Lecture 6: Wave or Particle? Duality

“A while back I bought an idiot’s guide to quantum mechanics, from which I learned only that I hadn’t yet reached the rank of an idiot.” — Joe Bennett, The Press, Christchurch, July 4, 2007

Learning Goals

• Describe the dual particle and wave nature of light and other quantum particles
• Forget all preconceived notions of what you have learned about light — most are wrong
• Describe what a quantum particle is
• Argue how a quantum particle can create interference patterns

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A Brief History of Light

Era	Proponent	Model	Evidence
17th C	Isaac Newton	Light as particles	Sharp shadows; light does not curve around objects
Early 19th C	Thomas Young	Light as waves	Two-slit experiment showed light interfering with itself — unequivocal proof Newton was wrong
Late 19th–Early 20th C	Planck, Einstein	Light as quantum particles	Blackbody radiation; photoelectric effect demanded a particle description

The particle-wave ambigram was created by Douglas Hofstadter in 2001 — a visual representation of this duality.

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Light Is a Quantum Particle

The Single-photon Experiment

(Video demonstration from Anton Zeilinger’s research group, University of Vienna)

When light is made dimmer and dimmer and shone onto a single-photon detector (CCD):

• You do not see a uniformly dim illumination (what a wave would produce)
• You see bright dots hitting only one detector at a time, randomly distributed — like raindrops falling
• Every experiment on dim light shows this particle nature

We call the particle of light a photon. But it is not a classical particle (like a billiard ball) — it is a quantum particle with complex behavior.

Single-Slit Simulation

(Video simulation available in the course material)

When photons pass through a single wide slit:

1. At first (~200 photons), the pattern appears completely random
2. As more photons accumulate (~2000+), a structure emerges — a particle pattern (classical pattern) with no wave-like interference
3. Some photons appear very far from the center — a spread characteristic of diffraction, even with a single slit

Fundamental tenet of quantum mechanics: We can predict the probability for a given event to occur, but we can never predict precisely what the result of a single experiment will be. Randomness is built into the fabric of nature.

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Feynman’s Quantum Theory of Light

Richard Feynman devised a simple set of rules (the path-integral method) for how photons act, described in his book QED: The Strange Theory of Light and Matter.

“Nothing I will tell you is incorrect, watered down, or simplified. You will be told everything about quantum mechanics the way practicing physicists view quantum mechanics.”

Two Critical Concepts

1. An event: Requires measurable initial and final conditions (sources and detectors)
2. Alternative ways: The different ways an event can occur

Developing the ability to distinguish these two concepts is critical for success.

The Arrow Rules

Feynman’s method uses arrows (amplitude vectors) with rules for how they change:

Action	Effect on Arrow
Photon travels through space	Arrow rotates like a stopwatch hand — the rate of rotation determines the photon’s color
Photon reflects	Arrow shrinks by a factor (e.g., 0.2 for glass) and rotates 6 hours (180°)
Photon transmits	Arrow shrinks by a transmission factor (e.g., 0.9798 for glass)
Alternative ways to same event	Add the arrows (vector addition), then square the length → probability

Key Facts About Arrow Rotation

• Color determines rotation rate: Blue rotates faster than green, which rotates faster than red
• The rotation rate changes continuously across the spectrum (including ultraviolet and infrared)
• In glass, the photon moves more slowly, so it takes more time → the arrow rotates more
• Typical rotation rate: approximately 15,000 revolutions per cm in air

Example: partial reflection from a Single Surface

Consider red photons reflecting off glass:

• Arrow for reflection off the top surface: length $0.2$, rotated by 6 hours
• Arrow for reflection off the bottom surface: length $0.2\times 0.9798\times 0.9798=0.192$ (transmits through top, reflects off bottom, transmits back out)

Maximum (arrows aligned):

$$\text{probability}=(0.2+0.192{)}^2=(0.392{)}^2=0.1537=15.37\%$$

Minimum (arrows opposite):

$$\text{probability}=(0.2-0.192{)}^2=(0.008{)}^2=0.000064=0.0064\%$$

As glass thickness changes, the relative rotation changes → the probability of reflection oscillates between 0% and ~16%.

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The Puzzle of Partial Reflection

One problem with Newton’s particle theory: how does a particle “know” whether to reflect or transmit?

• During the day, we see through windows (transmission dominates)
• At night, we see our reflection (reflection becomes visible when transmission light is dim)
• Partial reflection is always present — it’s just hard to see when overwhelmed by transmitted light

Practical Questions from the Lecture

1. What is a source of red photons? Laser pointers — an excellent source of monochromatic (single-color) light. Attenuated to one photon at a time.

2. What is a photon detector? CCD devices (in digital cameras and smartphones) — triggered by single photons. Before CCDs: photomultiplier tubes that amplify a single photon into billions of electrons.

3. Why does the arrow turn like a stopwatch? This is the quantum rule: a photon’s color is determined by how fast the arrow rotates per unit distance traveled.

4. Why does the arrow rotate more in glass? Light travels slower in glass, so the photon takes more time to cover the same distance → more rotation.

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Summary & Deep Questions

• Light always arrives in discrete bundles of the same size — called photons
• These photons are quantum particles, not classical particles — they follow bizarre quantum rules
• The arrow method provides a computational framework for predicting quantum behavior, even without advanced mathematics

Questions to Ponder

1. How do particles interfere with each other to act like waves? Can interference disappear if light is dimmed to one photon at a time?
2. Can a single photon “interfere with itself”? What does that even mean?
3. If light acts like raindrops hitting a surface, why don’t we feel them on our skin?
4. If light is a particle, what does it mean to say it has a wavelength?

Historical Note

The modern theory of light started with Planck (blackbody radiation), but the truly bizarre quantum nature was solved by:

• Einstein (photoelectric effect)
• Bose and Einstein (Bose-Einstein condensation → ultimately led to the laser)

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Key Takeaways

1. Light is fundamentally a quantum particle (photon): Dim light experiments show discrete detection events, not continuous dimming. Photons arrive one at a time, like raindrops — but they are not classical particles.

2. wave-particle duality is real but subtle: Newton (particles), Young (waves), and modern quantum theory all captured partial truths. The photon is neither a classical particle nor a classical wave — it is a quantum object that exhibits both behaviors depending on the experiment.

3. Feynman’s arrow method provides an accessible framework: Photon events are described by “arrows” (probability amplitudes) that rotate, shrink on reflection/transmission, and add vectorially. The probability is the squared length of the final arrow.

4. Color is rotation rate: A photon’s color is determined by how fast its probability amplitude arrow rotates as it travels — a deeper, more physical definition than wavelength alone.

5. Randomness is fundamental, not a limitation of measurement: We can calculate probabilities precisely, but never the outcome of a single quantum event. This is a profound departure from classical determinism.