A eigenstate of addition of angular momenta

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Homework Statement


[tex]l~m\rangle[/tex]
[tex]l=l_1+l_2[/tex]
[tex]l_1=2,l_2=1[/tex]
Find eigenstates(of[tex]L_z[/tex]) [tex]|2~0\rangle[/tex]


Homework Equations





The Attempt at a Solution


[tex]|l=3~m=3\rangle=|l_1=2~m_1=2\rangle|l_2=1~m_2=1 \rangle[/tex]. I do [tex]L_-=L_{1-}+L_{2-}[/tex] 3times.
So I get [tex]|3~0\rangle[/tex]= (omit)
Then How can I find [tex]2~0\rangle[/tex]?
I know that [tex]2~0\rangle[/tex] is orthogonal to [tex]|3~0\rangle[/tex]. But I don't know how to use that property to derive a result.
Please give me an answer.
 

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