The concept of a stress tensor is fundamental in classical continuum mechanics, encapsulating internal forces per unit area that a material can sustain. In quantum many‑body systems, however, the notion of pressure and stress has traditionally been limited to scalar or local quantities derived from the equation of state or from density functional approximations. In this work we introduce a fully tensorial representation of electronic internal forces that arises naturally from the quantum mechanical description of electrons in condensed matter systems. We define the Electron Quantum Stress Tensor (EQST) as a second‑rank tensor operator constructed from the kinetic and interaction contributions to the many‑electron Hamiltonian. By deriving explicit expressions for the EQST in terms of the electron density, its gradients, and the exchange–correlation potential, we establish a direct link between the microscopic wavefunction and macroscopic mechanical observables. The EQST framework provides a new lens for interpreting phenomena in two‑dimensional electron gases, quantum Hall fluids, and other strongly correlated electronic materials, offering a potential route toward uncovering novel mechanical properties of electrons that could have implications for future Nobel‑level discoveries.
The concept of a stress tensor is fundamental in classical continuum mechanics, encapsulating internal forces per unit area that a material can sustain. In quantum many‑body systems, however, the notion of pressure and stress has traditionally been limited to scalar or local quantities derived from the equation of state or from density functional approximations. In this work we introduce a fully tensorial representation of electronic internal forces that arises naturally from the quantum mechanical description of electrons in condensed matter systems. We define the Electron Quantum Stress Tensor (EQST) as a second‑rank tensor operator constructed from the kinetic and interaction contributions to the many‑electron Hamiltonian. By deriving explicit expressions for the EQST in terms of the electron density, its gradients, and the exchange–correlation potential, we establish a direct link between the microscopic wavefunction and macroscopic mechanical observables. The EQST framework provides a new lens for interpreting phenomena in two‑dimensional electron gases, quantum Hall fluids, and other strongly correlated electronic materials, offering a potential route toward uncovering novel mechanical properties of electrons that could have implications for future Nobel‑level discoveries.
Correlated electron systems present a profound challenge in condensed matter physics due to the intricate interplay between charge, spin, orbital, and lattice degrees of freedom. Understanding their rich macroscopic phase diagrams, which include phenomena like high-temperature superconductivity, colossal magnetoresistance, and Mott insulation, necessitates a bridge between the microscopic quantum interactions and emergent collective behavior. This paper introduces and formalizes the concept of quantum fugacity as a critical thermodynamic parameter capable of establishing this connection. Extending the classical notion of fugacity—a measure of a real gas's effective partial pressure—to the quantum many-body realm, quantum fugacity accounts for both particle density and the profound influence of quantum statistics and strong electron correlations. We develop a theoretical framework to calculate quantum fugacity from microscopic Hamiltonians, demonstrating its utility in mapping quantum phase transitions and discerning the driving forces behind symmetry-breaking phases. Through this approach, we show how variations in quantum fugacity serve as indicators for the onset of novel phases, providing a unified perspective on the phase behavior of diverse correlated electron materials. This concept offers a powerful new tool for designing and predicting materials with desired quantum functionalities.
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.
The equation of motion technique is used to derive quantum operators corresponding to the microscopic stress tensor and pressure. The technique is demonstrated for a one-dimensional system. Stress tensor and related operators are obtained for a nonrelativistic, many-body, Coulomb system. The stress tensor operators occur as a momentum flux density in continuity equations which are the quantum analog to Newton's second law (F = ma). The derivation also produces operators with obvious classical analogs such as linear and angular momentum densities in a context which makes their physical role evident.
There has been an enduring interest and controversy about whether or not one can define physically meaningful energy density and stress fields, e(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 r; all issues of nonuniqueness are relegated to terms that involve only the density n(r) and are equivalent to an effective single-particle problem with ground state wave function 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 $\ensuremath{-}\frac{1}{2}s{\ensuremath{\nabla}}^{2}s$ 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(r) is equilibrated to minimize fluctuations with the same chemical potential at all points 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.
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.
Quantum mechanics has long described particles such as electrons using a complex wavefunction ψ(x) that encodes probability amplitudes for position and momentum. Conventional formulations treat ψ(x) as a smooth, featureless entity governed by the Schrödinger equation and interpreted through Born's rule. This paper proposes a novel hypothesis: the wavefunction ψ(x) possesses an intrinsic internal structure, or texture, denoted Tψ(x). This texture is a scalar or tensorial field derived from spatial derivatives and phase gradients of ψ(x) itself. We develop a mathematical formalism in which Tψ(x) is defined explicitly, explore its physical implications, and show how it could alter the dynamics of quantum systems. By integrating Tψ(x) into the Hamiltonian as an additional potential term V_T, we derive modified evolution equations that reduce to standard quantum mechanics when the texture is flat. We discuss potential experimental signatures accessible via high‑resolution electron holography, where interference patterns may reveal subtle modulations attributable to Tψ(x). If verified, this internal texture paradigm would represent a significant refinement of quantum theory, offering new insight into the foundations of the wavefunction and possibly paving the way toward a Nobel Prize‑worthy discovery.