Suppose that an ion having s = 1, l = 0 is surrounded by an array of charges.  Its Hamiltonian is H = D[Sz2 - 1/3 s(s + 1)I] + G(Sx2 - Sy2) .  Assuming D/3 > G > 0.
(a)  Construct the energy matrix for this Hamiltonian.
(b)  Solve exactly for the energy eigenvalues.
(c)  Sketch the energy levels.
Given:   I is the identity matrix.

In this problem the units are chosen such that ħ = 1.



The angular momentum operators {Jx, Jy, Jz} are central to quantum theory.  States are classified according to the eigenvalues of these operators when J is conserved by the respective Hamiltonian H.
(a)  What condition(s) is (are) necessary for all eigenstates of H to be eigenstates of J
An eigenstate of J is usually specified by |j,mz>,
where J2|j,mz>  =  j(j + 1)ħ2|j,mz> and Jz|j,mz> = mzħ|j,mz>.
(b)  We can substitute Jx or Jy for Jz in (a).  However a state cannot be simultaneously an eigenstate of Jz and Jx.  Derive the commutation relation for the angular momentum operators Jx and Jz, (i.e.  [Jx,Jz] = -iħJy) from the definition of the linear momentum operator.
(c)  Prove that it is indeed possible for a state to be simultaneously an eigenstate of J2 = Jx2 + Jy2 +Jz2 and Jz



The matrix  σx is defined by
Prove the relation exp(iασx) = I cosα + i σx sinα, where I is the 2 x 2 identity matrix.