There is a discussion of this in appendix C of Perkins.
Recall that in quantum mechanics, to the measurable angular momentum is associated an operator, just like any other measurable quantity. In fact, angular momentum is more complex because there is a group of operators associated with it. Conceptually, it is easiest to begin with the operators for the angular momentum about each of the x, y, and z axes; I'll call these three operators Jx, Jy, and Jz.
Earlier I wrote out Jz in terms of the coordinates:
The fact that the operators don't commute means that a simultaneous measurement of more than one is impossible. Another operator that does commute with any of the three is the square of the total angular momentum:
Another pair of useful operators is
Now suppose we have a system with two (or more) angular momenta given by the states |j1, m1> and |j2, m2>, where the j's and m's are the corresponding quantum numbers (m's measured for a common axis), and we want to determine the angular momentum of the whole system, |J, M>. Classically we would simply add the vectors to get J = J1 + J2. But quantum mechanically, we can't simultaneously know all the components of angular momentum, so we can't perform the vector addition. In fact, the rules of quantum mechanics require that the x and y components of angular momenta take on all possible allowed values.
Therefore, the addition of |j1, m1> and |j2, m2> can result in a state with |j1 - j2| < J < j1 + j2. The z components are known though, and their addition yields the z component of the total angular momentum, M = m1 + m2. The result is that the whole system is a superposition of states with the same M and different J's: