Vectors in the subspace E(*j*_{1},*j*_{2}) may be expanded in terms of or basis vectors.
The
basis vectors of one basis can be written as linear combinations of basis vectors of the
other basis.

.

The expansion coefficients are
called the Clebsch-Gordan coefficients. Sometimes the Clebsch-Gordan coefficients are
written in terms of **the Wigner 3-j symbols**.
We have

**Properties of the Wigner 3-j symbols:**

We can permute the columns of the 3-j symbol. An even permutation does not alter its value. An odd permutation multiplies the initial value by . Moreover,

.

This implies

or

When adding the angular momenta **j**_{1}, **j**_{2},
and **j**_{3}, many different states of the system may correspond to the
same value of *j* and *m*.

Let

Then

The **Racah W coefficients and 6-j symbols** transform
from one scheme of adding **three** angular momenta to another.

The **9-j symbols** transform from one scheme of adding
**four** angular momenta to another.

Links:

- Racah coefficients, 6-j symbols, 9-j symbols
- 3-j, 6-j, and 9-j symbol calculators (1)
- 3-j, 6-j, and 9-j symbol calculators (2)

We will now establish a connection between the Clebsch-Gordan coefficients and the matrix elements of operators that transform like the spherical harmonics under rotation.

Angular momentum and rotations are closely linked. Let us, for a moment, return to the
rotation matrices. An arbitrary rotation can be accomplished in three steps, known as **Euler rotations**, and it is therefore characterized by three
angles, known as **Euler angles**. We may proceed as
follows.

First rotate the body ccw about the *z*-axis by an angle a.
Then rotate ccw about the *y’*-axis by an angle b.
Finally rotate ccw about the *z’*-axis by an angle g.

The rotation matrix is *R*(a,b,g)=*R _{z’}*(g)

*R*(a,b,g)*=R _{z’}*(g)

=

=

=

=

The rotation operator *U*(*R*(a,b,g)) therefore
may be written as

*U*(*R*(a,b,g))=*U(R _{z}*(a))

It only involves rotations about space-fixed axis. We already know the rotation operators for rotations about space-fixed axes. (See notes.)

For a spin ½ particle *U*(*R*(a,b,g))=*U(R _{z}*(a))

using

.

The matrix elements of the second rotation are purely real. This is a result of
choosing the *y’*-axis for our second Euler rotation.

For a system with angular momentum * J* the matrix elements of the rotation
operator in the subspace E(

*U ^{j} *

In E matrix elements between states with
different values *j* vanish. In E the matrix of the rotation operator is block
diagonal. *U ^{j}(R)* is referred to as the (

The matrix elements of the rotation operator, for the rotation *R*(a,b,g) are

The only part which mixes different *m* values is the middle rotation.
We can
define a new matrix *d ^{j}* with elements

,

which contains the nontrivial part of the rotation matrix *U ^{j}(a,b,g)* .
The matrix elements are also listed on the standard table of Clebsch-Gordan
coefficients.

We can establish a connection between the rotation operator and the spherical harmonics by constructing the state vector ( This state vector is an eigenvector of the position operator.)

Let and Then

and

(Matrix elements between states with different values *l* vanish.)

We have

We therefore have

or

This is the connection between the rotation operator and the spherical harmonics.
When *m*
= 0 then

**How do the spherical harmonics transform under rotation?**

Let

Then

But , therefore

The function can also be viewed as
an operator in coordinate space. (It could, for example, be the operator of an experiment
that measures an emission pattern and projects it onto a known distribution.)
The coordinate representation of this operator operating on the state vector |*y*>
is *y**( r),*
just as the coordinate representation of the operator X operating on the state
vector |

Rotating the observable we obtain

. (See notes.)

[*U(R)| r>=|Rr>*,

*U ^{T}(R)Y_{lm}(n)y(r)=Y_{lm}(Rn)y(Rr), *

*U ^{T}(R)Y_{lm}(n)U(R)y(r)=U^{T}(R)Y_{lm}(n)y(R^{-1}r)=Y_{lm}(Rn)y(r)=Y_{lm}(n')y(r)*]

We may therefore write

We define as an **irreducible tensor operator of rank k**
any set of 2

*T ^{k}_{q} *is an irreducible or spherical tensor of rank

or, equivalently

or

Considering the infinitesimal form of this expression we have

or, multiplying out the terms,

This yields

.

**The two commutation relations can also be taken as a definition of a spherical
tensor of rank k.**

Examples of tensor operators:

- A scalar operator is an irreducible tensor of rank 0. It consist of one component and
commutes with all components of the angular momentum
. It therefore fulfills the commutation relations that define a spherical tensor of rank 0.**J** - The spherical components of a vector operator
are defined as**A**Using our definition of a vector operator we can show that these components satisfy the commutation relations that define a spherical tensor of rank 1. For example,

etc.

A vector operator therefore is a spherical tensor operator of rank 1.

- A Cartesian tensor is defined
through

Letand**A**be two vectors, then**B***T*are the elements of a Cartesian tensor. Cartesian tensors are reducible, they can be decomposed into objects that transform differently under rotation. For_{ij}=A_{i}B_{j}*T*we have_{ij}=A_{i}B_{j}is a scalar, represents three independent components which transform like a vector.

represents the components of a traceless, symmetric tensor with 5 independent components.

*T*has been decomposed into three objects that transform like the spherical harmonics with_{ij}*l*= 0, 1, and 2 respectively. It has been decomposed into irreducible spherical tensors.

**Why do we care about tensor operators? **

The matrix elements of tensor operators with respect to angular momentum eigenstates satisfy

The double bar matrix element is independent of *m, m’*, and *q*.
This
is the **Wigner-Eckart theorem**.

(In the above equation *(2j'+1) ^{-1/2}* is factored out of the
double bar
matrix element. This factoring is not unique. Different books may factor out a different
factor.)

The matrix element is written
as the product of two factors. The first factor is a Clebsch-Gordan coefficient which
depends only on the way the system is oriented with respect to the *z*-axis.
It does
not depend on the physical nature of the particular tensor operator. The second term
depends on the physical nature of the operator and the system, but it does not depend on
the magnetic quantum numbers *m, m’*, and *q*. To evaluate for the various combinations of *m,
m’*, and *q *we need to only evaluate one matrix element. The others are
related via the Clebsch-Gordan coefficients.

Certain selection rules for tensor operators follow directly from the selection rule of
the Clebsch-Gordan coefficients. The matrix element is zero unless *m’=m+q* and

Proof of the Wigner-Eckart theorem:

Using the commutation relations that define a tensor operator we have

yields

or

- Similarly we evaluate the matrix elements of

We find the same recursion relations for the as we found for the Clebsch-Gordan
coefficients with *j _{1}=j, j_{2}=k*, and