Lecture 4: The Two-Slit Experiment (Stern-Gerlach Analog)
Learning Goals
- Work confidently with Stern-Gerlach analyzers and analyzer loops, understanding the critical differences between them
- Experience quantum weirdness first-hand — quantum interference in action
- Identify situations where quantum interference occurs and where it is destroyed (the principle of complementarity)
prerequisite: Be clear on the difference between Stern-Gerlach analyzers (which measure) and analyzer loops (which erase measurements).
The Two-Slit Analog: analyzer loop with Both Branches Open
The Setup
- Source: Atoms in known +z state
- Analyzer loop: Oriented along x (rotated 90° from z), both branches open
- Final analyzer (B): Oriented along z
- Detectors: D1 (top/+z exit) and D2 (bottom/−z exit)
The Central Question
What happens when atoms go through a fully open analyzer loop? Does it truly leave the atom unchanged, or does the measurement inside the loop affect the outcome?
Two Competing Analyses
Analysis A: Classical Probability Reasoning
Step-by-step, treating each branch as a definite path:
- Atom leaves source in +z state
- Enters analyzer A (oriented along x) — forced into +x or −x with probability ½ each
- Passes through gate (both open), enters eraser with definite x-state
- Eraser redirects atom — emerges with the same x-state
- Enters analyzer B (oriented along z) — now measuring z on an atom with a known x-state
- Since x ⟂ z, both outcomes equally likely: 50% D1, 50% D2
Conclusion: Both detectors should detect the same number of atoms.
Analysis B: Quantum (Which-Path) Reasoning
Treating the analyzer loop as a black box:
- Atom leaves source in +z state
- Enters analyzer loop — we cannot know what happens inside
- Because both paths are indistinguishable, the analyzer loop must leave the atom unchanged (otherwise we could deduce which path it took)
- Atom emerges from analyzer loop still in +z state
- Enters analyzer B (oriented along z) — measured as +z every time
- 100% D1, 0% D2
Conclusion: All atoms go to D1.
The Confrontation
Monumental disagreement! The two analyses predict completely different results. One must be wrong.
The Experiment
(Video demonstration:
sge_exp_1-10_al_classical_vs_quantum.html)
Result: Analysis B (quantum) is correct — 100% of atoms go to D1.
What’s Wrong with Analysis A?
Analysis A was completely logical, but its fatal assumption was:
“The atom goes through one branch or the other.”
If the atom took the +x branch or the −x branch (not both), then Analysis A’s conclusion would be inescapable. Since the experiment contradicts A, we must conclude:
The atom does NOT take the positive x path OR the negative x path. It somehow takes BOTH paths!
As Sherlock Holmes said in The Sign of Four (1890):
“When you have eliminated the impossible, whatever remains, however improbable, must be the truth.”
Watching the Atoms: Pass-Through Detectors
Part 1: Watching Both Branches
To test the “both paths” idea, we place pass-through detectors on each branch — detectors that can sense an atom while letting it pass through.
Question: If the atom goes through both branches, wouldn’t both detectors flash simultaneously?
Result: This never happens. Whenever we detect an atom, we always find it in only one branch.
And crucially, the final result changes: now D1 and D2 each get ~50% of atoms — the classical probability result returns!
Part 2: Watching a Single Branch
What if we watch only one branch (replace the other detector with a straight pipe)?
Result: Exactly the same as watching both branches — 50/50. Why?
- If the atom is detected on the +x branch → it has a definite +x state → 50% D1, 50% D2
- If the atom is not detected → we still know it went through the −x branch → same result
(Video demonstration:
sge_exp_1-11_watching.html)
The crucial insight: It’s not that the detectors “disturb” the atom — it’s that any way of obtaining which-path information, even in principle, destroys quantum interference.
The “Can I Watch?” Summary
The Quantum Mystery in a Nutshell
| Scenario | Which-path info? | Result |
|---|---|---|
| Both branches open, no watching | No | 100% D1 (interference) |
| Both branches open, watching one or both | Yes | 50% D1, 50% D2 (no interference) |
What We’ve Learned
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When we DON’T know which path: The atom somehow takes both paths, and the analyzer loop acts as if it doesn’t exist. All atoms go to D1.
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When we DO know which path (by watching): We see the atom go through one path or the other — but the final results change to 50/50.
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Nature is clever: When we think we’re clever enough to catch her in the act, she changes the results. Watched atoms behave differently than unwatched atoms.
The Formal Name: Quantum Interference
This phenomenon is called quantum interference. It is:
- Incredibly strange, but 100% true
- We can describe how nature acts, but not why
- This led Feynman to his famous statement: “No one understands quantum mechanics” — meaning no one understands how nature can act this way, even though we can calculate the results with high accuracy
This is the quantum mystery. Now you know what it is.
The Principle of Complementarity
The analyzer loop experiment reveals a deep principle:
Complementarity: Quantum interference and which-path information are mutually exclusive. You cannot have both simultaneously.
- If which-path information is available (even in principle) → no interference → classical probability applies
- If which-path information is erased → interference emerges → quantum behavior dominates
This is why the original analyzer loop (no watching) works: by “erasing” the which-path information through identical branches, it forces the atom to remain in its original state, and quantum interference is preserved.
The Delayed-Choice Variation
As a preview of even deeper strangeness:
We can set up a pass-through detector on one branch with adjustable sensitivity — so it only works some fraction of the time, disturbing the atoms less.
What happens? The results become a weighted mixture of the quantum and classical outcomes, proportional to how often we obtain which-path information. This foreshadows the Wheeler delayed-choice experiment, where the decision to observe or not can be made after the particle has already passed through the region of interference.
Key Takeaways
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The quantum two-slit analog produces interference: When both paths through the analyzer loop are open and indistinguishable, all atoms emerge in their original state — the analyzer loop is completely transparent.
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Which-path information destroys interference: Any measurement (even passive detection on a single branch) that reveals which path the atom took collapses the quantum behavior and restores classical probability (50/50).
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The atom takes “both paths” when unobserved: Classical reasoning (Analysis A) fails because it assumes the atom goes through one definite branch. The correct quantum description requires accepting that in the absence of which-path information, the atom’s path is indeterminate — it effectively traverses both.
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Complementarity: You cannot simultaneously observe quantum interference and know which path the particle took. These are complementary aspects of reality.
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This is the quantum mystery: Nature behaves in a way that defies classical intuition. We can calculate results with extraordinary precision, but why nature works this way remains deeply puzzling.