I would like to express the wave functions for three identical particles, each with orbital angular momentum $L=1$ and spin angular momentum $S=1/2$, in terms of single-particle wave functions. In other words, I would like to obtain the Clebsch-Gordan coefficients for this problem.
The problem is discussed in Sakurai's Quantum Mechanics textbook around p. 375 and in Greiner and Muller's "Quantum Mechanics: Symmetries, 2nd Edition" on p. 300. I know I have to find the spin wave functions and the orbital angular momentum wave functions separately, and then combine them to get fully antisymmetric wave functions. I have the spin wave functions (four symmetric, 2 mixed symmetric under exchange of particles 1 and 2, and 2 mixed antisymmetric under exchange of 1 and 2), but I haven't been able to get a small enough number of angular momentum wave functions to get just 20 fully-antisymmetric total wave functions.
In Sakurai's book, p 376, eq'n (6.5.20), we see that the 20 states can be decomposed into 2 states with total angular momentum $j=1/2$, 4*3=12 with $j=3/2$, and 6 with $j=5/2$. Could anyone fill in how Sakurai got 6 for $j=5/2$, 12 for $j=3/2$, and 2 for $j=1/2$?
Most importantly, referring to my comments on Y Macdisi's answer below, could anyone answer the following question: Is the orbital angular momentum state with single particle wave functions $\alpha=1$, $\beta=0$, and $\gamma=0$ related in some way to the wave function with $\alpha=-1$, $\beta=0$, and $\gamma=0$, or $\alpha=1$, $\beta=-1$, and $\gamma=-1$, and so on? I would love it if I could just keep the first set of values for $\alpha$, $\beta$, and $\gamma$, but I see no good reason to do that.
If anyone happens to know where this problem is discussed more fully, I would appreciate a reference. Or, if anyone knows how to do this, I would appreciate help in knowing what the $j=3/2$ and $j=5/2$ states could be in terms of single-particle orbital ang. momentum states and spin states. I think I actually have the normalization factors already, I just need to know what the single-particle states are.
Best Answer
The L=1 rep is a 3 dimensional irep of $SU(2)$; the S=1/2 rep is a 2 dimensional irep. Combining angular momentum (L=1) and spin (S=1/2) gives the tensor product rep $3 \otimes 2$ which is 6 dimensional. You are looking for the third exterior product of this : $A^3(3 \otimes 2)$ which is 20 dimensional and decomposes into $6 \oplus 4 \oplus 4 \oplus 4 \oplus 2$ irreps. These correspond to j=5/2,3/2,3/2,3/2,1/2 .