Two non-equivalent electrons. Representation of coupled momenta

Equation (1.14) defines the wave function of uncoupled momenta. It obeys the following orthonormality condition  

The condition S(n, n ) for many-electron atoms is normally ensured only approximately unlike the other d(a,a), which are rigorous. The principle of the orthogonality of the wave functions reflects the fact that at one time only one state described by a given set of exact quantum numbers is realized, an electron cannot occupy simultaneously several physical states. 

Spin-orbit interaction (1.21) couples the angular and spin momenta 1 and s. The main part of the spin-orbit interaction (1.21) is described by the operator 

It follows from the Eq. (10.2) that the spin-orbit interaction may be interpreted as the interaction of the appropriate momenta. If two momenta 1 and s interact with each other, then the corresponding quantum numbers become approximate. Only their vectorial sum 1 + s = j and its projection 

Each wave function forms a complete basis of functions, therefore, Eqs. (10.4) and (10.5) can be considered as the transformation from one representation to the other. 

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