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Just when I think I've understood addition of angular momentum, I find a problem that completely questions everything I think I know. Okay so here's the problem: question 3/II/32D on page 68 http://www.maths.cam.ac.uk/undergrad/pastpapers/2005/Part_2/list_II.pdf [Broken] .

For the possible values of J and M as I understand it, [tex]j_1+j_2\geq J\geq|j_1-j_2|,~j_1+j_2\geq M\geq-(j_1+j_2)[/tex] and therefore for two j=1 systems surely J=2,1, or 0 and M=-2,-1,0,1,2?

Thus we must find expressions for the states [tex]|1~ 2\rangle ,~|1 ~1\rangle ,~|1 ~0\rangle ,~|1 ~-1\rangle ,~and |1 ~-2\rangle ,~[/tex]?

So letting [tex]|1~m_1\rangle |1~m_2\rangle=|m_1\rangle|m_2\rangle[/tex] where [tex]m_i=\pm 1[/tex], we start with the top state J=2:

[tex]|2~2\rangle=|1\rangle|1\rangle[/tex]

Then apply [tex]J_-[/tex] to get [tex]|2~ 1\rangle=\frac{1}{\sqrt{2}}(|0\rangle|1\rangle+|1\rangle|0\rangle)[/tex].

Orthog. combination gives us a J=1 state: [tex]|1 ~1\rangle=\frac{1}{\sqrt{2}}(|0\rangle|1\rangle-|1\rangle|0\rangle)[/tex]. But if we try and apply [tex]J_+[/tex] to this in order to get [tex]|1 ~2\rangle[/tex] we just get zero, why? How do we get [tex]|1 ~2\rangle[/tex]?

For the possible values of J and M as I understand it, [tex]j_1+j_2\geq J\geq|j_1-j_2|,~j_1+j_2\geq M\geq-(j_1+j_2)[/tex] and therefore for two j=1 systems surely J=2,1, or 0 and M=-2,-1,0,1,2?

Thus we must find expressions for the states [tex]|1~ 2\rangle ,~|1 ~1\rangle ,~|1 ~0\rangle ,~|1 ~-1\rangle ,~and |1 ~-2\rangle ,~[/tex]?

So letting [tex]|1~m_1\rangle |1~m_2\rangle=|m_1\rangle|m_2\rangle[/tex] where [tex]m_i=\pm 1[/tex], we start with the top state J=2:

[tex]|2~2\rangle=|1\rangle|1\rangle[/tex]

Then apply [tex]J_-[/tex] to get [tex]|2~ 1\rangle=\frac{1}{\sqrt{2}}(|0\rangle|1\rangle+|1\rangle|0\rangle)[/tex].

Orthog. combination gives us a J=1 state: [tex]|1 ~1\rangle=\frac{1}{\sqrt{2}}(|0\rangle|1\rangle-|1\rangle|0\rangle)[/tex]. But if we try and apply [tex]J_+[/tex] to this in order to get [tex]|1 ~2\rangle[/tex] we just get zero, why? How do we get [tex]|1 ~2\rangle[/tex]?

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