= Exchange-coupled spin-one-half particles in an infinite square well
{title2=$E_{rs,S}=(r^2+s^2)\varepsilon-\tfrac J2[S(S+1)-3/2]$}
For well width $a$, put $\varepsilon=\pi^2\hbar^2/(2ma^2)$ and $J=\lambda\hbar^2$ 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 $0<J\ll\varepsilon$, the three lowest distinct <energy levels> are the orbital-$11$ singlet at $2\varepsilon+3J/4$, orbital-$12$ triplet at $5\varepsilon-J/4$, and orbital-$12$ singlet at $5\varepsilon+3J/4$, with multiplicities $1,3,1$. Distinguishable particles instead allow an orbital-$11$ triplet, making the lowest three levels $2\varepsilon-J/4$, $2\varepsilon+3J/4$, and $5\varepsilon-J/4$, 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.
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