# Direct products to coupled states

1. Dec 5, 2011

### mtszyk

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 $|1\, m_1 \rangle \otimes |1\, m_2 \rangle$ are there and what possible coupled states $|1\, 1; L\, M\rangle$ are there?

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

3. The attempt at a solution
So, I know the nine product states are simply vary $m_1$ and $m_2$ 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 $L_{max}=l_1 + l_2$ and $M=m_1 + m_2$, so L goes from 0 to 2 and M corresponds, totaling 9 values as expected. This makes sense to me because the $L$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!