Direct products to coupled states

[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:

Homework Statement

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?

Homework 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]

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!

 

[mark726]

Hello there,

A coupled state refers to a quantum state that is a combination of two or more individual states. In the case of two spinless particles, the possible coupled states are represented by the notation |1\, 1; L\, M\rangle, where L is the total angular momentum and M is the total angular momentum projection.

In your attempt at a solution, you are correct in stating that L_{max}=l_1 + l_2, where l_1 and l_2 are the individual angular momenta of the two particles. However, M is not simply the sum of m_1 and m_2. Instead, it can take on values from -L to L, with each value corresponding to a different coupled state.

To determine the possible coupled states, we use the Clebsch-Gordan coefficients (CG coefficients) in the equation you provided. These coefficients represent the coupling of two angular momenta, and can be found in tables or calculated using specific formulas.

For example, if we have two particles with l=1, then the possible coupled states are |1\, 1; 0\, 0\rangle, |1\, 1; 1\, 1\rangle, |1\, 1; 1\, 0\rangle, |1\, 1; 1\, -1\rangle, |1\, 1; 1\, -2\rangle, |1\, 1; 0\, -1\rangle, |1\, 1; 0\, -2\rangle, |1\, 1; -1\, -1\rangle, |1\, 1; -1\, -2\rangle. These correspond to the possible values of L and M for the coupled states.

I hope this helps clarify the concept of coupled states and how to represent them. Let me know if you have any further questions. Good luck with the rest of your problem!