9.5: Spins and Exchange

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Now notice something strange. The exchange interaction has split the \({\bf S}=1\) states from the \( {\bf S}=0\) states. We could write the potential as \(\hat{V} = J_{nl} − (2\hat{S} − 1)K_{nl}\), even though the Hamiltonian does not act on the spin! This is because the sign of the exchange integral depends on the (anti)symmetry of the spatial wavefunction. Thus we can write the matrix element as

\[\langle \Phi |J_{nl} − (2S − 1)K_{nl}|\Phi \rangle \nonumber\]

This ‘exchange interaction’ appears to depend on the spin - the triplet states have lower energy than the singlet (this is one of Hund’s rules for determining energy levels in atoms). It is this type of exchange force which keeps spins aligned in a ferromagnet, not the magnetic interaction itself.