Spin-orbit interaction and the vector

We now illustrate the usefulness of the coupled representation by considering the problem of spin-orbit interaction in one-electron atoms. Tlie anomalous Zeeman effect shows that each electron with intrinsic spin angular momentum hs possesses an associated magnetic moment given by 

This expression is only really useful in hydrogenic systems where the electrostatic potential V(r) is accurately known. 

To obtain a simple description of one-electron atoms we must find wavefunctions which are simultaneous eigenfunctions of the main Hamiltonian equation (3.9), and of the much smaller spin-orbit interaction,, equation (3.59). An alternative way of stating this problem is that we are seeking a representation (i.e. a set of wavefunctions) in which the total Hamiltonian diagonal. 

In this representation the expectation value of the spin-orbit interaction has the well-defined value 

The following explicit expressions for the wave-functions representing a one-electron atom may be obtained by using the Clebsch-Gordan coefficients given in Table 3.3  

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