Malus's law

In plain terms

Light can be polarised, meaning that its wave oscillates in one fixed direction across its line of travel, and a polariser is a filter that passes only light oscillating along its own axis. Pass polarised light through a second polariser tilted at some angle to the first and ask how much gets through. Malus measured it in 1808, and the transmitted fraction is the square of the cosine of the angle between the two axes, so aligned polarisers pass everything, crossed ones at right angles pass nothing, and at forty-five degrees exactly half gets through.

A century later quantum mechanics asked what happens when a single photon, one particle of light, meets that tilted polariser. A photon can't be split, so it either passes or it doesn't, and the answer is the same cosine squared, now as the probability of passing. The Born rule is the general law of quantum mechanics that turns a state into probabilities for outcomes, and here it already was, as Malus had measured it before anyone knew there was such a rule, before the photon, and before probability entered physics at all.

That makes Malus's law the oldest quantitative fact any theory of quantum probability must reproduce, and the simplest test any such theory can face.

In CSD

CSD's machine-checked corpus derives this law rather than assuming it. Among the states of a single photon's polarisation, those that lead to the photon passing form a region, and that region occupies exactly the cosine-squared fraction of the whole state space when the space is measured with the programme's forced measure, the one even-handed notion of volume it admits. It's the smallest complete instance of the entire programme, the one case where the full argument, regions and volumes and all, can be followed by hand on the back of an envelope.

That's also its danger, flagged wherever the corpus uses it. Polarisation is a two- level system, and the two-level case is deceptively special, as its state space is a sphere, which makes the geometry so tractable that it's easy to mistake the special case for the general one. It's the smallest instance, the best pedagogy, and the worst extrapolation.

Mathematically

The transmitted fraction equals the cosine squared of the angle between the polariser settings, which is identically the Born weight for the corresponding spin measurement.

In the corpus it's a volume ratio, the Fubini-Study measure of the pass region over the total, computed exactly on CP^1.

The name

Etienne-Louis Malus (1775-1812) was a soldier-scientist of the Napoleonic era, a member of the Ecole Polytechnique's first cohort and then an engineer on the Egyptian campaign, where he survived plague at Jaffa and never fully recovered his health.

The discovery is the stuff of legend, and for once the legend is documented. In 1808, in his rooms on the Rue d'Enfer, he idly examined the sunset reflecting off the windows of the Luxembourg Palace through a calcite crystal, and he noticed the doubled image dimming and brightening as he rotated the crystal. Reflected light was polarised, and nobody had known. He named polarisation, won the Academy prize for double refraction, and was dead at thirty-six.

Related entries

• Born ruleQuantum probabilities aren't a law of nature in CSD, as they're the sizes of regions in the space of states, measured with a ruler nobody was free to choose.
• Stern-Gerlach experimentAtoms sent through an uneven magnetic field split into two clean beams instead of a smear, and sorting them twice along different axes undoes the first sorting, so the whole strangeness of quantum measurement sits in one 1922 apparatus.
• Mach-Zehnder interferometerSplit a single particle down two paths and recombine them, and where it lands depends on both paths, including the one you can prove it never took.
• Bloch sphereThe states of the simplest quantum system form an ordinary ball, which is a lovely accident of low dimension and the single most misleading picture in quantum mechanics.
• Born volume ratioIn every finite dimension the Born weight of an outcome equals a ratio of volumes in the space of quantum states, for every outcome and every state, with no genericity condition attached.