Coupling of Ket Spaces and the Addition of Angular Momenta

In these notes we discuss the coupling of ket spaces of systems to form the ket spaces of larger systems. The constituent systems may be independent of each other in some physical circumstances or in some models, while in other circumstances or more realistic models they may become coupled. We will see many examples of this process in this course, including the coupling of spin and orbital degrees of freedom for a single particle, the coupling of the electronic degrees of freedom in an atom with the degrees of freedom of the nucleus, the coupling of matter degrees of freedom with those of the electromagnetic field, and many more. In this process it frequently happens that the systems being coupled possess angular momentum operators, that is, the action of rotations on these systems is defined. Recall that the existence of angular momentum operators implies the existence of rotation operators, and vice versa. In this case, it turns out that rotations are also defined on the coupled system, and that the angular momentum in the coupled system is just the sum of the angular momenta of the constituent systems. Likewise, the rotations that act on the coupled ket space are the products of the rotations acting on the constituent spaces. In such cases there arises the problem of constructing the standard angular momentum basis in the coupled space in terms of the standard angular momentum bases in the constituent spaces. This problem is very common in applications, and is important for computing and classifying energy eigenstates and for many other purposes.