What is the (J=0,M=0) state of a system of 3 angular momenta j1=j2=j3=1?

Answer 1

Classically, I'd think it corresponds to an equilateral triangle of j's. To work it out, set up the highest (J=3) state then use lowering and orthogonality to get to the other states that you want (cf Shankar, "Principles of Quantum Mechanics"). You get the coupled states as linear combinations of the states in the uncoupled basis, where the coefficients are Clebsch-Gordon coefficients.

One trick to notice is that , starting with any two of the j's and then adding the third, you cannot have J=2 and add the third to get zero, nor can you have J=0 and get 0 when you add the third one. Only J=1 will add with the third j=1 to give zero total. So you can start by finding the J=1 states for two of the j's. Then you know that the state you are looking for is some (probably fully antisymmetric) combination of these with the third j.