Wigner-Eckart theorem

1. Wigner-Eckart theorem:
The Wigner-Eckart theorem gives the M dependence of matrix elements of spherical tensor
operators in a basis of good angular momentum. If I and M are the angular momentum quantum
numbers for the total angular momentum operator appropriate for the space of a particular Tq
k,
Where α is shorthand for all quantum numbers other than I, M, the total angular momentum
quantum numbers for the system. The factor F (α,J,k, α’.J’) is independent of the quantum numbers
M, q, M'. The dependence on M, q, M' sits solely in the Clebsch-Gordan coefficient. The factor F
(α,J,k, α’.J’) is usually written in terms of the so-called "double-barred" or "reduced" matrix element,
The factor
1
√(2𝐽′
+1)
is factored out of the M, q, M' independent factor for convenience. (It makes the
absolute value of the reduced or double-barred matrix element invariant to bra, ket interchange).
Also, the Wigner-Eckart theorem greatly reduces the labor of calculating matrix elements. We need
to choose only one (convenient!) set M, q, M' to calculate the full matrix element. The rest then
follow through the Clebsch-Gordan coefficient. A very important special case is the one for
rotationally invariant or scalar, k = 0, operators. Because <JM00|J’M’> = 1δJJ’, δMM’, the matrix
element of a scalar (k = 0) operator is independent of M. The important case is that of a Hamiltonian
operator of an isolated system. Because the properties of the systemcannot depend on the
orientation of the frame of reference from which it is viewed, the Hamiltonian must be a spherical
tensor of rank I = O. The energies will be independent of the quantum number M. The (2J + I)-fold
degeneracy of the energy levels can be removed only through the action of external perturbations.
More important for calculation purposes, we can choose any convenient M, such as M = J, to
calculate the energies of the isolated (unperturbed) system. We now briefly give a few double-barred
matrix elements of some simple operators. The "extra" ,J(2l' + 1) factor, part of the definition of the
double barred matrix element, leads to the fact that the reduced matrix element of the simple unit
operator is not unity.

2.  Diagonal Matrix Elementsof Vector Operators

3.  Proof of the Wigner-Eckart theorem:

5. This is called Wigner-Eckart theorem.