= Kinetic-energy estimate from wavefunction nodes
{title2=$\langle K\rangle\sim\hbar^2n^2/(ma^2)$}
A <wavefunction> with extent $a$ and many nearly equally spaced internal <nodes of a wavefunction> has local <wavevector> $k\sim\pi n/a$. Thus $\langle K\rangle=\hbar^2\int|\psi'|^2/(2m\int|\psi|^2)$ scales as $\hbar^2n^2/(ma^2)$. Exact coefficients require the envelope and <boundary conditions>. An order-one node-count offset is needed for low states: a nodeless localized state still has positive <kinetic energy> by the <Heisenberg uncertainty principle>.
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