Clebsch-Gordan Coefficients for three spin-1 particles?

[dipole]

I'm doing a problem where I need to know the coefficients to change from the
$\vec{J} = \vec{J}_1 + \vec{J}_2 + \vec{J}_3$ to the {$\vec{J}_1, \vec{J}_2, \vec{J}_3$} for three spin-1 particles, but I'm having trouble finding a table or reference for this... surely every time someone needs to write such a wave function they don't do all the algebra by hand, so where can I find a table to do this?

 

[vela]

You combine the angular momenta two at a time.

 

[dipole]

You combine the angular momenta two at a time.

This doesn't really help me...

For my situation, suppose I want to find all the states with $m = 2$. Well, there are three possibilities:

$\mid j = 3, m =2 \rangle$

and then two distinct states with $\mid j = 2, m = 2 \rangle$ which correspond to a
symmetric and anti-symmetric state, presumably. How can I construct these by just coupling $j_{12}$ with $j_3$ (where $j_{12}$ is the coupled-states of $j_1$ and $j_2$)? How do I even start and how do I know which linear combinations to couple to which? It's very confusing. :(

 

[vela]

Well, give it a shot. There's no shortcut method if that's what you're looking for.

 

[paranormal]

I understand your frustration in finding the Clebsch-Gordan coefficients for three spin-1 particles. These coefficients are essential in understanding the angular momentum states of composite systems, and it is important to have accurate and reliable data.

There are several resources available for finding Clebsch-Gordan coefficients, including tables and online calculators. One reliable source is the National Institute of Standards and Technology (NIST) database, which provides a comprehensive table of Clebsch-Gordan coefficients for various quantum numbers and spin states.

Another useful resource is the book "Atomic, Molecular, and Optical Physics Handbook" by Gordon W.F. Drake, which includes a comprehensive table of Clebsch-Gordan coefficients for various combinations of spins.

Additionally, there are several online calculators available that can quickly and accurately calculate the Clebsch-Gordan coefficients for specific quantum states. These calculators use algorithms based on the Wigner-Eckart theorem to determine the coefficients.

In conclusion, while it may be tedious to manually calculate the Clebsch-Gordan coefficients for three spin-1 particles, there are reliable and easily accessible resources available to assist you in your research. I suggest exploring the NIST database, the "Atomic, Molecular, and Optical Physics Handbook," or online calculators to find the coefficients you need for your problem.