Suppose that an ion having S=1, L=0 is surrounded by an array of charges.  Its Hamiltonian is H=D[S_z²-1/3 S(S+1)]+G(S_x²-S_y²) .  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:    .

Solution (a) You can solve this problem without knowing anything about the physical quantities represented by the operators S2, L2, Sx, Sy, and Sz.  L2 and S2 represent the operators for the square of the orbital angular momentum and the spin angular momentum of a spin 1 particle and Sx, Sy, and Sz represent the operators for the Cartesian components of the spin.  The eigenvalues of S2 are S(S+1) and the eigenvalues of L2 are L(L+1).  In this problem the units are chosen such that h/2p=1. Sz is diagonal.  The basis picked is an eigenbasis of Sz.  The eigenvalues of Sz are 1,0,-1. H=D[Sz2-1/3 S(S+1)]+G(Sx2-Sy2).  D and G are numbers, Sx2, Sy2, and Sz2 are matrices and S(S+1) is a number times the identity matrix. . S=1, S(S+1)=2, 1/3 S(S+1)=2/3. . (b) . Either  . The eigenvalues are The eigenvector corresponding to the eigenvalue  is   yield   . The eigenvector corresponding to the eigenvalue  is    yield   The eigenvector corresponding to the eigenvalue  is   yield   (c) We have found the eigenbasis of H. The eigenvalues are .

The angular momentum operators {J_x, J_y, J_z} 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,m_z>, where J²|j,m_z>=j(j+1)²|j,m_z> and J_z|j,m_z>=m_z|j,m_z>.
(b) We can substitute J_x or J_y for J_z in (a).  However a state cannot be simultaneously an eigenstate of J_z and J_x.  Derive the commutation relation for the angular momentum operators J_x and J_z (i.e. [J_x,J_z]=-i J_y) 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=J_x²+J_y²+J_z² and J_z.

Solution: (a) It is possible to specify a common eigenbasis of two operators if they commute.  If one of the operators is non degenerate, then all of its eigenvectors are also eigenvectors of the other operator.  In order for all eigenstates of H to be eigenstates of J2 and Jz we need [J2,H]=0 and [Jz,H]=0 and H is non degenerate.  (States with different j or mz must have different energy eigenvalues E.) (b) Assume the classical relationship . For the quantum mechanical operators we then have                                                                                      0                0        0                                                    0 .  (In general .) (c) , then it is possible to find simultaneous eigenstates. .

Problem:

The matrix is defined by .  Prove the relation , where I is the identity matrix.

Solution: Let be the matrix of the Hermitian operator Sx in the {|1>,|2>} basis. . Therefore Sx2=I, Sxn(n=even)=I,  Sxn(n=odd)=ISx=Sx. .    .