6. Consider a system of angular momentum l = 1. A basis of its state space is formed by the three eigenvectors of L2: |+ 1), |0), | 1), whose eigenvalues are, respectively, +h, 0, and -h, and which satisfy: L+|m) = hv2|m1) L+|1) = L-| 1) = 0 This system, which possesses an electric quadrupole moment, is placed in an electric field gradient, so that its Hamiltonian can be written: Wo H = " (L - L) where Lu and L, are the components of L along the two directions Ou and Ov of the xOz plane that form angles of 45 with Ox and Oz; wo is a real constant. a. Write the matrix representing H in the {|+ 1), |0}, | 1)} basis. Vhat are the stationary states of the system, and what are their energies? (These states are to be written |E1), |E2), |E3), in order of decreasing energies.) b. At time t = 0, the system is in the state: 1 |b(0)) = + 1) |- 1)] What is the state vector (t)) at time t? At t, Lz is measured; what are the probabilities of the various possible results? c. Calculate the mean values (La)(t), (L,)(t) and (L2)(t) at t. What is the motion performed by the vector (L)? d. At t, a measurement of L is performed. (i) Do times exist when only one result is possible? (ii) Assume that this measurement has yielded the result h2. What is the state of the system immediately after the measurement? Indicate, without calculation, its subsequent evolution.