This is the hallmark of a spin-1 system, and our goal in this assignment is to fully explore it. 1. To start, we need to pick a basis, and given our experience with the S, basis in spin-1/2, we should use that here, too. Write down the following three things: (i) the three basis states (call them [1), |0), and |-1), corresponding to measuring +h, 0, and -h respectively), as kets and in matrix notation; (ii) the eigenvalue equations (all three of them since we have three eigenvalues) for the operator S2; and (iii) the matrix representation of the operator 2. Recall the raising and lowering operators we defined in class: S. = Sx + isy and S_ = 5x - iSy- After painstaking theoretical work, you finally figure out what they do when operating on our basis states:! S.[1) = 0 $ |1) = Vzh|0> S,10) = V2h/1) $_10) = V2hl - 1) S,1 - 1) = V2h|0) $ 1 - 1) = 0. Use this to write out the matrix representation of S, and S_, and then use their definition above to find the matrix representation of Sx and Sy (that was our ultimate goal here). 3. Now that we have S, and Sy, we need to find their eigenvalues and eigenvectors. Do so, and write out the eigenstates in ket form and in matrix form. 4. We now have a complete theory of spin-1, which means we can finally so some interesting experiments with those photons. Suppose you arrange your apparatus to prepare the photons in the state ly) = [1) -310) + 2i] -1). . You prepare 1000 photons this way, and measure their spin component S2. What values do you measure, and how many photons of the 1000 yield each result? . You collect all the photons that had mesaurements of S, = -h. You then measure S, on those; what do you get, and what is the number of photons for each possible result? . If, on the original set of 1000 photons, you had instead measured Sy, what would have been the average (i.e., expectation value) of all results