 Exchange-coupled spin-one-half particles in an infinite square well

For well width $a$, put $\epsilon ={\pi }^2{\hslash }^2/(2m{a}^2)$ and $J=\lambda {\hslash }^2$ when the interaction is $-\lambda {\mathbf{S}}_1\cdot {\mathbf{S}}_2$. The spin-dot-product eigenvalue gives exchange energies $3J/4$ for the singlet state and $-J/4$ for the triplet states. Identical fermions require symmetric spatial orbitals with the singlet and antisymmetric orbitals with the triplet. Two particles in the same orbital can therefore only be singlets.

For $0<J\ll \epsilon$, the three lowest distinct energy levels are the orbital-$11$ singlet at $2\epsilon +3J/4$, orbital-$12$ triplet at $5\epsilon -J/4$, and orbital-$12$ singlet at $5\epsilon +3J/4$, with multiplicities $1,3,1$. Distinguishable particles instead allow an orbital-$11$ triplet, making the lowest three levels $2\epsilon -J/4$, $2\epsilon +3J/4$, and $5\epsilon -J/4$, with multiplicities $3,1,6$. The sixfold level includes both orbital orderings. For negative $J$, the spin-sector orderings reverse within each fixed orbital sum.