quantum mechanics - How to use Clebsch-Gordan coefficients for 3 particles?

I have a Hamiltonian for 3 particles of spin 1 that I boiled down to: \begin{equation} k(\textbf{S}^2+\cdots), \end{equation} where: \begin{equation} \textbf{S}=\textbf{S}_1+\textbf{S}_2+\textbf{S}_3. \end{equation} I read somewhere on the internet that the Clebsch-Gordan coefficients were the thing to use to figure out the value of $\textbf{S}^2$, but I'm not seeing it. Could someone explain to me how to use the Clebsch-Gordan coefficients for this case? - In particular, I understand (more or less) Clebsch-Gordan coefficients for adding two angular momenta, but here, I have 3 particles. Is there an equivalent of Clebsch-Gordan coefficients for three angular momenta $j_1,j_2,j_3$?

Answer

Clebsch-Gordan coefficients let you treat n spins (or generaly - any n particles with arbitrary angular momentum) as a single composite system. The coefficients are simply the matrix element of basis transformation from seperated to composite system.

For 2 particles with total angular momentum eigenvalues $ l_1,l_2 $, such that for example $ m_1=-l_1,-l_1+1,...,+l_1 $ the composite system is -

$$ (l_1) \otimes (l_2) \sim (l_1+l_2) \oplus ...\oplus(|l_1-l_2|) $$

2 spin half particles reduce to - $$ (\frac{1}{2})\otimes(\frac{1}{2})\sim (1)\oplus(0) $$

the left size decribes 2 particles, each has spin 1/2 ,and you can decribe the sany system configuration by combination of those spins -

$$ |++\rangle , |+-\rangle, |-+\rangle, |--\rangle $$

the right side describes the system as composite system, which can be described by it's total angular momentum and it's total $z$ component -

$$ |S_{tot}=1,m_s=1\rangle ,|S_{tot}=1,m_s=0\rangle , |S_{tot}=1,m_s=-1\rangle , |S_{tot}=0,m_s=0\rangle ,$$

As I said, the Clebsch-Gordan coefficients are the basis transformation matrix elements, for example, one of them is - $$ \langle++|S_{tot}=1,m_s=1\rangle $$

The way to add 3 spins is first to add 2 of the and then to add the third one to the composite system -

$$ [(\frac{1}{2})\otimes (\frac{1}{2})]\otimes (\frac{1}{2})\sim [(1)\oplus(0)]\otimes(\frac{1}{2})\sim [(1)\otimes(\frac{1}{2})]\oplus[(0)\otimes(\frac{1}{2})]\sim (\frac{3}{2})\oplus (\frac{1}{2})\oplus(\frac{1}{2})$$

(notice the first addition. the later one is "conditional" - if S1+S2=1 then ... if S1+S2=0 then)

so the new basis, in order of $ |S_{tot},S_{1+2},m_{tot}\rangle $ is

$ |\frac{3}{2},1,-\frac{3}{2}\rangle,|\frac{3}{2},1,-\frac{1}{2}\rangle,|\frac{3}{2},1,\frac{1}{2}\rangle,|\frac{3}{2},1,\frac{3}{2}\rangle,|\frac{1}{2},1,\frac{1}{2}\rangle,|\frac{1}{2},1,-\frac{1}{2}\rangle,|\frac{1}{2},0,-\frac{1}{2}\rangle,|\frac{1}{2},0,\frac{1}{2}\rangle $

this is enough for most applications (finding energy spectrum and such)