The chapter starts from physical Hamiltonians containing more than one angular momentum and then develops the basis language needed to treat those systems.
This first item is intentionally qualitative. The detailed definitions, calculations and examples are developed in the following items and in Chapter 7.
The opening section introduces interaction terms involving two angular momenta: exchange interactions, spin-orbit coupling, local anisotropy and the Zeeman interaction.
The common issue is that the Hamiltonian contains more than one angular momentum, so the local labels alone are not always the most useful basis labels.
The chapter constructs the coupled basis from angular momenta that first live in independent Hilbert subspaces.
The local basis keeps the individual projections visible; the coupled basis keeps the total angular momentum and total projection visible.
This roadmap should be used only to keep the logic of the chapter visible. It does not replace the worked algebra, the derivations or the examples.
The mathematical details begin in the next item, with the interaction Hamiltonians and the addition problem they motivate.