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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  [Pg.165]

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  [Pg.165]

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 [Pg.166]

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 [Pg.166]

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  [Pg.166]

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 [Pg.166]


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