## ABSTRACT

Classically, if we have a system with angular momentum J
_{1} and a second system with angular momentum then the total angular momentum of the combined systems is simply J
_{2} + J
_{2}. The corresponding situation in quantum mechanics is where we have one system in a state with a total angular momentum quantum number j_{1} and component m_{1} along some axis, and a second system in a state with total angular momentum quantum number j_{2} and component m_{2} along some axis; the problem is, what can we say about the total angular momentum of the two systems in this situation. The answer, unfortunately, is not as simple as it is in classical mechanics.