I started working on this topic and thought I would post it for other to contribute their ideas. The Clebsch-Gordan coefficients basically tell you how to move between two different representations of spin. I am trying to work them out logically. For starters I am doing the very simplest case: how to represent a spin-one particle as two spin one-half particles. The +1 and -1 spin states of course can only be up-up and down-down. It's the other two states that are interesting. We have in the two-particle representation: up-down (call it A>) and down-up (call it B>) We have to combine these to get the states l=1,m=0 (call it 10>) and l=0,m=0 (call it 00>) I think I know the answer: A> + B> gives 10>, and A> - B> gives 00> I wonder how we would argue that it has to work this way? I would find it convincing if there was an argument using an external magnetic field, and showing how the respective states behaved properly under its influence. In particular, the state 00> should do absolutely nothing under an external field, so I wonder how we would show that the difference of the A and B states behaves this way?