomography) use this. A patient receives a small dose of a tracer, often a sugar molecule labelled with fluorine-18, which emits positrons. Each positron annihilates with a nearby electron; a ring of detectors catches the two back-to-back 0.511 MeV photons, and a computer traces the line between them. Millions of such lines map where the tracer collects, for example in active tumours. This is a direct medical use of quantum physics and relativity together.
Worked exampleExample 2: The photons in a PET scanner
An electron and a positron, both nearly at rest, annihilate into two photons. Find each photon's energy and wavelength, and explain why the photons go in opposite directions.
- Total energy available = 2mₑc² = 2(0.511 MeV) = 1.022 MeV, shared equally: 0.511 MeV each.
- λ = hc/E = 1240 eV·nm ÷ 511 000 eV = 2.43 × 10⁻³ nm = 2.43 pm.
- Before, the total momentum is about zero. Each photon carries momentum h/λ, so to keep the total zero they must have equal momentum in opposite directions.
Each photon has 0.511 MeV and λ = 2.43 pm (gamma rays), travelling back to back. (Notice that 2.43 pm is the Compton wavelength h/mₑc from the last lesson: not a coincidence, since E = mₑc² gives λ = h/(mₑc).)
Just for funThe physicist William Henry Bragg is said to have joked that physicists used the wave theory of light on Mondays, Wednesdays and Fridays, and the particle theory on Tuesdays, Thursdays and Saturdays.
Hands-on activityProbability builds a pattern
You need: two six-sided dice, paper and a pencil.
In quantum physics, single events are random but large numbers form a predictable pattern. Model this:
- Draw a horizontal axis labelled 2 to 12 (the possible sums of two dice).
- Roll both dice and put a dot above the sum. Do this 10 times. Can you see a pattern?
- Keep going to 100 rolls (a family member can help roll and record).
- Compare your dot plot with the predicted probabilities: 7 is most likely (6 ways out of 36), 2 and 12 least likely (1 way each).
What to look for: after 10 rolls the dots look scattered almost at random; after 100 a clear peak at 6 to 8 appears. In the single-photon experiment, each photon is like one roll: you cannot predict it, but the wave's intensity predicts the pattern. (Dice are only an analogy: dice are random because we don't know the exact throw; quantum randomness appears to be fundamental.)
Your turn
1Name two experiments that show light behaves as a wave and two that show it behaves as a particle.
Wave: e.g. Young's double-slit interference, diffraction gratings, thin-film colours, polarization. Particle: e.g. photoelectric effect, Compton effect, blackbody spectrum, pair production.
2In the single-photon double-slit experiment, what part of the result is particle-like and what part is wave-like?
Particle-like: each photon is detected at one point, whole. Wave-like: the photons' landing points build up an interference pattern that only a wave passing through both slits can explain.
3What does the intensity of the light wave at a point on the screen tell us in the probability interpretation?
The probability of detecting a photon at that point (high intensity = likely, zero intensity = never).
4What is the minimum photon energy for producing an electron-positron pair? Find the matching wavelength.
1.022 MeV. λ = 1240 eV·nm ÷ 1.022 × 10⁶ eV = 1.21 × 10⁻³ nm (1.21 pm).
5A 3.0 MeV gamma photon produces an electron-positron pair. What is their combined kinetic energy?
3.0 − 1.022 = about 2.0 MeV (1.98 MeV).
6Why must annihilation of an electron and positron at rest produce at least two photons rather than one?
Total momentum before is zero. A single photon always carries momentum (h/λ), so it cannot have zero momentum. Two photons moving in opposite directions can have zero total momentum.
7What happens to the interference pattern if detectors reveal which slit each photon passes through?
It disappears (you get two overlapping single-slit patterns). Gaining particle (path) information destroys the wave interference: complementarity.
More practice: warm-up
1What is wave-particle duality?
Light (and matter) shows wave properties in some experiments and particle properties in others.
2Who proposed the principle of complementarity?
3What is the antiparticle of the electron called?
Positron emission tomography
5In a PET scan, what is the energy of each annihilation photon?
More practice: on your own
1Repeat Example 1 for a 1.0 nW beam. Would only one photon be in a 1 m apparatus at a time?
N = 1.0 × 10⁻⁹ ÷ 3.14 × 10⁻¹⁹ = 3.2 × 10⁹ photons/s; spacing = 3.00 × 10⁸ ÷ 3.2 × 10⁹ ≈ 0.094 m. About 10 photons are in a 1 m apparatus at once, so this beam is too bright for a true one-at-a-time experiment.
2Explain why the dark fringes in a single-photon experiment are evidence that each photon somehow "uses" both slits.
With one slit open, photons do land at the positions of the dark fringes. Opening the second slit reduces the number landing there to zero. Something about each photon must be affected by both slits: the probability wave passes through both and cancels (destructive interference) at the dark fringes.
3A gamma photon of 2.0 MeV creates an electron-positron pair. If they share the extra energy equally, what is the kinetic energy of each?
Extra = 2.0 − 1.022 = 0.978 MeV; each gets about 0.49 MeV.
4Why does a PET scanner use a full ring of detectors?
The two annihilation photons travel in opposite directions along a line through the source. Detecting both at the same moment on opposite sides of the ring identifies that line; many lines intersecting locate the tracer.
5How did the probability interpretation change the scientific idea of predictability?
Classical physics was deterministic: exact knowledge of the present predicts the future exactly. Quantum physics predicts only probabilities for single events (where one photon lands), though patterns for many events are predicted precisely. Many physicists, including Einstein, found this hard to accept.
Challenge
1Pair production cannot happen in empty space, only near a nucleus (or other particle). Use conservation laws to explain why, considering a photon making a pair at rest.
Suppose a photon produced a pair that is at rest (the minimum-energy case). The photon had momentum E/c ≠ 0, but the pair at rest has zero momentum: momentum is not conserved. Even if the pair moves, a careful calculation shows energy and momentum cannot both be conserved for photon → e⁻ + e⁺ alone. A nearby nucleus absorbs the leftover momentum (while taking almost no energy because it is so massive), allowing both laws to hold.
2A proton-antiproton pair annihilates at rest into two photons. The proton's rest energy is 938 MeV. Find the wavelength of each photon.
Each photon gets 938 MeV. λ = 1240 eV·nm ÷ 9.38 × 10⁸ eV = 1.32 × 10⁻⁶ nm (1.32 fm).
3A student claims: "Light is a particle; the interference pattern happens because photons bump into each other." Use the single-photon experiment to refute this.
In the single-photon experiment photons pass through the apparatus one at a time (e.g., spaced ~94 m apart in a 1 m apparatus), so they cannot bump into each other, yet the interference pattern still builds up. The pattern must come from each photon's own wave (probability) behaviour, not from photon-photon collisions.
Quiz
1. Which experiment provides evidence for the wave nature of light?
2. In a single-photon double-slit experiment, each photon
3. According to the probability interpretation, a bright fringe is a place where
4. The minimum energy of a photon that can produce an electron-positron pair is
5. Why do the two photons from electron-positron annihilation at rest travel in opposite directions?
6. What happens to the interference pattern when detectors determine which slit each photon passes through?
Big idea (F3.1, F1.1): light is neither just a wave nor just a particle. It travels as a wave (giving interference) but is detected in whole photons, and the wave tells us the probability of where photons land. Pair production and annihilation show energy turning into matter and back, which PET scanners use.
How to help: the dice activity is a good conversation about probability. Ask: "Can you predict one roll? Can you predict the pattern of 100 rolls?" Then connect: that is how photons behave in the double-slit experiment.
Common difficulties:
- Thinking the interference pattern comes from photons interacting with each other. It forms even one photon at a time.
- Thinking duality means light switches between being a wave and a particle. It is one kind of thing with both properties.
- Forgetting that pair production needs at least 1.022 MeV (two rest energies), not 0.511 MeV.
- PET is sometimes confused with MRI. PET uses positrons and gamma photons; MRI uses magnetic fields and radio waves.