Properties of angular momentum operators

Problem:

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

Problem:

Show that if any operator commutes with two of the components of an angular momentum operator, it commutes with the third.

Problem:

Consider the three spin matrices

image

(a)  Calculate the commutator of Sx and Sy.
(b)  What are the possible values we can get if we measure the spin along the x-axis?
(c)  Suppose we obtain the largest value when we measure the spin along the x-axis.  If we now measure the spin along the z-axis, what are the probabilities for the various outcomes?

 Solution:

Problem:

Consider the vector space E(k,j) spanned by vectors with the same k and j but different indices m.
(a)  What is the dimension of this space?
(b)  The space is globally invariant under the action of all components of J.  Any component of J acting on a vector in E(k,j) yields another vector in E(k,j).  What are the matrices of Jz, J+, and J- in E(k,j)?
(c)  Write down the matrices of J2, Jz, Jx, and Jy for j = 0, ½, and 1.

Solution: