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gn0m0n

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Specifically, I am looking at Griffiths, p. 155-157. The eigenstates for spin 1/2 systems are easy enough to understand, although the derivations of them as actually presented in the context of magnetical orbital angular momentum through the lowering and raising operators was a bit obscure to me. I suppose I am even there struggling to recognize the physical significance of the operators (this is p. 147); if anyone can elucidate them at all I would much appreciate it.

Anyway, then we get to spin and Griffiths states that the "algebraic theory of spin is a carbon copy of the theory of orbital angular momentum," which I generally get. I also understand that we can then represent any spin 1/2 particle's state as a linear combination of the two vectors |1/2 1/2> and |1/2 -1/2>, ie, |s m> = ax(+) + bx(-) where x(+) is the spin-up eigenvector and x(-) is the spin-down eigenvector (right so far?). This can also be represented as the spinor (a b)^T (sorry, I'm making up how to convey the notation as I go along - this is the column vector with entries a, b) - now this essentially gives the coefficient of each basis vector? Ie, here, x(+) and x(-), right?

Fair enough, but then he says "the spin operators become 2x2 matrices"... can anyone expand on that? Is it that these operators are not conveying any action performed on the system but simply the observable, e.g. S_x=spin component in the x-direction? What do these operators act on? What about S(+) and S(-), the spin raising and lowering operators - these do not represent observables so what do they represent? I know they move us up and down the "ladder" of states but... how?

Again, it is mainly the Pauli spin matrices I want to understand better... For one thing, I don't understand how or why they are different, given that we choose the z-axis arbitrarily (although I do suspect that the indeterminacy principle has to do with it - perhaps once an axis is chosen we assume measurements are made first on that axis, which we call the z, then any subsequent spin measurements along either other axis will require these different matrices/operators?).

For x(+) we have simply (1 0)^T but for x(+)^(x) we have (1/sqrt(2) 1/sqrt(2)) ... why? What is physical meaning of this latter one? I know the physical meaning of first is that it is entirely "along the |1/2 1/2> vector", right? Now the latter one means we'd have equal probability of finding the spin in the x direction to be 1/2 or -1/2... but don't we also have equal probability (before we measure) to find spin in z direction to be 1/2 or -1/2?

As you can see, I'm going a little crazy here. Someone please save my sanity!

Sorry for the long post - I'm trying to be crystal clear.

Thanks.