You will have to apply the Wigner-Eckart theorem mostly to vector operators.  It then becomes the projection theorem.

Let J be the total angular momentum of a system.  A scalar observable A commutes with all components of the angular momentum, [A,J]=0.  In the standard basis {|k,j,m>} the matrix elements of a scalar observable <k’,j’,m’|A|k,j,m> are non zero only if j=j’ and m=m’, and they are independent of m.

In a subspace E(k,j) the (2j+1)´(2j+1) matrix is diagonal and all is elements are equal.

The same holds for another scalar observable B.  Therefore, if we restrict ourselves to a subspace E(k,j) , we can obtain the matrix elements of B between any two vectors in this subspace by just multiplying the matrix of A by a constant.

Let P(k,j) be the projector into the subspace E(k,j). P(k,j) is a Hermitian operator.  We can express the above statement in the following way:

P(k,j)BP(k,j) = l(k,j)P(k,j)AP(k,j).

A vector observable V is defined through .  If we restrict ourselves to a subspace E(k,j), then the Wigner-Eckart theorem tells us that the matrix elements of any component of a vector observable V’ between any two vectors of this subspace can be obtained from the matrix elements of the same component of a vector observable V by just multiplying by a constant.

P(k,j)V’P(k, j)=m(k,j)P(k,j)VP(k,j).

In particular, for any vector observable V we have

P(k,j)VP(k,j)=a(k,j)P(k,j)JP(k,j).

Inside the subspace E(k,j) we have

<k,j,m’|V|k,j,m>=a(k,j)<k,j,m’|J|k,j,m>.

This is the projection theorem.  The projection theorem is a special case of the Wigner-Eckart theorem.

Inside each subspace, all matrix elements of V are proportional to the corresponding matrix elements of J.  (Note: In E(k,j) we have non-zero off-diagonal elements of V in the {|k,j,m>} basis, unlike for a scalar observable, but these off-diagonal elements are also proportional to the corresponding off-diagonal elements of J.  Also note, that the matrix elements of any function of J are zero between vectors in different subspaces E(k,j).  We cannot show this for a general vector operator V.)

The Wigner-Eckart theorem as applied to a vector operator can be written as

Note we have picked the same j for the bra and the ket but not the same k.  We also have

Here we pick the same j and the same k for the bra and the ket.  The are independent of k and k’. Therefore

or

with 

To find a(k’,k,j) we evaluate

With the spherical components of J being

and the spherical components of V being

we have

by the Wigner-Eckart theorem.  The coefficients c_jm do not depend on k and k’.  Furthermore c_jm is independent of m, since is a scalar operator.  The c_jm=c_j also do not depend on the physical nature of the vector operator V.  So if we set V=J and k’=k we have

Therefore

We have

In the subspace E(k, j) we have

i.e. the projection theorem.

Problem:

When calculating the splitting of atomic energy levels in a weak magnetic field  (Zeeman effect) we have to evaluate the matrix elements of the operator L_z+2S_z between eigenstates of H₀, J², and J_z.  Evaluate these matrix elements for the 2p orbitals.

• Solution:

  For the 2p orbitals we have l=1.  We can therefore have j=½, or j=3/2.

  • For j=3/2 we can have m=3/2, ½, -½, -3/2.

    The projection theorem states that the matrix elements of L_z+2S_z are proportional to the matrix elements of J_z.  The matrix elements of J_z are

    The proportional constant is

    

    so

    

     

    We therefore have

  • For j=1/2 we can have m=½, -½.

    Again the matrix elements of L_z+2S_z are proportional to the matrix elements of J_z. .

    Now the proportional constant is since

    

     

    We therefore have