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 J^{2}|j,m_{z}> = j(j + 1)ħ^{2}|j,m_{z}>
and J_{z}|j,m_{z}> = m_{z}ħ|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}] = -iħJ_{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
J^{2 }= J_{x}^{2 }+ J_{y}^{2 }+J_{z}^{2}
and J_{z}.

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

Consider the three spin matrices

(a) Calculate the commutator of S_{x} and S_{y}.

(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 S_{x}, namely ħ, 0, or –ħ.

(c) The largest possible eigenvalue is ħ.

To find the corresponding eigenvector expanded in terms of the eigenvectors of S_{z}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.

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 J _{z},
J_{+}, and J_{-} in E(k,j)?**

(c) Write down the matrices of J^{2}, J_{z}, J_{x}, and J_{y} 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**J**. 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.^{2}and J_{z} - Details of the calculation:

(a) The dimension of the space is 2j + 1.**J**^{2}and J_{z}have common eigenvector. Each eigenvalue of J^{2}characterized by j is 2j + 1 fold degenerate, there are 2j + 1 possible values of m.

(b) J_{z}|k,j,m> = mħ|k,j,m>.

J_{+}|k,j,m> = [j(j+1) - m(m+1)]^{½}ħ|k,j,m+1>.

J_{-}|k,j,m> = [j(j+1) - m(m-1)]^{½}ħ|k,j,m-1>.

The matrix elements in E(k,j) therefore are

<k,j,m|J_{z}|k,j,m'> = mħδ_{m m'},

<k,j,m|J_{±}|k,j,m'> = [j(j+1) - m'(m'±1)]^{½}ħ δ_{m 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 havej = 1:

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