Connecting Malus's law and the single photon

Problem

In the polariser experiment it is observed that single photons either pass through or are absorbed, while for an intense beam Malus’s law $I' = I\cos^2\alpha$ holds. Explain how the two descriptions connect, and why the outgoing photon “is not the same” as the incident one.

Solution

The single photon is probabilistic. A photon polarised at an angle $\alpha$ with respect to the polariser’s axis cannot pass “halfway”: it either passes through entirely or is absorbed. Quantum mechanics prescribes that it passes with probability

$P(\text{passes}) = \cos^2\alpha$

The intense beam recovers Malus. With $N$ identical photons, on average $N\cos^2\alpha$ pass through. The intensity is proportional to the number of transmitted photons:

$I' = I\,\cos^2\alpha$

The macroscopic Malus law emerges as the statistical average over the huge number of individual all-or-nothing events. It is a fine example of how continuous classical behaviour is the average of discrete quantum events.

Why it is “not the same” photon. The outgoing photon is polarised along the polariser’s axis, no longer at $\alpha$: the measurement has projected (collapsed) its state. It has lost all memory of the initial polarisation — a second polariser parallel to the first would now let it pass with probability 1. The quantum state has changed: in this sense it is not “the same” photon as before.

$\ev{\text{Malus} = \text{statistical average of quantum "all-or-nothing"; the outgoing photon is re-polarised by the measurement}}$

Links

Topics: Fisica quantistica Concepts: Polarizzazione e legge di Malus · Fotone · Dualismo onda-particella Objects: Polarizzatore