# A eigenstate of addition of angular momenta

1. Nov 2, 2011

### rar0308

1. The problem statement, all variables and given/known data
$$l~m\rangle$$
$$l=l_1+l_2$$
$$l_1=2,l_2=1$$
Find eigenstates(of$$L_z$$) $$|2~0\rangle$$

2. Relevant equations

3. The attempt at a solution
$$|l=3~m=3\rangle=|l_1=2~m_1=2\rangle|l_2=1~m_2=1 \rangle$$. I do $$L_-=L_{1-}+L_{2-}$$ 3times.
So I get $$|3~0\rangle$$= (omit)
Then How can I find $$2~0\rangle$$?
I know that $$2~0\rangle$$ is orthogonal to $$|3~0\rangle$$. But I don't know how to use that property to derive a result.