[SOLVED] Quantum Mechanics - Spin 1. The problem statement, all variables and given/known data Problem is attached. 2. Relevant equations 3. The attempt at a solution The first part is seemingly straight forward. Measurements are +/- hbar/2, both with probability (1/sqrt)^2 = 1/2 of being observed. For the next part I have written the operator S_x as: S_x = 1/2 [S+ + S-] Where S+ and S- are the raising and lowering operators respectively. ie. S+ = S_x + i S_y and S- = S_x - i S_y Then using: S+ | s,m > = [s(s+1 - m(m+1)]^(1/2) hbar | s,m+1 > S- | s,m > = [s(s+1 - m(m-1)]^(1/2) hbar | s,m+1 > With s = 1/2 (this is clear from the first part, since we have the eigenvalues of S_z (m) = -1/2 and 1/2 and -s <= m <= s in integer steps. I obtain S+ | 1/2 > = 0 S- | 1/2 > = hbar | -1/2 > S+ | -1/2 > = hbar | 1/2 > S- | - 1/2 > = 0 So S_x | 1/2 > = hbar/2 | -1/2 > S_x | -1/2 > = hbar/2 | 1/2 > and S_x | psi > = hbar/2 1/sqrt ( | 1/2 > + | -1/2 > ) So the measurement is simply hbar/2 For the final part (where I become stuck!) S_y | 1/2 > = 1/2i (S+ - S-) | 1/2 > = ihbar/2 | -1/2 > S_y | -1/2 > = 1/2i (S+ - S-) | 1/2 > = -ihbar/2 | 1/2 > So I am required to find a state vector psi such that: S_y | psi > = -hbar/2 | psi > 1/2i (S+ - S-) | psi > = -hbar/2 | psi > (S+ - S-) | psi > = -ihbar | psi > But is it even possible to construct a state vector out of the spin-up and spin-down eigenvectors to give this result? I can't seem to do it???