The state space E(*k,j,m*) can be considered a direct sum of the orthogonal
subspaces E(*j*,*m*), where *m* varies in integral jumps from *-j* to *+j*
and *j* takes on all the values actually found in the problem. The dimension of each
subspace is *g(j)*. (Here each vector in a subspace E(*j*,*m*) has the same
*j* and *m* but a different index *k*.)

We can also 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*. The dimensions of these subspaces are *2j+*1, and the subspaces are **globally
invariant** under the action of all components of * J*. Any component of

**What are the matrices of J_{z}, J_{+},
and J_{-} in E(k,j)?**

.

.

.

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

,

.

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.

*j=0*:The subspaces E(

*k,j)*are 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.

- For an arbitrary
*j*we have.

.

.

,

with the center of the charge distribution at the origin of the coordinate system. Here

.

A nucleus with a quadrupole moment has a quantum mechanical charge density *r*_{a}JM(**r**)µ|y_{a}JM(**r**)|^{2}, which depends on
the quantum numbers *J *and *M*. The quadrupole moment and the energy of the
nucleus in an electric field gradient therefore depend on *J* and *M*.

Consider a nucleus with a quadrupole moment and with angular momentum *l=*1.
Assume that the Hamiltonian of the nucleus may be written as

,

where *L _{u}* and

(a) Write the matrix of *H* in the { *|*1*>, |0>, |-*1*>*
} basis. What are the stationary states of the system and what are their energies?
Write
these states as *|E _{1}>, |E_{2}>*, and

(b) At time *t=0* the system is in the state . What is the state vector* |y(t)>* at time* t*? At time* t* *L _{z}*
is measured? What are the probabilities of the various possible results?

(c) Calculate the mean values *<L _{x}>(t), <L_{y}>(t)*,
and

(d) At time* t L _{z}^{2}* is measured. Do times exist when only
one result is possible? Assume that the measurement yielded h

- Solution:
(a) . .

.

.

.

.

.

in the {

*|1>, |0>, |-1>*} basis..

.

.

Assume is positive. Then

.

.

.

(b) .

.

.

=0.

.

=0.

or

.

(d) .At and

*|y(t)>*is an eigenvector of*L*with eigenvalue h_{z}^{2}^{2}.At and

*|y(t)>*is an eigenvector of*L*with eigenvalue 0._{z}^{2}After a measurement of h

^{2}the state of the system is the eigenstate of*L*. This state evolves like_{z}^{2}*|y(0)>*.

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

Solution:

- Concepts:

The commutation relations for the Cartesian components of any angular momentum operator - Reasoning:

Commutation relations are what defines a vector operator as a angular momentum operator. We define angular momentum through [J_{i},J_{j}] = ε_{ijk}iħJ_{k}. -
Details of the calculation:

Let i ≠ j,k j ≠ k and let i, j, k be cyclic ( x, y, z or y, z, x or z, x, y ).

Then [J_{i},J_{j}] = iħJ_{k}, J_{i}J_{j}- J_{j}J_{i}= iħJ_{k}.

Let A be an operator.

Assume [J_{i},A] = 0, [J_{j},A] = 0. Then [A, J_{k}] = (1/iħ)[A,J_{i}J_{j}- J_{j}J_{i}].

iħ[A, J_{k}] = [A,J_{i}J_{j}] - [A,J_{j}J_{i}] = [A,J_{i}]J_{j}+ J_{i}[A,J_{j}] - [A,J_{j}]J_{i}- J_{j}[A,J_{i}] = 0.

Consider the so-called spin Hamiltonian , for a system of spin 1. Show that the
Hamiltonian in the *S _{z} *basis is .
Find the eigenvalues of this Hamiltonian.

- Solution:
For a spin 1 system we choose as the basis vectors the eigenstates of

*S*, {_{z}*|1>, |0>, |-1>*} with eigenvalues respectively. In this basis the matrix for*S*is_{z}.

The matrices for

*S*and_{x}*S*are_{y}.

Therefore

,

.

.

We have .

The eigenvalues and eigenvectors are:

.