Probing quantum mechanical energy on a local scale

We explore the usefulness of using quantum mechanical energy densities, and investigate the computation of physically meaningful results despite the non-uniqueness of these quantities. In the case of the kinetic energy density, we define two different forms, the one form involving the -hbar²/2m phi(x) del² phi(x), and the other involving +hbar²/2m |phi(x)|². Since the difference of these two approaches includes a boundary term, we specifically explore the kinetic energy current densities at the boundaries and demonstrate why the first form gives physically more meaningful results than the second form.

We carry these notions over to many electron systems, within the framework of density functional theory, where we consider the Coulomb energy. Once again, we note the non-uniqueness in defining the Coulomb energy density, the one form involving the square of the electric field vector, and the other involving the product of the Coulomb potential with the density. We demonstrate why the first form gives physically more meaningful results than the second form.

This sets the stage for extracting even more useful information from the computational studies of real material systems using standard methods.