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The question of "realism" has to do with whether the result of the measurement exists before it is measured. I have a bowl of fruit in my kitchen that has some apples in it. I don't know how many are left; it is either two, three, or four. But I don't believe that the number of actual apples in my actual bowl is ambiguous, nor that the wavefunction of the bowl will collapse into e.g. the three-apple state when I go to look at it, nor that I can keep the number of apples ambiguous by pulling one off the top while carefully not looking at the bottom of the bowl. But all of these counterintuitive statements are things people do say about quantum-mechanical states. These ideas are particularly associated with the "Copenhagen interpretation" of quantum mechanics, in which a special process called a "measurement" produces an instantaneous global change in a wavefunction.

The question of "realism" has to do with whether the result of the measurement exists before it is measured. I have a bowl of fruit in my kitchen that has some apples in it. I don't know how many are left; it is either two, three, or four. But I don't believe that the number of actual apples in my actual bowl is ambiguous, nor that the wavefunction of the bowl will collapse into e.g. the three-apple state when I go to look at it, nor that I can keep the number of apples ambiguous by pulling one off the top while carefully not looking at the bottom of the bowl. All of these counterintuitive statements are things people do say about quantum-mechanical states. These ideas are particularly associated with the "Copenhagen interpretation" of quantum mechanics, in which a special process called a "measurement" produces an instantaneous global change in a wavefunction.

The amazing thing about Wikipedia — the thing that makes it trustworthy enough that people treat it incorrectly as an oracle — is that if it's wrong, you can click "edit" and fix it. The next sentence you quote says that this interpretation is present in "all sorts of physics textbooks and papers and whatnot." You found one.

Bell's 1964 paper is much easier to find now than it used to be. (When I went down by Bell's Theorem rabbit hole as a grad student twenty years ago, I had to use secret library jujitsu to find a copy; the journal that published it failed after about a year.) If you have questions about how things are defined, going backwards to original sources is generally the way to go.

I did wind up watching the video while I was writing this. I had apparently watched the first half earlier and stopped, so I didn't go back to the first half, but your questions are all about the ending. The video's statements at the end about what has or hasn't been proved are all consistent with my understanding, but the point they are making is pretty subtle, and I don't think your question summarizes it correctly. As I wrote at the top of this question, I think it's fine to sit with the idea that your previous understanding is different from the actual state of things, even if you can't yet summarize the actual state of things to your own satisfaction.

The amazing thing about Wikipedia — the thing that makes it trustworthy enough that people treat it incorrectly as an oracle — is that when it's wrong, you can click "edit" and fix it. The next sentence you quote says that this interpretation is present in "all sorts of physics textbooks and papers and whatnot." You found one.

Bell's 1964 paper is much easier to find now than it used to be. (When I went down my Bell's Theorem rabbit hole as a grad student twenty years ago, I had to use secret library jujitsu to find a copy; the journal that published it failed after about a year.) If you have questions about how things are defined, going backwards to original sources is generally the way to go.

I did wind up watching the video while I was writing this. I had apparently watched the first half earlier and stopped, so I didn't go back to the beginning, but your questions are all about the ending. The video's statements at the end about what has or hasn't been proved are all consistent with my understanding, but the point they are making is pretty subtle, and I don't think your question summarizes it correctly. As I wrote at the top of this answer, I think it's fine to sit with the idea that your previous understanding is different from the actual state of things, even if you can't yet summarize the actual state of things to your own satisfaction.

$$ \left( \frac{\hat p}{2m} + \hat V(x) \right)\psi(x,t) = \hat E\psi(x,t), $$

$$ \left( \frac{\hat p^2}{2m} + \hat V(x) \right)\psi(x,t) = \hat E\psi(x,t), $$