A three-level system has a Hamiltonian represented by the matrix H = 0 b 0 0 b0a (1) where a, b, and c are positive real numbers with a >b. There are two other observables P and Q represented by the following matrices 100 4 0 0 P = 020 Q 003 00 i 0-0 (2) For all parts of the problem below, we will assume the system starts out at t = 0 in the state |(t = 0)) = a) First, find the eigenvectors and eigenvalues of the Hamiltonian H. (3) b) Suppose that you immediately measure P at t=0. What are the possible outcomes and with what probabilities? c) Assume that you measured P = 2 at t = 0 (right after preparing the state). What is the state for all times t after the measurement? d) Assume that you measured P = 1 at t = 0 (right after preparing the state). What is the state for all times t after the measurement? If you wait some more time T and you measure P again, what are the possible outcomes and with what probabilities? Explain your answer in the context of Eq 3.73 of the book. e) Now suppose that you immediately measure Q at t=0 (there is no P measurement anymore). What are the possible outcomes and with what probabilities? Would the answer change if you waited some time T before measuring Q?