Discuss the below:
Q: Let's consider an ion (in an effective spin state corresponding to s = 1) at a crystal lattice site. As for the effective potential seen by the ion, assume spin Hamiltonian of the form:
H = alpha*S^2_z + beta*(S^2_z - S^2_y)
where alpha and beta are some real constants and |alpha| >> |beta|
a. Choosing the three independent eigen states of S_2 (corresponding to s = 1) as basis, express H_0 = alpha*S^2_z (unperturbed Hamiltonian) and H' = beta*(S^2_z - S^2_y) (perturbation) by suitable 3 x 3 matrices.
b. For the full Hamiltonian H = H_0 + H', obtain the energy eigen values and the corresponding eigenstates which are correct up to 1st-order in beta.