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Malus' law state that the intensity of light through a polarized filter is given by $$I=I_0\cos^2(\theta)$$ where $I_0$ is the original intensity and $\theta$ is the angle between the polarization of the original beam of light and the axis of the polarizer.

This law was formulated in the 18th century mainly on empirical grounds. Nowadays we can derive something similar for the probability of transmission of a photon (instead of the intensity) using quantum mechanics. Clearly quantum mechanics recovers Malus' law for a large number of photons.

I am trying to understand if Malus' law is indeed a classical law or is it purely quantum mechanical. I could not find a straightforward answer. Is there a way to derive such a law from classical grounds?

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The definition of terms "classical" and "quantum" is not always clear-cut. Of course one might say that everything is quantum and nothing is classical, but I think the distinction is worth making; it draws our attention to those parts of quantum theory that are well-captured by classical concepts.

The notion of a continuous field, characterised by a small set of real numbers at each point, and able to be observed without disturbing it, is "classical". Classical electromagnetism (Maxwell's equations; Lorentz force equation) is, then, classical physics.

If we take it that a linear polarizer transmits just the part of the electric field in a wave which is aligned in a given direction, then the $\cos(\theta)$ law for the amplitude, and therefore $\cos^2(\theta)$ for the intensity, follows immediately. In this sense it is classical.

If, on the other hand, we look into the physical interaction in the polarizer that causes it to have this kind of transmission, then in some cases we would struggle to come up with a good model employing only classical concepts.

In the case of radio waves and a polarizer just consisting of a set of parallel conducting wires, a classical picture is quite good. In the case of light waves in a birefringent material you can also use classical models of the fields, but you have to take the birefrigence itself as a given. In other cases the light-matter interaction is more 'quantum'. The inverted-commas are there to remind that the issue of specifying whether or not some process is classical is not a precise idea.

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  • $\begingroup$ Thanks do you have anything I could check for the parallel conducting wires radio wave version? $\endgroup$
    – Mauricio
    Oct 13, 2022 at 9:41
  • $\begingroup$ It probably gets a mention in some e-m or electronics textbooks, but I don't know where. The electric field component along the wires causes a current and gets absorbed. The electric field component in the other direction can pass through. The wire spacing has to be less than a wavelength. $\endgroup$ Oct 13, 2022 at 9:57

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