### 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

(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:

• Concepts:
The eigenvectors and eigenvalues of the angular momentum operator, spin 1
• Reasoning:
We are asked to predict the outcomes of measurements on a spin 1 particle.
• Details of the calculation:
(a)  The commutator is

(b)  We can measure one of the eigenvalues of Sx, namely ħ, 0, or -ħ.
(c)  The largest possible eigenvalue is ħ.
To find the corresponding eigenvector expanded in terms of the eigenvectors of Sz we solve the system of equations
.
We obtain a = b/√2 = c.
The normalized eigenvector is |1>x = ½|1> + (1/√2)|0> + ½|-1>.
The probabilities for measuring ħ and -ħ are ¼, and the probability of measuring 0 is ½.

#### 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:

• Concepts:
The eigenvectors and eigenvalues of the angular momentum operator
• Reasoning:
The the vector space E(k,j) spanned by vectors with the same k and j is a subspace of the state space E(k,j,m) spanned by the eigenvectors of a complete set of commuting observables, where two of those observables are J2 and Jz.  We can consider E(k,j,m) to be the direct sum of all orthogonal subspaces E(k,j) spanned by vectors with the same k and j but different indices m.
• Details of the calculation:
(a)  The dimension of the space is 2j + 1.  J2 and Jz have common eigenvector.  Each eigenvalue of J2 characterized by j is 2j + 1 fold degenerate, there are 2j + 1 possible values of m.
(b)  Jz|k,j,m> = mħ|k,j,m>.
J+|k,j,m> = [j(j+1) - m(m+1)]½ħ|k,j,m+1>.
J-|k,j,m> = [j(j+1) - m(m-1)]½ħ|k,j,m-1>.
The matrix elements in E(k,j) therefore are
<k,j,m|Jz|k,j,m'> = mħδm m',
<k,j,m|J±|k,j,m'> = [j(j+1) - m'(m'±1)]½ħ δm m'±1.
The matrix elements are independent of the value of k, they depend only on j and m, they are "universal".  They do not depend on a particular physical system.
(c)  j = 0:
The space E(k,j) is one-dimensional, the only possible value for m is zero.  All matrices reduce to one element, which is zero.

j = ½:
In the { |+½>, |-½> } basis we have

j = 1:
In the { |1>, |0>, |-1> } basis we have