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The polarising filter

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To see the theme of measurement concretely, an experiment within reach of a desk drawer is enough: a polarising filter (polarised sunglass lenses are one example). A polariser is a material that lets through only light with electric field parallel to a certain direction d^\uv{d} — the axis of the filter — and filters out the rest.

Light polarised at an angle α\alpha strikes the filter with axis d^\uv{d}: it emerges polarised along d^\uv{d} and with reduced intensity according to Malus’s law.

The “wave” face: Malus’s law

"Wave" behaviour — Malus's law

Linearly polarised light with field E⃗\vv{E} forming an angle α\alpha with the filter’s axis d^\uv{d} emerges with intensity ∣I⃗′∣=∣I⃗∣ cos⁡2α\ev{\abs{\vv{I}'} = \abs{\vv{I}}\,\cos^2\alpha} and with polarisation rotated until it aligns with d^\uv{d}.

For an intense beam this is the complete description: the intensity decreases continuously from ∣I⃗∣\abs{\vv I} (filter aligned, α=0\alpha = 0) to zero (filter crossed, α=90°\alpha = 90°).

The “photon” face: measurement

If, however, we send the photons one at a time, ”cos⁡2α\cos^2\alpha of the intensity” no longer makes sense: a single photon either passes or does not pass. The filter then measures the polarisation, and the outcome is probabilistic:

  • the photon passes with probability cos⁡2α\cos^2\alpha;
  • the photon is absorbed with probability sin⁡2α\sin^2\alpha.

If it passes, its polarisation changes: it becomes parallel to d^\uv{d}, “aligning” with the filter. The emerging photon is no longer “the same” as before.

Here is theme 4 in action

The polariser is the clearest example of quantum measurement theory:

  1. Polarisation is a quantised observable: the outcome is binary (passes or not), not continuous.
  2. The measurement is probabilistic: we cannot predict whether a specific photon will pass, only the probability cos⁡2α\cos^2\alpha.
  3. The photon, if it passes, changes state: before it had polarisation α\alpha, afterwards 0°0° with respect to d^\uv{d}. It is the “cat after the photo”.
  4. Over many photons the probabilities translate into intensities and we recover the classical Malus law (the “wave”).

Four themes — quantisation, duality, uncertainty, measurement — intertwine in a single elementary experiment.

The role of probability

The probabilistic interpretation is due to Max Born (1926): the “square” of the amplitude of a particle’s wave represents the probability of finding it at a point. For the polariser, the square of the cosine of the angle becomes the probability that a photon passes. Einstein never fully accepted this idea — “God does not play dice with the universe”, he wrote — but a century of experiments has proved him wrong.

Einstein versus Bohr: dice, ghosts, entanglement

The most famous debate of twentieth-century physics, between Albert Einstein and Niels Bohr, lasted thirty years and redefined what “understanding nature” means. Bohr, father of the Copenhagen interpretation (1927), maintained that quantum mechanics is a complete theory: probability is not the physicist’s ignorance, it is the correct description of the microscopic world. Einstein was never convinced. His objections:

  • “God does not play dice with the universe” (1926, letter to Max Born). Bohr’s reply: “Stop telling God what to do”.
  • The ideal experiment of the 1927 Solvay Congress: a clock in a box that weighs an outgoing photon, to violate the time/energy pair. Bohr countered within a night, using Einstein’s own general relativity to find the error in the calculation.
  • The EPR paradox of 1935 (Einstein, Podolsky, Rosen): two particles that “talk” to each other instantaneously across space would violate relativity — his “spooky action at a distance”.

In the 1960s John Stewart Bell turned the EPR paradox into a testable inequality. The real experiments — Aspect 1982, Hensen 2015 — measured the quantum value with very high precision: nature really does violate Bell’s inequality. Classical “local realism”, however intuitive, is simply false; entanglement is a real phenomenon (Kumar 2010). Bohr was right on the factual level, Einstein wrong, but his objection stimulated the most important line of research into the foundations of twentieth-century quantum mechanics.

Topics: Fisica quantistica Concepts: Polarizzazione e legge di Malus · Fotone

Related exercises: Raccordo Malus e singolo fotone · Problema — Polarizzatore e conteggio di fotoni · Problema — Collasso della funzione d’onda al polarizzatore

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