There has been an enduring interest and controversy about whether or not one can define physically meaningful energy density and stress fields, $e\left(\mathbf{r}\right)$ and ${\sigma }_{\alpha \beta }\left(\mathbf{r}\right)$, in quantum systems. A key issue is the kinetic energy since the well-known forms, $\frac{1}{2}{\left|\mathbf{\nabla }\mathrm{\Psi }\right|}^2$ and $-\frac{1}{2}\mathrm{\Psi }{\nabla }^2\mathrm{\Psi }$, lead to different densities, and analogous issues arise for interaction energy terms. This paper considers the ground state of a system of many interacting particles in an external potential, and presents a resolution to the problems in steps. (1) For the kinetic energy all effects of exchange and correlation are shown to be unique functions defined at each point $\mathbf{r}$; all issues of nonuniqueness are relegated to terms that involve only the density $n\left(\mathbf{r}\right)$ and are equivalent to an effective single-particle problem with ground state wave function $s\left(\mathbf{r}\right)=\sqrt{n\left(\mathbf{r}\right)/N}$. (2) Interactions can be considered in two ways: in terms of potentials acting on particles or in terms of the interaction fields, e.g., the Maxwell form in terms of electric fields. In each case, the problem reduces to a mean field part that is a function of the density and a part due to correlation that is uniquely defined; however, it is different for the two cases. (3) The final results follow from the nature of energy and stress. Because the energy determines the ground state itself through the variational principle, the energy density approach leads directly to kinetic energy in the form $-\frac{1}{2}s{\nabla }^{2}s$ and interactions in terms of potentials acting on the particles. This leads naturally to density functional theory and provides an interpretation in which the energy density $e\left(\mathbf{r}\right)$ is equilibrated to minimize fluctuations with the same chemical potential at all points $\mathbf{r}$. On the other hand, stress is related to forces, and the only physically acceptable expressions for the stress field involve the combination $\frac{1}{2}[s{\nabla }^{2}s-{\left|\mathbf{\nabla }s\right|}^2]$, as derived by Schrödinger, Pauli, and others, and Coulomb interactions in terms of electric fields, not potentials. Together these results lead to well-defined formulations of energy density and stress fields that are physically motivated and based on a clear set of arguments.

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• Electronic structure
• First-principles calculations
• Band structure methods
• Density functional theory
• Embedded atom model
• Variational wave functional methods