Lecture 7: Exploring the Quantum Nature of Light

“If quantum mechanics hasn’t profoundly shocked you, you haven’t understood it yet.” — Niels Bohr, Nobel Prize Winner in Physics, 1922

Learning Goals

• Determine how light, while acting as a particle, manages to partially reflect off glass
• Use the quantum theory to describe partial reflection
• Determine the physical origin of the colors on soap bubbles and oil slicks
• Describe how one can measure the wavelength of light simply by measuring an angle

Prerequisites: Familiarity with the photon as a particle of light and with the quantum rules from Lecture 6.

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The Problem of Partial Reflection

Why Classical Intuition Fails

One might think glass has “holes” (like a screen) — photons pass through holes when they transmit and bounce off the mesh when they reflect. But:

• Thicker glass can transmit more than thinner glass — hard to explain with a hole model
• Reflection probability oscillates with thickness — this is what is actually observed
• Polishing glass makes it more reflective — inconsistent with the idea of holes letting photons through

So if glass does not have holes, how does a particle manage partial reflection?

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Reflection from a Glass Slab

Thought Experiment Setup

This is a thought experiment — it cannot be actually performed, but we can imagine how it would work and use the quantum theory to explain the results.

• Light is incident vertically (normal incidence) — the diagrams show paths at small angles only for visual clarity
• We consider light reflecting off a glass slab and determine how the arrow method explains the observed behavior

Two Paths for Reflection

For a photon to be detected above the glass (reflection event):

Path	Description	Arrow Length
Top surface	Reflects directly off the top of the glass	$0.2$
Bottom surface	Transmits through top, reflects off bottom, transmits back through top	$0.2\times 0.9798\times 0.9798=0.192$

The bottom path involves: transmit (×0.9798) → reflect (×0.2) → transmit (×0.9798).

Reflection Probability

Maximum (arrows aligned):

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

Minimum (arrows opposite):

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

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Transmission Through Glass

Two Paths for Transmission

For a photon to be detected below the glass (transmission event):

Path	Description	Arrow Length
Direct	Transmits straight through (no reflections)	$1\times 0.9798\times 0.9798=0.9600$
Two reflections	Transmits → reflects off bottom → reflects off top → transmits out bottom	$1\times 0.9798\times 0.2\times 0.2\times 0.9798=0.0384$

Transmission Probability (Two-Path Approximation)

Maximum:

$$P_{\text{max}}=(0.9600+0.0384{)}^2=(0.9984{)}^2=0.9968=99.68\%$$

Minimum:

$$P_{\text{min}}=(0.9600-0.0384{)}^2=(0.9216{)}^2=0.8493=84.93\%$$

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The Probability Problem

conservation of Probability

A photon must either reflect or transmit — the probabilities should sum to 100%:

Case	Reflection	Transmission	Sum
Max reflection / Min transmission	$15.37\%$	$84.93\%$	$100.30\%$
Min reflection / Max transmission	$0.0064\%$	$99.68\%$	$99.69\%$

The sums don’t add up to exactly 100%! This is not round-off error — there is a genuine problem with only considering two paths.

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Resolving the Problem: Multiple Reflections

The Missing Paths

Our quantum rules say: identify all alternative ways an event can occur. We neglected paths with more than one internal reflection!

For reflection (odd number of reflections):

• 1 reflection: $0.2\times 0.9798^2=0.192$ (already calculated)
• 3 reflections: $0.2^3\times 0.9798^2=0.2\times 0.2\times 0.2\times 0.9798^2=0.00768$
• 5 reflections: $0.2^5\times 0.9798^2=0.0003072$
• General: $2n+1$ reflections → arrow length $0.96\times (0.2{)}^{2n+1}$

For transmission (even number of reflections):

• 0 reflections: $0.9798^2=0.9600$ (already calculated)
• 2 reflections: $0.9798^2\times 0.2^2=0.0384$ (already calculated)
• 4 reflections: $0.9798^2\times 0.2^4=0.001536$
• 6 reflections: even smaller

Case 1: Minimal Reflection (Complete Cancellation)

When the arrow rotations all line up (integer number of revolutions for the glass transit):

• All bottom-reflection arrows point in the same direction
• Their sum (geometric series):

$$0.96\times [0.2+0.2^3+0.2^5+\dots ]=0.96\times 0.2\times \frac{1}{1-0.04}=0.2$$

• The top-reflection arrow (0.2, rotated by 6 hours) points opposite to this sum
• Total: $0.2-0.2=0$ → zero reflection!
• Transmission: $0.96\times [1+0.04+0.04^2+\dots ]=\frac{0.96}{1-0.04}=1$ → 100% transmission!

Case 2: Maximal Reflection

When the transit rotation corresponds to 6 hours:

• Bottom-reflection arrows alternate in direction
• Their sum:

$$0.96\times [0.2-0.2^3+0.2^5-\dots ]=\frac{0.96\times 0.2}{1+0.04}=0.1846$$

• Top reflection (0.2) adds to this → total: $0.2+0.1846=0.3846$
• Probability: $0.3846^2\approx 14.8\%$ (maximum possible for glass with 4% per-surface reflection)

Corresponding transmission:

$$P={\left(\frac{0.96}{1+0.04}\right)}^2=0.9231^2\approx 85.2\%$$ $$14.8\%+85.2\%=100\%✓$$

Probability is conserved when all multiple reflections are included!

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The Physical Origin of Soap Bubble and Oil Slick Colors

(Images: soap bubble by Iman Sadeghi; oil slick by Jim Freericks)

The colors on soap bubbles and oil slicks come from partial reflection of light — exactly the quantum phenomenon we have been analyzing.

How It Works

1. A particular thickness of film may reflect red light completely while transmitting blue or green light
2. The color that is primarily reflected or transmitted depends on the thickness of the film
3. As thickness changes (due to fluid movement, evaporation, or gravity):
   • The relative rotation between different reflection paths changes
   • Different colors (different rotation rates) constructively or destructively interfere
   • Different parts of the film display different colors

Key Facts

• Soap films and oil layers behave just like glass — they have interfaces with air and/or water
• The film thickness acts like the glass thickness in our calculations
• Different colors (photons with different rotation rates) will have different constructive/destructive interference conditions for the same thickness
• This is how interference colors arise in thin films

Measuring Wavelength from an Angle

Since the constructive/destructive interference condition depends on:

• Film thickness
• Angle of observation
• Rotation rate (color) of the photon

One can determine the wavelength of light simply by measuring the angle at which a particular color constructively reflects.

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Summary

• The quantum arrow method successfully explains partial reflection — the photon explores all possible paths simultaneously
• Including all multiple-reflection paths is essential for probability conservation
• The oscillation of reflection probability with glass thickness is a direct consequence of the arrow rotation rule
• Soap bubbles and oil slicks are natural demonstrations of thin-film interference, explained by the same quantum theory
• Different colors reflect or transmit preferentially depending on film thickness — each color has a different rotation rate

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Key Takeaways

1. Partial reflection is explained by quantum path integrals: A photon does not “decide” to reflect or transmit — it takes all possible paths simultaneously, and the probability amplitudes for these paths combine (via vector addition of arrows) to determine the observable probability.

2. Multiple reflections are essential for completeness: The initial two-path calculation gives probabilities that don’t sum to 100%. Including all possible multiple-reflection paths (using geometric series) restores probability conservation — a beautiful validation of the theory.

3. The arrow rotation rule is the key to understanding thin-film interference: Different colors (photon rotation rates) have different constructive/destructive interference conditions for the same film thickness, producing the vivid colors of soap bubbles and oil slicks.

4. Transmission and reflection are complementary: When transmission is maximized, reflection is minimized, and vice versa — the total probability is always conserved. This emerges naturally from the quantum rules.

5. The geometric series provides exact results: The sum of all multiple-reflection paths converges to a geometric series, yielding exact probabilities that satisfy conservation laws. This is a preview of more advanced path-integral calculations.