Kinetic-energy estimate from wavefunction nodes ($⟨K⟩\sim {\hslash }^2{n}^2/(m{a}^2)$)

A wavefunction with extent $a$ and many nearly equally spaced internal nodes of a wavefunction has local wavevector $k\sim \pi n/a$. Thus $⟨K⟩={\hslash }^2\int |{\psi }^{\prime }{|}^2/(2m\int |\psi {|}^2)$ scales as ${\hslash }^2{n}^2/(m{a}^2)$. 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.