Kathryn0505 said:
A superposition state is the sum of all the possibilities of all the states it could be in. Until we actually measure and know what state it is in. But that's not to say that it's not in a certain state until we measure? We just aren't sure yet. Just because we don't know yet doesn't mean something's not a certain way. (That was probably poorly worded).
The wording is good enough. It's clear what you mean, and what you're saying here is not the way it is. The terms are not the possible states the system might
be in right now. The first clue to that is that you can expand the state in several different bases. For example, if we write the eigenvalue equations for two observables A and B as
Au_a=au_a
Bv_a=bv_b
a state \psi can be expressed either as
\psi=\sum_a\langle u_a,\psi\rangle u_a
or
\psi=\sum_b\langle v_b,\psi\rangle v_b
Each u_a is the state that the system can be in
immediately after a measurement of the observable A that had the result a, or to put it differently,
the mathematical representation of the statistical properties of an ensemble of systems of the same type that were all prepared by a measurement of A that had the result a. The statistical properties of such an ensemble are as simple as they can get, as far as measurements of A are concerned. If we measure A again, we will get the same result with certainty.
The above doesn't conclusively prove that the system isn't always in a specific but unknown eigenstate, but Bell inequality violations do exactly that. The assumption that any measurement is simply "finding out what state the system is in" has consequences called Bell inequalities, which QM predicts do
not hold. Experiments have proved that assumption wrong by showing that Bell inequalities don't actually hold in the real world.
Most QM books include derivations of some sort of Bell inequality. I like the derivation in Isham's book (
215,
216) of a particular Bell inequality for spin-1/2 particles, called the CHSH inequality. That's a really good book by the way. It's probably exactly what you need right now, so I think you should consider buying it. (I guess you'll have to if you also want to see his proof that QM predicts a violation of the inequality. I think it's on the next page and can't be previewed at Google Books).
Kathryn0505 said:
Briefly, as far as entanglement goes, measuring one particle gives information about the other. But does it change what the second one is doing? Or just give us information about it?
I think you need to think of an entangled "pair" as a single entity.