Matrix elements of spherical tensor operators the Wigner-Eckart theorem

Matrix elements of spherical tensor operators the Wigner-Eckart theorem 

We now consider how to evaluate the matrix elements of a spherical tensor operator, written as (rj, j, m Tk(A) rj, / . m ) where r] and if denote any further quantum numbers required to characterise the states (for example, vibrational quantum numbers v). If we now rotate the bra, operator and ket through Euler angles o using equations (5.44) and (5.102), the result must be unaffected. Thus  

We now integrate over all co, making use of equation (5.100), and divide each side by 87T2. The result is 

The quantity in braces in the second line of (5.122) is independent of the projection quantum numbers because it is a summation over all possible values of n, n and p. Thus we have 

This is the Wigner-Eckart theorem, a very important result which underpins most applications of angular momentum theory to quantum mechanics. It states that the required matrix element can be written as the product of a 3- j symbol and a phase factor, which expresses all the angular dependence, and the reduced matrix element (rj, j T/ (d) if. j ) which is independent of component quantum numbers and hence of orientation. Thus one quantity is sufficient to determine all (2j + 1) x (2k + 1) x (2/ + 1) possible matrix elements (rj, j, mfIkq(A) rj, jf m ). The phase factor arises because the bra (rj, j, m transforms in the same way as the ket (— y m rj, j, —m). The definition of the reduced matrix element in equation (5.123), which is due to Edmonds [1] and also favoured by Zare [4], is the one we shall use throughout this book. The alternative definition, promoted by Brink and Satchler [3], 

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