Does measurement really affect polarization?

Question

The answer must be yes but I have trouble dealing with the explanation given here.

As you can see the light is polarized at an angle theta to x axis. After it passes the polarizer A which is along x axis the light gets polarized in that direction and if you put yet another analyzer behind A you get all the light passing through it. Now the author says that this eigenstate e_p can be decomposed in the following form:

This is what the author writes explaining how the analyzer A disturbed the light

After passing through the analyzer, the light is completely polarized along e_x. If we place, after the first analyzer A, a second analyzer A', having the same axis, all the photons which traversed A will also traverse A'. According to what we have just seen in point (ii), this means that, after they have crossed A, the state of the photons is the eigenstate characterized by e_x. There has therefore been an abrupt change in the state of the particles. Before the measurement, this state was defined by a vector E(r, t) which was collinear with e_p. After the measurement, we possess an additional piece of information (the photon has passed) which is incorporated by describing the state by a different vector, which is now collinear with e_x. This expresses the fact, already pointed out in § A-2, that the measurement disturbs the microscopic system (here, the photon) in a fundamental fashion.

Here's my argument. Weren't photons in the x state always in the x state the whole time? The initial state contained a superposition of both states. So how come the analyzer A disturbed the light? It only stopped the e_y component. All the analyzer did was block out the photons with the y state and thus there was no "abrupt change" in the state of light at least of that in the x direction. Where am I wrong if I am?

Answer 1

Perhaps your difficulty is this: a beam of polarized light does not contain a combination of $x$-polarized photons and $y$-polarized photons. Rather, each incident photon is polarized along the "p" direction. In this author's picture, each individual photon is modified.

Answer 2

Weren't photons in the x state always in the x state the whole time?

No. They were in the $\hat e_p$ state before they went through the polarizer. How do we know? Because if you put a polarizer lined up in the $\hat e_p$ direction then 100% of the photons went through. So they originally were in a state that is 100% transmitted by a $\hat e_p$ oriented polarizer. And later they don't have this property of having 100% transmission through a $\hat e_p$ oriented polarizer. So they were changed by the $\hat e_x$ polarizer.

That's the key. They were in a state that had a property (100% transmission by a polarizer oriented in the $\hat e_p$ direction) and then they weren't in that state because they no longer had that property.

Sure, your text seems to be focusing on the new state's new property (100% transmission by a polarizer oriented in the $\hat e_x$ direction) but the key is that this is a new property, one it didn't have before and that it lost properties it had before.

The initial state contained a superposition of both states.

The word superposition and all the ideas of it are more misleading than helpful. A superpositions of two states is just another states. The state that is polarized in the direction $\hat e_x$ is just a superposition of the states polarized in the $\hat e_p$ direction and states polarized in the $\hat e_q$ direction (where $\hat e_q$ is orthogonal to $\hat e_p$).

So how come the analyzer A disturbed the light?

The polarizer absorbs states polarized in the direction $\hat e_y$ and lets states polarized in the direction $\hat e_x$ through unaffected. And since the theory is linear, this is enough information to tell us what it does to absolutely every state. But there really are lots of different states, not just those two.

It only stopped the e_y component.

There is no only about it. You had to say what it did to both states polarized in the direction $\hat e_y$ and states polarized in the direction $\hat e_y$ before you'd given enough information to say how it treats an arbitrary state.

All the analyzer did was block out the photons with the y state

It did no such thing since you never sent a photon polarized in the direction $\hat e_y$ through the polarizer, so it didn't do that. That would be like saying a convicted bank robber robbed a bank while in prison because you think if he had been at the bank he would have robbed it. He wasn't at the bank, he was in jail.

Similarly, you didn't have a photon polarized in the direction $\hat e_y$ go into the polarizer, so the polarizer didn't absorb such an incoming photon.

and thus there was no "abrupt change" in the state of light at least of that in the x direction.

The change is you created a state of polarization in the $\hat e_x$ direction when before there wasn't one.

Where am I wrong if I am?

You are wrong to think superposition is about being in states rather than just being a mathematical fact that you can express a vector as a linear combination of basis vector, which is all it is. People say superposition and all they mean is complex linear combination. And the result of a superposition is just another state that is 100% just as fundamental and meaningful and basic as the things you added together. Just like the orthonormal vectors $\hat e_p$ and $\hat e_q$ are just as fundamental and meaningful and basic as the orthonormal vectors $\hat e_x$ and $\hat e_y.$

There is zero magic about superposition. The real key is that since operators are linear you can fulling specify them by saying what they do on basis vectors. Like how you can specify a matrix by giving the columns (which are exactly what each basis vector is sent to).

So you can use a basis that easy to describe for some operator and then use that easy description to get a description of any state. But this isn't magical.

As for these interactions you can also get the rate of transmission from the coefficients of the linear combination. But these rates are definitely the rates at which the initial state is changed to the new state. And we know it is a change since properties that used to exist (100% transmission by certain polarizers) no longer exist.