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arXiv:2501.08432 · Physical review. B./Physical review. B

Energy density and stress fields in quantum systems

Richard M. Martin, N. Chetty, Dallas R. Trinkle

Published Nov 13, 2025• 0 citations
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Abstract

There has been an enduring interest and controversy about whether or not one can define physically meaningful energy density and stress fields, $e(\mathbf{r})$ and ${\ensuremath{\sigma}}_{\ensuremath{\alpha}\ensuremath{\beta}}(\mathbf{r})$, in quantum systems. A key issue is the kinetic energy since the well-known forms, $\frac{1}{2}{|\mathbf{\ensuremath{\nabla}}\mathrm{\ensuremath{\Psi}}|}^{2}$ and $\ensuremath{-}\frac{1}{2}\mathrm{\ensuremath{\Psi}}{\ensuremath{\nabla}}^{2}\mathrm{\ensuremath{\Psi}}$, 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 $n(\mathbf{r})$ and are equivalent to an effective single-particle problem with ground state wave function $s(\mathbf{r})=\sqrt{n(\mathbf{r})/N}$. (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 $\ensuremath{-}\frac{1}{2}s{\ensuremath{\nabla}}^{2}s$ 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 $e(\mathbf{r})$ 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 $\frac{1}{2}[s{\ensuremath{\nabla}}^{2}s\ensuremath{-}{|\mathbf{\ensuremath{\nabla}}s|}^{2}]$, as derived by Schr\"odinger, 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.

DOI
10.1103/49lf-cbyk

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There has been an enduring interest and controversy about whether or not one can define physically meaningful energy density and stress fields, e(r)e(\mathbf{r})e(r) and \ensuremathσ\ensuremathα\ensuremathβ(r){\ensuremath{\sigma}}_{\ensuremath{\alpha}\ensuremath{\beta}}(\mathbf{r})\ensuremathσ\ensuremathα\ensuremathβ​(r), in quantum systems. A key issue is the kinetic energy since the well-known forms, 12∣\ensuremath∇\ensuremathΨ∣2\frac{1}{2}{|\mathbf{\ensuremath{\nabla}}\mathrm{\ensuremath{\Psi}}|}^{2}21​∣\ensuremath∇\ensuremathΨ∣2 and \ensuremath−12\ensuremathΨ\ensuremath∇2\ensuremathΨ\ensuremath{-}\frac{1}{2}\mathrm{\ensuremath{\Psi}}{\ensuremath{\nabla}}^{2}\mathrm{\ensuremath{\Psi}}\ensuremath−21​\ensuremathΨ\ensuremath∇2\ensuremathΨ, 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 r\mathbf{r}r; all issues of nonuniqueness are relegated to terms that involve only the density n(r)n(\mathbf{r})n(r) and are equivalent to an effective single-particle problem with ground state wave function s(r)=n(r)/Ns(\mathbf{r})=\sqrt{n(\mathbf{r})/N}s(r)=n(r)/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 \ensuremath−12s\ensuremath∇2s\ensuremath{-}\frac{1}{2}s{\ensuremath{\nabla}}^{2}s\ensuremath−21​s\ensuremath∇2s 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(r)e(\mathbf{r})e(r) is equilibrated to minimize fluctuations with the same chemical potential at all points r\mathbf{r}r. On the other hand, stress is related to forces, and the only physically acceptable expressions for the stress field involve the combination 12[s\ensuremath∇2s\ensuremath−∣\ensuremath∇s∣2]\frac{1}{2}[s{\ensuremath{\nabla}}^{2}s\ensuremath{-}{|\mathbf{\ensuremath{\nabla}}s|}^{2}]21​[s\ensuremath∇2s\ensuremath−∣\ensuremath∇s∣2], as derived by Schr"odinger, 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.
Based on the Schrodinger equation, exact expressions for the non-relativistic particle energy in the local external field and the external field potential are derived as inhomogeneous density functionals. On this basis, it is shown that, when considering more than two noninteracting electrons, the energy of such a system cannot be an inhomogeneous density functional. The result is extended for the system of interacting electrons. This means that the Hohenberg-Kohn lemma which assert that in the ground state to each inhomogeneous density corresponds only one potential of the external field cannot be a justification of the existence of the universal density functional in the general case. At the same time, statements of the density functional theory remain valid when considering any number of noninteracting ground-state bosons due to the Bose condensation effect.
We derive a functional form for the energy of interacting many-particle systems from first principles. Our main contribution is that we define the effective zero-point energy of an interacting system in terms of fluctuations of momentum and the Coulomb electric field. An uncertainty equation can be derived, relating the interaction energy in terms of the kinetic energy and local charge density. The Lieb-Thirring bound is then applied to constrain the kinetic energy which takes particle (anti-)symmetry into account, and the resulting functional form applies to fermionic as well as bosonic systems; in the uniform density case, the functional form agrees with the entire range of available Quantum Monte Carlo (QMC) data of many-particle systems, to within chemical accuracy. Furthermore, the functional form shows promising agreement with QMC phase transition data found in the literature, and in the high density case, where QMC data is not as readily available for a BCC Wigner state, predicts the stability of a BCC Wigner crystal agreeing qualitatively with Unrestricted Hartree-Fock. Seen in this way, many-particle wavefunctions are minimum uncertainty states in coherence with the electromagnetic bath, where the coupling is a self-generated scalar potential.
I show how the Fermi and Bose pressures in quantum systems, identified in standard discussions through the use of thermodynamic analogies, can be derived directly in terms of the flow of momentum across a surface by using the quantum mechanical stress tensor. In this approach, analogous to classical kinetic theory, pressure is naturally defined locally, a point which is obvious in terms of the stress-tensor but is hidden in the usual thermodynamic approach. The two approaches are connected by an interesting application of boundary perturbation theory for quantum systems. The treatment leads to a simple interpretation of the pressure in Fermi and Bose systems in terms of the momentum flow encoded in the wave functions. I apply the methods to several problems, investigating the properties of quasi continuous systems, relations for Fermi and Bose pressures, shape-dependent effects and anisotropies, and the treatment of particles in external fields, and note several interesting problems for graduate courses in statistical mechanics that arise naturally in the context of these examples.
It is shown that interacting many-particle quantum systems can be described in terms of fully correlated nnn-particle densities, which determine uniquely the potential acting on the system and satisfy a minimum principle with respect to the ground-state energy. This leads to a generalization of ordinary density functional theory in terms of nnn-particle densities which allows the direct and self-consistent treatment of correlation effects within electronic structure methods for atoms, molecules, and solids.
I show how the pressure in Fermi and Bose systems, identified in standard discussions of quantum statistical mechanics by the use of thermodynamic analogies, can be derived directly in terms of the flux of momentum across a surface by using the quantum mechanical stress tensor. In this approach, which is analogous to classical kinetic theory, the pressure is naturally defined locally. The approach leads to a simple interpretation of the pressure in terms of the momentum flow encoded in the wave functions. The stress-tensor and thermodynamic approaches are related by an interesting application of boundary perturbation theory for quantum systems. I investigate the properties of quasi-continuous systems, the relations for Fermi and Bose pressures, shape-dependent effects and anisotropies, and the treatment of particles in external fields, and note several interesting problems for graduate courses in statistical mechanics.
Both quantum bound state equilibrium and classical statistical mechanical equilibrium are characterized by d/dt partial density(x,t)=0. For a quantum particle, W(x,t)=exp(-iEt)W(x) and d(x,t)=W*W. Density is linked to fluid flow and arguments of pressure balance such that the overall average fluid flow is zero are made. This pressure-force balance is common in classical statistical mechanics and used to calculate the variation of density with x i.e. obtain a differential equation with d d(x,t) /dt partial. A similar balance equation is used in (1) as the authors try to derive the Schrodinger equation using a fluid dynamical approach. In this note, we try to argue that one may think of the two equilibria in terms of an energy quantity which remains constant throughout space. This energy quantity depends directly on the type of interactions in the system at any point x. For a classical ideal gas, balanced two body time reversal scattering represents the interaction. V(x), a potential, does not change this fundamental manner of interacting at any given point, although it does affect density. As a result the energy related constant at each x is T, related to average kinetic energy. We argue that this already points to the dominant type of interaction creating the statistical interaction i.e. one that depends only on p in interactions which conserve kinetic energy. This “happens” to be equivalent to maximizing Shannon’s entropy, but we argue this is particular to this scenario in the sense that not all interacting systems will have this equivalence. In other words, we argue that one must be careful with regards to the maximization of entropy as a fundamental rule. In particular, we consider the single particle bound quantum state. The interaction exists between the particle and the potential V(x), but one may consider a distribution of free particle momentum probabilities i.e W(x)=wavefunction = Sum over p a(p)exp(ipx). The form exp(ipx) comes from separate arguments as is discussed in the note, but the point is that there are no two body scattering reactions occurring at a given x, neither is there acceleration from on x to another (except for a vrms mathematical quantity). Nevertheless the “free particle” exp(ipx) interacts at each x with V(x). As a result of this interaction, one has free particles exp(ipx) which are no longer associated with pp/2m, but rather with an Eave which is a constant for every p. In other words the identity of exp(ipx) is unusual because it is linked to exp(ipx) and pp/2m, but also to Vk1 a(p-k1), Vk2 a(p-k2) etc where V(x)=Sum over k Vk exp(ikx). Thus each x point is associated with a constant Eave and this defines the equilibrium. (It is also equivalent to conservation of energy at each x.) In such a case, one does not maximize entropy to find a(p) values. In the two cases, the common point seems to be considering the interactions at each x and finding a constant “energy” type quantity which does not vary with x. This we argue appears to define equilibrium for both the classical statistical and quantum bound states. Instead of considering density which does not change in time at each x, there is constant energy which not only does not change in time, but has a constant value at each x which seems to be an alternative rule.