A Transition probabilities along the z direction

A system constituted by two spins, / and S, is considered. The Hamiltonian is given by 

In the coupled system I-S there are four possible eigenstates according to the values of m/z (first term in the ket) and msz (second term in the ket), as shown in Fig. 3.8A. The transition probabilities in this system are wf, wf, wq and W2- 

We shall now further rearrange Eq. (V.5) so as to distinguish terms which leave the total z component, Mz = m/z + msz, unchanged, terms for which AMz = 1, and terms for which A Mz = 2. This distinction is important because these terms will contribute to wo, w and wf, and w2, respectively. The direction cosines can be expressed by the polar angles, according to the relationships 

Note that the F, functions are just the spherical harmonics of order i. We can now express the transition probabilities. Let us start from 

The only part of H extracting H—) from —h) is that containing /+ , and therefore 

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