Isospin and Clebsch Gordan coefficients

Question

If I have to combine 2 spin $\frac{1}{2}$ particles I do with Clebsch Gordan coefficients so that for example $|{\frac{1}{2},\frac{1}{2}}\rangle|{\frac{1}{2},\frac{-1}{2}}\rangle=\frac{1}{\sqrt{2}}(|{1,0}\rangle+|{0,0}\rangle)$. Now if I have to calculate C-G coefficients for this new state and another state for example $|{\frac{1}{2},\frac{1}{2}}\rangle$, how I have to do?

Answer 1

I recommend to you "Lie Algebra in particle physics" by Howard Georgi, section 3.5, he gives a very detailed example.

The idea is to start with the highest spin state: in your example it's $\big|1,1\big\rangle = \big|\frac{1}{2},\frac{1}{2}\big\rangle\big|\frac{1}{2},\frac{1}{2}\big\rangle$ and then act with the lowering operator $J^-$.

for instance for the first time you get \begin{align} J^-\big|1,1\big\rangle &= J^-\left(\big|\frac{1}{2},\frac{1}{2}\big\rangle\big|\frac{1}{2},\frac{1}{2}\big\rangle\right)\\ \big|1,0\big\rangle&=\sqrt{\frac{1}{2}}\left(\big|\frac{1}{2},-\frac{1}{2}\big\rangle\big|\frac{1}{2},\frac{1}{2}\big\rangle+|\frac{1}{2},\frac{1}{2}\big\rangle\big|\frac{1}{2},-\frac{1}{2}\big\rangle\right) \end{align}