Can I Couple Angular Momenta Into One Big One?

[suyver]

I have three angular momenta $l_1,l_2,l_3$ which I want to couple into one big one:

$$L\equiv l_1+l_2+l_3.$$

Can I just do this by coupling $l_1,l_2$ into $L'$ and then couple $L',l_3$ into $L$?

I would guess that I could equally couple $l_2,l_3$ into $L'$ and then couple $l_1,L'$ into $L$ and this would give the same result. Correct?
My reason to assume this: the different $l_i$ work on different parts of the system.

 

[Dr Transport]

That is the correct way to do it, couple two of the angular momenta then couple the third. If memory serves me correctly, this was done by Wigner and also by Racah...Look in Wigners book or Tinkhams book on Group Theory and Quantum Mechanics, it is all there.

 

[spdf13]

I've been doing a lot of reading on this subject lately. You are completley correct in stating that you can first add j1 and j2, and then add this to j3. Likewise you can first add j2 and j3 and add this to j1. These product eigenstates are related through the wigner 6J coefficients.

 

[suyver]

Thanks for answering, all! I was quite sure that I was right, but I thought it never hurts to ask. After all, there are no stupid questions (only stupid people )