Bring the chapter together: the Schrödinger equation in space, separable potentials and the cubic box, central potentials and the conservation of angular momentum, the radial equation, the effective potential, the labels of states and the two-body problem. No worked example sits above the answers.

A map of a city is flat, but the city has height. Moving from one dimension to three is adding the height back: the streets are the same, there are simply more ways to get anywhere.

A separable potential gives products and summed energies. A central potential gives $\frac{u(r)}{r}Y_{\ell m}$, with $u$ obeying a one-dimensional equation in the effective potential, $u(0)=0$, and energies independent of $m$.

• Cubic box: $E\propto n_{x}^{2}+n_{y}^{2}+n_{z}^{2}$, with degeneracies from rearrangements.
• Near the origin $u\propto r^{\ell+1}$; only $s$ states reach the centre.
• Coulomb $V_{\text{eff}}$: minimum at $r_{0}=\hbar^{2}\ell(\ell+1)/mk$, depth .
• A spherical well binds only if $V_{0}a^{2}>\pi^{2}\hbar^{2}/8m$.