= Two-spin Heisenberg Hamiltonian in an opposing longitudinal field
{c}
For two spin-one-half particles,
$$
H=\alpha\mathbf S^{(1)}\mathbin\cdot\mathbf S^{(2)}
+B(S_z^{(1)}-S_z^{(2)})
$$
has two eigenvalues $\alpha\hbar^2/4$ and two further eigenvalues
$$
-\frac{\alpha\hbar^2}{4}
\pm\sqrt{B^2\hbar^2+\frac{\alpha^2\hbar^4}{4}}.
$$
For positive $\alpha$, the zero-field limit has a unique <spin-one-half singlet state> ground state and a triply degenerate <spin-one-half triplet state> excited level.
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