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Homework Help: Direct products to coupled states

  1. Dec 5, 2011 #1
    I'm a bit unclear on exactly what a coupled state really means and how to represent it, so here's what I have:

    1. The problem statement, all variables and given/known data
    Consider the coupling of two spinless l=1 particles,
    What possible product states [itex]|1\, m_1 \rangle \otimes |1\, m_2 \rangle[/itex] are there and what possible coupled states [itex]|1\, 1; L\, M\rangle [/itex] are there?

    2. Relevant equations
    [itex]|j_1\, j_2;j\, M \rangle = \sum_{m_1, m_2}(j_1m_1;j_2m_2|jm) |1\, m_1 \rangle \otimes |1\, m_2 \rangle[/itex]

    3. The attempt at a solution
    So, I know the nine product states are simply vary [itex]m_1[/itex] and [itex]m_2[/itex] from -1 to 1, but what are L and M for the coupled states? The only thing I could think of and make the number of each representation match is have [itex]L_{max}=l_1 + l_2[/itex] and [itex]M=m_1 + m_2[/itex], so L goes from 0 to 2 and M corresponds, totaling 9 values as expected. This makes sense to me because the [itex]L[/itex]s need not be in the same direction, but I'm really just grasping at straws.

    The next part of the problem asks to solve for some CG coefficients using a method analogous to class, but if I understand this part I'm pretty certain that I can do the other part.

    Thanks for your time!
  2. jcsd
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