The Hamiltonian operator for a 2-state system is given by H = a1|1){1| + a2|2) (2| + b (|1) (2| + |2) (1|) , where a1, az and b are real numbers, and the kets |1) and |2) form an orthonormal basis. Find the energy eigenvalues and the corresponding eigenkets, as linear combinations of |1) and |2). Deduce these results in the case where a1 = -a2 = b = a.