http://en.wikipedia.org/wiki/Wigner%E2%80%93Eckart_theorem
The Wigner–Eckart theorem is a theorem of representation theory and quantum mechanics. It states that matrix elements of spherical tensor operators on the basis of angular momentum eigenstates can be expressed as the product of two factors, one of which is independent of angular momentum orientation, and the other a Clebsch-Gordan coefficient. The name derives from physicists Eugene Wigner and Carl Eckart who developed the formalism as a link between the symmetry transformation groups of space (applied to the Schrödinger equations) and the laws of conservation of energy, momentum, and angular momentum.[1]
The Wigner-Eckart theorem reads:
where 
is a rank 
spherical tensor, 
and 
are eigenkets of total angular momentum 
and its z-component 
, 
has a value which is independent of 
and 
, and 
is the Clebsch-Gordan coefficient for adding 
and to get 
.
In effect, the Wigner–Eckart theorem says that operating with a spherical tensor operator of rank on an angular momentum eigenstate is like adding a state with angular momentum to the state. The matrix element one finds for the spherical tensor operator is proportional to a Clebsch-Gordan coefficient, which arises when considering adding two angular momenta.
Example
Consider the position expectation value 
. This matrix element is the expectation value of a Cartesian operator in a spherically-symmetric hydrogen-atom-eigenstate basis, which is a nontrivial problem. However, using the Wigner–Eckart theorem simplifies the problem. (In fact, we could get the solution right away using parity, but we’ll go a slightly longer way.)
We know that 
is one component of 
, which is a vector. Vectors are rank-1 tensors, so is some linear combination of 
for 
. In fact, it can be shown that 
, where we defined the spherical tensors[2] 
and 
(the pre-factors have to be chosen according to the definition[3] of a spherical tensor of rank k. Hence, the 
are only proportional to the ladder operators). Therefore
which is zero since both of the Clebsch-Gordan coefficients are zero.
References
- ^ Eckart Biography– The National Academies Press
- ^ J. J. Sakurai: “Modern quantum mechanics” (Massachusetts, 1994, Addison-Wesley)
- ^ J. J. Sakurai: “Modern quantum mechanics” (Massachusetts, 1994, Addison-Wesley)
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