# Addition of Angular Momementa in deeply bound situations, proton spin crisis

In Landau and Lipshitz's introductory book on Quanum Mechanics, "Quantum Mechanics Non-Relativistic Theory, Third Edition: Volume 3", chapter XIV (page 433 in the edition on Amazon) is "Addition of Angular Momenta". Interestingly, the first page is footnoted to the effect that the theory covered only applies in the limit of a binding force sufficiently weak that the angular momentum can be considered separately for the particles:

Strictly speaking, we shall always be considering (without explicitly mentioning the fact each time) a system whose parts interact so weakly that their angular momenta may be regarded as conserved in a first approximation.

All the results given below apply, of course, not only to the addition of the total angular momenta of two particles (or systems) but also to the addition of the orbital angular momentum and spin of the same system, assuming that the spin-orbit coupling is sufficiently weak.

Can someone provide a more complete explanation for this? I'm curious as to what fails as the binding force gets too strong. I'd like an intuitive explanation.

One of the reasons for wanting to know more about this is that I suppose it has something to do with the "proton spin crisis", that is, the question of how angular momentum is distributed among the components (quark, gluon &c.) of a proton. For example see:

A. W. Thomas, Int.J.Mod.Phys.E18:1116-1134,2009, "Spin and orbital angular momentum of the proton" 0904.1735

So it seems like an important part of understanding strongly bound states and it would be nice to better understand the limitations of the usual methods from an intuitive as well as through calculations.

Consider two independent systems, two distant atoms, for example. Each atom has is own angular momentum $J_1$ and $J_2$. If these atoms are different and distant (non interacting), then the QM variables are separated, the total wave function becomes a product of the two atomic wave functions, and the total system angular momentum is a sum of the two. The system energy is determined with the sum of atomic energies and depends on $J_1$ and $J_2$ in a simple way.