LS coupling and spin orbit interaction

[Clara Chung]

Why do you need a weak spin orbit interactions in individual electrons in order to use the LS coupling?
From what I know, we are free to combined angular momentum whenever [L1, L2]=0 and this should be true on the orbital angular momentum (and spin) of the electrons because they are differential operators of different electrons (different coordinates).
Moreover, even the above reason is enough for me, under spin orbit interaction, the orbital angular momentum and spin angular are good quantum numbers to indicate the state of an electron when using perturbation theory (they are n l s j mj).
So why?

 

[BigJDubs]

A:The LS coupling is a scheme used to describe the combination of the orbital angular momentum $\vec{L}$ and the spin angular momentum $\vec{S}$ of an electron.In the LS coupling, the total angular momentum $\vec{J}$ of the electron is expressed as$$ \vec{J} = \vec{L} + \vec{S} $$For this to work, one needs to assume that the interaction between $\vec{L}$ and $\vec{S}$ is weak, i.e. the spin-orbit interaction needs to be weak. This is because the spin-orbit interaction couples the two angular momenta and thus there is no clear separation between the two. If the spin-orbit interaction is not weak, then one needs to use a different scheme, such as the jj-coupling, to describe the combination of $\vec{L}$ and $\vec{S}$.