A composite system of three particles with different spin

[Demon117]

In general how does one find the basis of a composite system of particles with different spin. Say for example spin 1, spin 1/2, and spin 5/2.

My first thought was to consider the uncoupled basis wherein the particles with spin 1/2 and spin 5/2 have similar state kets. Would the basis for the particle of spin 5/2 "envelope" the basis of the particle with spin 1/2?

 

[Demon117]

For starters, the uncoupled basis would be

$\left\{|\frac{1}{2},1,\frac{3}{2}\rangle ,|\frac{1}{2},1,\frac{1}{2}\rangle<br /> ,|\frac{1}{2},1,-\frac{1}{2}\rangle ,|\frac{1}{2},1,-\frac{3}{2}\rangle<br /> ,|\frac{1}{2},0,\frac{3}{2}\rangle ,|\frac{1}{2},0,\frac{1}{2}\rangle<br /> ,|\frac{1}{2},0,-\frac{1}{2}\rangle ,|\frac{1}{2},0,-\frac{3}{2}\rangle<br /> ,|\frac{1}{2},-1,\frac{3}{2}\rangle ,|\frac{1}{2},-1,\frac{1}{2}\rangle<br /> ,|\frac{1}{2},-1,-\frac{1}{2}\rangle ,|\frac{1}{2},-1,-\frac{3}{2}\rangle<br /> ,|-\frac{1}{2},1,\frac{3}{2}\rangle ,|-\frac{1}{2},1,\frac{1}{2}\rangle<br /> ,|-\frac{1}{2},1,-\frac{1}{2}\rangle ,|-\frac{1}{2},1,-\frac{3}{2}\rangle<br /> ,|-\frac{1}{2},0,\frac{3}{2}\rangle ,|-\frac{1}{2},0,\frac{1}{2}\rangle<br /> ,|-\frac{1}{2},0,-\frac{1}{2}\rangle ,|-\frac{1}{2},0,-\frac{3}{2}\rangle<br /> ,|-\frac{1}{2},-1,\frac{3}{2}\rangle ,|-\frac{1}{2},-1,\frac{1}{2}\rangle<br /> ,|-\frac{1}{2},-1,-\frac{1}{2}\rangle ,|-\frac{1}{2},-1,-\frac{3}{2}\rangle<br /> \right\}$

The notation may be a little strange for some of you but it basically incorporates the spin states of each particle and the direction of orientation; this is mean't to save time.

 

[Vanadium 50]

You can also use 6-J symbols and do it all in one shot. One painful shot, but one shot.