Polarization measurements provide handy illustrations of complementary variables, wave function collapse, and other important quantum phenomena, so it will be worthwhile to consider a few such examples. As you may recall from chapter 2, a linearly polarized wave oscillates back and forth in a single dimension perpendicular to its direction of travel. For example, if a wave is traveling forward in front of you, its peaks and troughs might oscillate left and right (horizontal polarization), or up and down (vertical polarization), or at an angle 45-degrees from horizontal (45° polarization), etc. In quantum physics, precise polarization at any given angle is equivalent to a superposition of polarizations at other angles. Polarizations at 45° angles from each other are complementary variables: the closer a photon is to having a definite horizontal or vertical polarization, the “fuzzier” it is with respect to 45° or -45° polarization measurements, and vice versa.

The polarization of a photon can be measured using a linear polarizing filter, a semitransparent material that allows photons to pass through when they have the same orientation as the filter, but blocks photons with the opposite orientation. For example, a horizontal polarizing filter allows horizontally-polarized photons to pass through, but blocks photons that are vertically polarized. Photons with any other polarization state—for instance, photons that are polarized at an angle between vertical and horizontal—are considered to be in a superposition of horizontal and vertical polarization states, and there is no pre-determined fact of the matter what they will do when they hit a horizontal polarizing filter. As with any measurement of something in a superposition, it’s a matter of chance. Photons that are polarized at a small angle (e.g. 1 degree) relative to horizontal are *likely* to pass through the horizontal filter, but some will be blocked nonetheless.

According to quantum mechanics, the probability that a photon will pass through a polarizing filter is equal to cos^{2}(θ), where θ is the angle between the photon’s polarization and the orientation of the filter. For example, if a photon is polarized at an angle 30° from horizontal, its chance of passing through a horizontal filter is cos^{2}(30°) = ¾. If the photon is polarized at 60° relative to the orientation of the filter, its chance of passing through is cos^{2}(60°) = ¼, and so on.

Now, suppose we use a horizontally-oriented filter to measure the polarizations of a bunch of photons. (Of course, we’ll also have to use some sort of photon detector—e.g. our eyes—to see whether the photons pass through the filter.) If the photons have random polarizations, about 50% of them will pass through the filter, and 50% will be blocked. On the other hand, if many of the photons have similar polarizations, we may find that significantly more or less than 50% of the light passes through the filter.

Nevertheless, for each individual photon, there are only two possible outcomes of this measurement: either the photon passes through the filter, or it is blocked. In a sense, the filter “measures” whether the photon has horizontal or vertical polarization, and those are the only two options. When a photon hits the horizontal filter, its superposition collapses to one of those two polarization states. In other words, any photon that encounters the horizontal filter must acquire either a horizontal or vertical polarization, even if its polarization beforehand was indeterminate. If the photon acquires a horizontal polarization, it passes through the horizontal filter; otherwise it acquires vertical polarization and is blocked.

Each photon must decide, so to speak, whether it wants to become horizontally polarized (and pass through the filter) or vertically polarized (and be blocked). Thus, any photon that passes through the horizontal filter will have horizontal polarization afterwards. For this reason, if you stack two horizontal polarizing filters together, any photon that passes through the first filter will also pass through the second. But if you rotate the second filter by 90 degrees, so that it only allows vertically-polarized light to pass through (and blocks horizontally-polarized light), no photons will be able to pass through both filters. All of the light will be blocked, as demonstrated in the video below.

Now, here’s where things get really interesting. What happens if you insert a third filter, oriented at 45 degrees, between the horizontal and vertical filters? Stacked together, the horizontal and vertical filters block 100% of the light; but if you insert a 45° filter between them, some light gets through again! Why does that happen?

Quantum mechanics provides an explanation. There is also a classical physical (non-quantum) explanation why some light passes through the three filters, but the classical explanation cannot account for the fact that each individual photon either passes through the filters completely (with 100% of its original energy) or is completely blocked. Moreover, there are similar phenomena that cannot be explained at all by classical physics. For example, a quantum mechanical property called “spin,” which is closely analogous to angular momentum, exhibits behaviors similar to the ones described above, and there is no classical explanation for this. About 50% of the randomly-polarized incoming beam of light passes through the first filter (the horizontal one). All of the photons that make it through that first filter must have horizontal polarization afterwards, so they’ll all be blocked by the vertical filter if they encounter that next. On the other hand, if the horizontally-polarized photons instead hit a 45° filter, roughly half of them will collapse to the orientation required by that middle filter, because cos^{2}(45°) = ½. So half of the photons that came through the first filter will pass through the middle filter as well. But after passing through the middle filter, which is oriented a 45° angle, the photons don’t have horizontal polarization anymore. Because 45° degree polarization is equivalent to a superposition of horizontal and vertical polarization states, half of the photons that passed through the middle filter will collapse to vertical polarization when they encounter the third filter. Thus, some photons—about an eighth of the original number of photons (12.5% of the incoming beam of light)—will pass through all three filters.