A particle with spin s = 1/2 is prepared in a state with orbital angular momentum l = 1 (with orbital angular momentum L? = ((( + 1)h2 = 2h2). The relevant part of the Hamiltonian of the particle is given by aL . S, where L and S are the orbital angular momentum and spin operator, respectively. (a) Denote the eigenstate of { L2, Lz, S2, Sz} by [me, ms), where me and ms are eigenvalues of Lz and Sz, respectively. Denote the sum of the angular momenta J = L + S, and the eigenstates of {J2, Jz, L', S"} as (jm). Express all possible [jm) as linear combinations of [me, ms) . (b) Write down the Hamiltonian in the basis of {lim) }