In each subspace E(*j*_{1},*j*_{2}) the eigenvectors of *J*^{2} and *J*_{z}
are linear combinations of basis
vectors. We write

The are called the **Clebsch-Gordan coefficients**.

All vectors are also linear combinations of basis vectors. We write

,

.

and are orthonormal bases of E(*j*_{1},*j*_{2}).

,

**Properties of the Clebsch-Gordan coefficients**:

- i) .
- ii) .

We can find recursion relations for the Clebsch-Gordan coefficients. We have

.

Therefore we can write

*<j _{1},j_{2};m_{1}',m_{2}'|J_{+}|j_{1},j_{2};j,m>*

=

,

or

.

For *m=j* we have

.

More recursion relations can be found using *J-* and *J*^{2}.

The Clebsch-Gordan coefficients are only defined to within an arbitrary phase. We choose this phase so that

- i) all Clebsch-Gordan coefficients are real,
- ii) (stretched case).

All coefficients with the same *j*_{1}, *j*_{2}, and *j*
are now fixed through recursion relations.

For different *j* we choose

All coefficients are now fixed. Tables of the Clebsch-Gordan coefficients are available.

Links:

- Clebsch-Gordan coefficients (1), (2)
- Clebsch-Gordan coefficient calculator (1)
- Clebsch-Gordan coefficient calculator (2)