Bring the chapter together: the exchange operator, bosons and fermions, symmetric and antisymmetric states, Slater determinants and the Pauli principle, exchange energies, the helium atom, many-electron atoms and the self-consistent field. No worked example sits above the answers.

A choir sings from one score with no names on the parts: any singer can take any line, and the harmony comes out the same. Identical particles share one state in which no part belongs to anyone.

$\hat{P}_{12}\psi(x_{1},x_{2})=\psi(x_{2},x_{1})$ with eigenvalue $+1$ for bosons and $-1$ for fermions; $d(d\pm1)/2$ two-particle states; $E_{\pm}=E_{a}+E_{b}+J\pm K$.

• No two electrons in one spin-orbital; a subshell holds $2(2l+1)$.
• Helium: $\bar{Z}=Z-\tfrac{5}{16}$ and $E=-\bar{Z}^{2}$ hartree.
• Hund: largest $S$, then largest $L$, then $J=|L-S|$ below half filling and $L+S$ above.