On the positivity of the kinetic-energy density

Summary

We study the positivity of a mixed form of the kinetic-energy density:$\tau _m (r) = (1/2)(\hbar ^2 /2m)[ - Re \psi ^ * (r)\Delta \psi (r) + |\nabla \psi (r)|^2 ]$. We show that for local Schrödinger equations with monotonic and other simple potentials, the positivity of τ_m can be checked from its asymptotic behaviour. The case for non-local potentials is also discussed. We consider the fluctuations of τ_m and its role in the derivation of the virial theorem. Few explicit one-dimensional examples are displayed for illustrative purposes. We shortly discuss also the positivity of the quadratic differential formK(f)=−f″ f+(f′) ²≥0 for solutions of one-dimensional second-order differential equations.