Energy density and stress fields in quantum systems

A long-standing issue is whether or not it is possible to define energy density e(r) and stress fields σ_αβ(r) in quantum systems since the two forms of the kinetic energy -½ψ∇²ψ or ½|∇ψ|² lead to different densities and there are similar issues for interactions. This work presents a resolution in two steps. 1) Demonstration that all effects of exchange and correlation are unique functions at each point r; all issues of non-uniqueness involve only the density n(r). 2) Derivation of explicit forms based on the nature of energy and stress. Because minimization of the energy determines ψ, the appropriate density involves the terms in the hamiltonian: -½ψ∇²ψ and interactions in terms of potentials. This leads to DFT interpreted as equilibration of the energy density e(r). But energy is also determined by ψ, as is stress, and simple examples show that the only acceptable form is ¼[|∇ψ|²-ψ∇²ψ], as derived by Pauli, and interactions in terms of electric fields, not potentials. This leads a unique local pressure (-⅓Trσ(r)) and we propose forms for other components of the stress tensor. Since stress is a derivative of the energy this also leads to an energy density. Thus we find two distinct energy densities, each well-defined and appropriate for specified uses, and a unique stress (local pressure) density.