Lecture 5: complementarity

“The student suddenly says to himself, ‘I understand quantum mechanics,’ or rather he says, ‘I understand now that there isn’t anything to be understood.‘” — Freeman Dyson, Innovation in Physics, Scientific American (1958)

Learning Goals

• Review the two-slit experiment with analyzer-loops (watched and unwatched)
• Identify how one can erase which-way information and restore interference
• Describe how complementarity can be partially destroyed or reinstated

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Complementarity and Which-Way Information

Bohr’s Complementarity Principle

Niels Bohr developed the theory of complementarity, which states:

• If one designs an experiment that tests for either the particle or the wave properties of a quantum object, one cannot simultaneously measure the other (complementary) property.

A more modern formulation focuses on which-way information:

• If a quantum device has two paths that are indistinguishable from each other:
  • If we know which way the particle went → it adopts the properties associated with that path
  • If we do not know which way → the particle retains whatever properties it had prior to entering the device

This behavior is exactly what we exploit in the Stern-Gerlach analyzer loop.

Gentle Measurement and superposition

• Historically, it was thought that “watching” a particle inevitably disturbed it enough to change the experiment.
• Modern experiments have found ways to determine “which way” much more gently — so gently that the particle cannot be disturbed in the old classical sense, yet the quantum behavior still changes.
• This shows that knowing which-way information is the truly fundamental principle, not physical disturbance.

Superposition of quantum states: A particle can traverse two indistinguishable paths at the same time — not as a “half particle” splitting and rejoining, but as a whole particle simultaneously experiencing both paths, with a 50% chance to be found on each if we dared to look.

• There is no classical analog to this behavior, and the lack of a simple visualization is one reason students find quantum mechanics frustrating.

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Wheeler’s Delayed Choice Experiment

The Central Question

John Wheeler (Feynman’s Ph.D. adviser) asked:

When does a quantum particle realize it is being watched?

Can we set up an experiment and decide after the atom passes through the two paths whether or not we want to watch? The answer is yes.

This raises a deep philosophical question: does the choice get communicated to the quantum particle backward in time? The verdict is still out.

The Conspiracy Theory

A classical explanation (the “conspiracy theory”) proposes:

• The atom knows the arrangement of the apparatus in advance
• If the final analyzer is vertical → the atom takes both paths
• If the final analyzer is horizontal → the atom chooses one path, even if unwatched

Quantum mechanics predicts instead: an unwatched atom simply goes through both branches regardless of how we set up the experiment.

Four Scenarios of the Analyzer Loop

Scenario	Trailing Analyzer	Detection	Result
Unwatched (V)	Vertical	All atoms → D1	Atom preserves original $|+z⟩$ state
Watched (V)	Vertical	50% → D1, 50% → D2	Watching changes state to $|+x⟩$ or $|-x⟩$
Unwatched (H)	Horizontal	50% → D1, 50% → D2	Atom unchanged, randomly measured in x-basis
Watched (H)	Horizontal	50% → D1, 50% → D2	No difference from watched case — tempting but misleading

Key insight: The horizontal-analyzer results alone could be explained classically (atom has definite state at all times). We need the perpendicular-analyzer results to reveal quantum behavior.

The Delayed Choice Test

1. Start with a horizontal trailing analyzer
2. Atom passes through the analyzer loop (unwatched)
3. After atom exits the loop, quickly rotate the trailing analyzer to vertical

Predictions:

Theory	Prediction
Conspiracy	Atom chose a definite $|\pm x⟩$ state → 50% each at D1, D2
Quantum Mechanics	Atom preserved $|+z⟩$ state → 100% at D1, 0% at D2

Experimental result: Every time this kind of experiment is performed, quantum mechanics is verified to be correct. There is no evidence for such conspiracies.

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Partial Destruction of Complementarity

Classical vs. Quantum Predictions

For classical-like explanations, one can fit a classical equation to quantum observations:

$$P(+x)=\frac{1}{2}(1+\cot \theta )$$

Since $P(+x)\le 1$ (it is a probability), this requires $\theta \ge 45^{\circ }$. This classical constraint does not hold in quantum mechanics, showing we can rule out classical hidden-variable descriptions.

Tagging Atoms

Tagging atoms provides a concrete way to mark which-way information:

• Exciter: Shine light of the right energy to excite an atom from its ground state to a higher-energy state

• De-exciter: Shine light on an already-excited atom to coax it to emit a photon and return to the ground state

• An excited atom is distinguishable from a ground-state atom — they are no longer alike

• This means we have which-way information, destroying interference

• If the tag is later removed (the which-way information is erased), interference can be restored

(Video demonstration of this with light, using polarization as the tag, will be shown later in the course.)

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Key Takeaways

1. Complementarity is fundamentally about which-way information: If we know which path a quantum particle took, it behaves like a particle; if we don’t, it can exhibit wave-like superposition. Physical disturbance is not the root cause — information is.

2. Superposition has no classical analog: A quantum particle can simultaneously traverse two indistinguishable paths as a whole particle, not as a split entity. This defies classical intuition.

3. Delayed choice experiments rule out classical conspiracy theories: The choice of measurement can be made after the particle has passed through the interaction region, and quantum mechanics still correctly predicts the outcome — the particle does not “know in advance” which experiment will be performed.

4. Which-way information can be erased: Tagging atoms (making them distinguishable) destroys interference, but removing the tag can restore quantum behavior. This is the principle behind quantum eraser experiments.

5. The probabilistic nature is fundamental: We can predict probabilities of events, but never the precise outcome of a single quantum experiment.