1. We can define an operator 0 = = nxX + nY + nZ, where is a real unit 3- vector (with n + n + n = 1) and X, Y, Z are the Pauli matrices. This 3-vector indicates a direction in space, which corresponds to a possible axis along which one could measure a spin-1/2 particle. (Or equivalently, we can think of that direction as indicating a point on the Bloch sphere.) a. Show that 0 = 1 for any choice of n. b. Find the eigenvalues of O. How do they depend on n? C. We can specify a point on the Bloch sphere by two angles 0 and $, where the corresponding state vector is 1) = cos(0/2)| z) + sin(0/2) e |z). What is the 3-vector n corresponding to the angles 0, ? 1 d. Show that the state (1) from part c is an eigenvector of the operator O for that choice of 3-vector . What is its eigenvalue? e. Show that the state vector (7+) = sin(0/2)|12) - cos(0/2) e |z) is also an eigenvector of that same operator O. What is its eigenvalue?