= Three spin-one-half angular-momentum decomposition
{title2=$(1/2)^{\otimes3}=3/2\oplus1/2\oplus1/2$}
For three distinguishable <spin one-half> systems, the <Clebsch-Gordan decomposition> first combines two spins into $0\oplus1$, then couples the third: $0\otimes1/2=1/2$ and $1\otimes1/2=1/2\oplus3/2$. Thus the <tensor product> has one quartet and two doublets. A <Hamiltonian operator> depending only on the total $S^2$ and $S_z$ acts identically on the two doublet copies. In particular $H=\epsilon_tS^2/\hbar^2+\epsilon_zS_z/\hbar$ has <energy eigenvalues> $\epsilon_tS(S+1)+\epsilon_zM$, generically once for each quartet projection and twice for each doublet projection. Equal numerical energies have their multiplicities added.
Back to article page
