Matrix elements for composite systems

It is a common occurrence that we wish to evaluate the reduced matrix element of an operator which acts on only one part of a coupled scheme. For example, the general formula for the reduced matrix element of an operator Tk(A ) which acts only on part 1 of a coupled scheme j + j2 = j is  

To derive this result, we express a typical matrix element in terms of the coupled representation on the one hand and in terms of the decoupled on the other  

The rest of the derivation is performed by expressing the Clebsch-Gordan coefficients in terms of3-j symbols andmaking use of equation (5.92). The corresponding equation for a spherical tensor operator Tk(A2) which acts only on part 2 ofthe coupled scheme is 

Equation (5.136) results from this equation when we set Tk2 A2) and equation (5.138) follows when T4 (A ) = 1 and k =0. Another important result arises when k =k2 = k and K = 0. In this case Tjj is just the scalar product Tk(A ) Tk(A2), apart from a phase and a normalisation factor, see equation (5.110). In this case, we can use equation (5.94) to replace the 9-j symbol by a 6-j symbol and so arrive at the important result 

Finally, we consider the composite tensor Tk(Au B ) which is the tensor product of T4 (A ) and T/t2(fi ), both of which act on part 1 only of the coupled scheme  

Equations (5.136) and (5.13 8) are special cases of a more general result for the reduced matrix elements of a tensor operator T k, ki) obtained by coupling together T (Ai) and T H 2) which act on parts 1 and 2 respectively of the composite system. In this case, we require the reduced matrix element (71,72, 7 T- -(A i, k2)ll7i, 72- j )- We get it by considering the transformation from the coupling scheme ((J, ] )], (k, h)K) j to the coupling ((/, h)ji)j- The result is 

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