Lecture 9: The double slit Experiment

“We choose to examine a phenomenon which is impossible, absolutely impossible to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery… Any other situation in quantum mechanics, it turns out, can always be explained by saying, ‘You remember the case of the experiment with the two holes? It’s the same thing.‘”

— Richard Feynman, The Character of Physical Law (1965)

Learning Goals

Articulate the bizarre results of the two-slit experiment and calculate them using quantum rules.
prerequisite: You must be familiar with quantum rules for light, partial reflection, and single-slit diffraction.
After this lesson, you will be able to:
- Explain what the quantum mystery is.
Stretch goal: Generalize these techniques to more complicated situations.

Demonstration: single slit vs. Double Slit

Single Slit

A central broad region with additional, dimmer diffraction fringes on the wings.
Origin of this behavior was covered in the previous lecture.

Double Slit

The same single-slit fringes are present, but with additional bright fringes within each diffraction band.
The double-slit image is four times brighter than the single-slit image.

With single-photon-sensitive detectors, the initially random distribution of individual photon hits gradually builds up into the interference fringe pattern. Each photon arrives one at a time — yet collectively, they trace out a wave interference pattern.

(Video demonstrations — MIT “Double Trouble” and University of Leiden “Photons Can Do That?“)

The Quantum Mystery: Part 1

When the light intensity is dimmed so that only one photon is in the apparatus at any given time:

Individual photons hit the screen at seemingly random positions.
Over time, these individual hits build up to show the interference fringes that are characteristic of a wave pattern.

The Baffling Questions

Wave interference patterns normally arise when two sources of waves interfere constructively and destructively.
If there is only one photon in the apparatus at a time, how does the photon interfere with itself?
Does the photon go through one slit or the other? Or does it split and go through both?
Can a particle interfere with itself?

The Quantum Mystery: Part 2 — Watching the Photon

To find out what the photon is doing, we add detectors near the slits:

detector A at the top slit
Detector B at the bottom slit
Detector D on the screen (as before)

The Original Experiment (No Watching)

One event: Photon leaves the source → detected on the screen.
Two alternative ways: Through the top slit or through the bottom slit.
Two arrows are added → they can align (constructive) or cancel (destructive).
Square the final arrow → probability ranges from 0 to 4% (4× the single-slit maximum).

The Watched Experiment

Two events:
1. Photon → Detector A + Detector D
2. Photon → Detector B + Detector D
Each event has only one way it can occur → one arrow each.
Each arrow squared gives a constant probability. Add them → 2% everywhere.
The interference fringes disappear!

What Actually Happens

The photon always goes through either the top slit or the bottom slit.
It never breaks into two and goes through both slits simultaneously.
All three detectors never go off at the same time.
But the interference pattern is gone — replaced by a simple particle pattern (the sum of two single-slit particle patterns).

The act of observing the experiment changed its results.

Why This Feels Unfair

Nature does not reveal her secrets:

A particle acts like a wave (produces interference fringes) when unwatched.
When we try to observe how it does this, it becomes a simple classical particle.

Follow the quantum rules and you will never have a problem — just be sure to carefully determine each event and the alternative ways it can occur.

The Heisenberg Uncertainty Principle Explanation (Niels Bohr)

If we want to localize the photon’s position well enough to determine which slit it passed through:

We must hit it with a probe particle whose quantum wavelength is small enough to resolve the slit separation.
A small-wavelength particle has large energy and momentum.
When it scatters off the photon, the photon receives a huge “kick” — it is greatly disturbed.
This disturbance destroys the quantum interference.

Analogy: Trying to track a ping-pong ball by shooting BB pellets at it — each BB strike violently changes the ball’s trajectory.

Imperfect Detectors: A continuous Transition

What if we use a detector with less than perfect efficiency (e.g., 36%)?

Detected photons → act like particles (constant probability curve).
Undetected photons → act like waves (oscillating probability curve).
The efficiency determines how we weight each piece.
The pattern continuously transitions from full oscillation (0–4%) to flat (2%).

This demonstrates that quantum behavior is not “all or nothing” — it exists on a continuum controlled by measurement strength.

The Principle of complementarity

Niels Bohr (1927): Whenever we can tell which slit the particle goes through, the quantum interference disappears and we see the particle pattern.

Key facts:

Every experiment that has tried to determine how a quantum particle “interferes with itself” has failed.
No experiment has ever violated this principle.
Related ideas: quantum eraser and delayed choice experiments explore the boundaries of this principle.

Can We Truly Understand Quantum Mechanics?

The Pragmatic View

We have a set of quantum rules that work perfectly:

Identify the event(s).
Find the different alternative ways each event can occur.
Assign an arrow (probability amplitude) to each way.
Sum the arrows to get the final arrow.
Square the final arrow to get the probability.
Add probabilities from all events to get the total probability.

Every experiment yields to these rules with perfect accuracy.

The Philosophical View

We cannot predict what will happen to one particular photon — only the statistical distribution of many.
Quantum mechanics inherently contains an element of randomness.
Einstein’s objection: “God does not play dice with the universe.”

As Feynman said: “I think I can safely say that nobody understands quantum mechanics.”

He meant: no one truly understands how a single particle manages to “interfere with itself.” But we can predict what it does with extraordinary precision.

The takeaway: Treat quantum mechanics as a cookbook of rules that quantum particles follow. The predictive power is what matters for science — understanding the origin of the rules is a deeper problem we haven’t solved.

Beyond Two Slits: Three Times the Charm

Adding a third slit reinforces the same lessons:

The quantum interference effects are surprisingly robust — they occur in nearly every situation where we allow them.
They are also fragile — it’s easy to destroy them by adding a detector.

With three narrow slits (each giving 0.05 arrow size):

Unwatched: maximum probability = 2.25% (9× a single slit, since $3^{2} \times 0.0 5^{2} = 0.0225$ )
Shape of the pattern is the same as the two-slit case but with more complex interference structure.

Summary

The two-slit experiment is the preeminent example of quantum mechanics. It contains deep mysteries we may never fully unravel.
We can create abstract theories that predict quantum behavior with great accuracy.
You now possess the knowledge of what quantum mechanics is and how to describe its behavior.
Future topics building on this: quantum teleportation, quantum computing, quantum encryption, quantum entanglement.
Next module: quantum Zeno effect, interaction-free measurement, and the Hong-Ou-Mandel effect.

Key Takeaways

Single particles produce interference patterns: When photons pass through a double slit one at a time, they still build up an interference pattern — each photon somehow “interferes with itself.”
Observation destroys interference: Placing detectors at the slits to determine which path the photon takes causes the interference pattern to collapse into a simple particle pattern. The act of measurement changes the outcome.
The quantum rules work perfectly: Identify events, enumerate alternatives, assign arrows, sum, and square. This procedure predicts every experimental result with perfect accuracy, even if the underlying reality remains mysterious.
Complementarity is universal: No experiment has ever succeeded in simultaneously revealing both the particle’s path and its wave interference. This is not a technological limitation but a fundamental principle.
Imperfect measurement yields a continuum: Quantum behavior transitions continuously from wave-like to particle-like depending on measurement strength — it is not a binary, all-or-nothing phenomenon.

Historical note: While the original two-slit experiment with light dates to Thomas Young (early 1800s), it was Richard Feynman who popularized it as a teaching tool for quantum mechanics in the 1960s and beyond.

📝 夔嵬的笔记

探索

Lecture 9: The Double Slit Experiment