Eigenstates spin-orbit coupling

In light alkali atoms, Li and Na, the fine structure splitting of a low state is typically much larger than the radiative decay rate but smaller than the interval between adjacent states. In zero field the eigenstates are the spin orbit coupled tsjnij states in which and s are coupled. However, in very small fields and s are decoupled, and the spin may be ignored. From this point on all our previous analysis of spinless atoms applies. How the passage from the coupled to the uncoupled states occurs depends on how rapidly the field is applied. It is typically a simple variant of the question of how the m states evolve into Stark states. When... 

As spin-orbit coupling is weak in conjugated systems the total spin is a conserved quantum number. The low-lying energy eigenstates are singlet S = 0) and triplet (5=1) states. 

From Eq. (6.29) it is clear that we will require the eigenstates and eigenvalues of the spin-orbit coupling operator rr 1). Recalling the definitions of the spin operator s = her/2 and of the total angular momentum j = II2 + s as well as the equality (/ s) = (s 1) we can formulate an operator identity... 

The energy of the spin-orbit coupled eigenstates for a hydrogen atom is as follows. 

In the previous part of the chapter we expressed the problem of an electron in a local, central potential in terms of radial equations and eigenstates of orbital angular momentum. In generalising to the case where the electron obeys the Dirac equation (3.154) we remember that spin and orbital angular momentum are coupled. 

This coupling prevails in a many-electron configuration when the electrostatic interactions between all electron pairs are much stronger than all spin-orbit interactions of the individual electrons. Therefore, as before, H) alone can be taken into account in first-order perturbation theory, and the most appropriate quantum numbers for characterizing the energy eigenstates are... 

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