(5+10) Consider a system of polar molecules. The ground state and the lowest excited states are in the rotational degree of freedom, labeled by J, M, where J is the angular momentum quantum number and M = -J, ..., J is the rotational quantum number along the z-axis. The Hamiltonian of this system can be written as H = B(J^2), where J^2 is the angular momentum operator. (a) What is the eigenstate energy for the eigenstates of J = 0 and J = 1? (b) When a static electric field, E, is applied, we have an additional term H' = Ed(|0,0><1,0| + |1,0><0,0|), where d is the electric dipole moment. Write down a 4x4 matrix representation for the Hamiltonian in the subspace of J = 0 and J = 1, using the following basis: {|0,0>, |1,0>, |1,-1>, |1,1>}, and calculate its eigenenergies.
