In each subspace E(j₁,j₂) the eigenvectors of J² 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(j₁,j₂).

,

Properties of the Clebsch-Gordan coefficients:

• i) .
• ii) .

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

.

Therefore we can write

<j₁,j₂;m₁',m₂'|J₊|j₁,j₂;j,m>

=

,

or

.

For m=j we have

.

More recursion relations can be found using J- and J².

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 j₁, j₂, 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)