Problem — Collapse of the wave function at the polarizer

Problem

A photon polarised at $30^\circ$ with respect to the axis of a vertical polarizer is incident on the instrument. Discuss what happens to its polarisation, in the single event and on average over many repetitions.

Solution

Single event. The photon is not attenuated “a little”: it is indivisible. It either passes entirely or is absorbed entirely. There is no such thing as ”$\cos^2 30^\circ$ of a photon”. The outcome of the single event is genuinely random — not predetermined.

Probability (quantum Malus law). The probability that the photon crosses the polarizer is

$P = \cos^2\alpha = \cos^2 30^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} = 0{,}75$

$\ev{P = \cos^2 30^\circ = 0{,}75}$

Collapse. If the photon passes, its polarisation state collapses: it is no longer at $30^\circ$, but comes out polarised exactly along the axis of the filter. The measurement has projected the state onto the axis. If it does not pass, it is absorbed (together with its energy).

Average over many repetitions. Repeating with a great many identical photons at $30^\circ$, a fraction $0{,}75$ passes and $0{,}25$ is absorbed. The transmitted intensity is thus $I' = 0{,}75\,I$, recovering the classical Malus law as a statistical average of the discrete quantum events. The single photon is probabilistic; the ensemble reproduces the classical wave result.

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