For a single photon, its polarization can be a superposition of two orthogonal bases (e.g. horizontal or vertical). However, as I understand, once it has been measured, it must collapse into one of the possible bases. So does that mean that once a photon hits a polarizing beam splitter its wave function collapses? After all, there are only two options: either the photon goes through or it does not. Assuming the original polarization was 45 degrees the, the probability for both events is 50%. Or, if we do not measure which path it takes, does it simply exist in a superposition of the two possible paths, meaning that we have not affected the wave function.

Secondly, I am a bit confused by the idea that once we make a measurement the photon must collapse into one of the possible base states. For example, if we measure that the photon was reflected by the beam splitter, we might conclude that it is either horizontally or vertically polarized (depending on the configuration of the beam splitter) but it could very well be simply 45 degrees of polarization.

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    $\begingroup$ Do you understand that a measurement is characterized by on operator and it is the eigenstates of the operator that are selected? Here is one thing that might be causing your confusion: if you position the selector so that it selects for polarizations at $45^\circ$ from the horizontal then it's operator is different from the one when it is positioned to select for either horizontal or vertical polarization. And that means it has different eigenstates. $\endgroup$ – dmckee --- ex-moderator kitten Jul 21 '16 at 1:09
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    $\begingroup$ BTW, you've been noodling around this problem over the course of several questions, and I feel like you haven't really dug into a single good source yet. Taking a quick look at several different source is not a method I would recommend for learning this. Pick one source and follow through with it. There are lots of correct ways to introduce the subject, but taking bits from several different treatments is unlikely to be correct. $\endgroup$ – dmckee --- ex-moderator kitten Jul 21 '16 at 1:11
  • $\begingroup$ Once a photon has been measured, it's gone. It doesn't make much sense to talk about a collapse for atomic systems (where the system is still there, even if the mixed state is gone), but it makes absolutely no sense for photons. $\endgroup$ – CuriousOne Jul 21 '16 at 1:26

I don't know why this hasn't been answered yet, but the simple answer is: no, the wave function doesn't collapse when it goes through the polarizer. The "photon" remains in a superposition of two states.

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