This problem will familiarize you with uncertainty relations and minimum uncertainty states, and will train you in using matrix notation and angular momentum operators. a) reproduce the derivation of the uncertainty relation between the components jr and jy of an angular momentum vector for any state (0) by starting with the positive-definite quantity (jz – (jx) + im(jy – (jy))|4)||* > 0 (1) and writing it as a positive-definite quadratic expression in 1, as we did in class (rather than the schwartz inequality used in the textbook). b) for the uncertainty relation to be "saturated' (that is, to become an equality), the above quantity must vanish. assume a state that satisfies (jx+ixjv) = 0 and show that for this state (jx) = (ju) = 0. such a state is (a particular case of) a minimum uncertainty state for jy and jy. c) for spin 1 (j = 1, m = -1; 0; 1) the state 4) becomes a column 3-vector with components corresponding to m = 1 (top), m = 0 (middle) and m = -1 (bottom). write down the 3 x 3 matrices representing jr, jy and j, as well as the operators j+ = jr ijy. d) write down the equation (j. + ) = 0 in matrix form for the case of spin 1 and find the components of the vector in terms of . assume that is normalized and choose its overall phase such that all its components be real. (hint: it may be to express the above equation in terms of j+ and e) for the state that you found above, calculate ajx, ajy and (iz) in terms of , and show that the uncertainty relation is saturated.