Semiclassical sum rules

The expression ${\left|{\Psi }_n\mathrm{}\right(0\left)\right|}^2=\frac{{\left(2\mathrm{}\mu \right)}^{\frac{3}{2}}{E_n}^{\frac{1}{2}}\left(\frac{d{E}_n}{\mathrm{dn}}\right)}{4{\pi }^2}$, relating the square of the $n\mathrm{th}$ $s$-wave wave function at the origin to the bound-state reduced mass $\mu$ and the excitation energy $E_n$, is derived semiclassically. The relation is then used to obtain several sum rules for electron-positron annihilation and an expression for the contribution of a given flavor of heavy quark to the photon-nucleon total cross section.