How do arrows turn into the chance of an event?
Everything in this module now comes together. Quantum physics computes the chance of an event with three steps, the rule Feynman explained in QED:
1. Every way the event can happen gets an arrow, called a probability amplitude. Along each way, the arrow is built by multiplying the arrows of each step. 2. If the ways cannot be told apart, add their arrows. 3. The probability of the event is the square of the length of the final arrow.
Take the interferometer of the lab: a photon has two paths to reach detector D1. Each path gets an arrow of length 1/2 (each of the two beam splitters on the way shortens it by 1/√2). If the two paths have the same length, the two arrows point the same way: they add to length 1, and the probability is 1^2 = 100%. Make one path longer and its arrow turns.
One path is made longer by exactly half a wavelength. What is the chance that the photon reaches D1? Half a wavelength turns one arrow by 180°: the two arrows point in opposite directions and cancel. The probability is 0, and the photon always goes to D2.
And if we can tell which path the photon took, for example because the paths leave a mark on it? Then the ways can be told apart, and the rule changes: square each arrow first, then add the probabilities. Each path gives (1/2)^2 = 1/4 at each detector: 50% on each side, whatever the lengths. The interference is gone.
This is exactly what you will see in the lab: interference when the paths cannot be told apart, none when they can. Probabilities from coins, arrows that add and turn, and squares of lengths: that is all the mathematics behind module 5. Before going there, module 4 explains the instruments on the optical table.
Two indistinguishable paths with phase difference φ: ψ = ½ (1 + e^iφ) at D1, and P(D1) = |ψ|^2 = ½ (1 + cos φ) = cos^2(φ/2).
At D2 the arrows meet with an extra half turn: P(D2) = sin^2(φ/2). The two add up to 1, as probabilities must.
Distinguishable paths: P = |ψ1|^2 + |ψ2|^2 = ¼ + ¼ = ½, with no dependence on φ.
This is the Born rule: probability = |amplitude|^2.
For each indistinguishable way, multiply arrows along it; add the arrows of all the ways; the probability is the square of the length. That is quantum physics in one rule.