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A Differential Geometric Model for Energy-Time Integration in Many-Electron Quantum Systems

begum yildirim

Published May 19, 2026• 0 citations
Publisher

Abstract

We present a novel mathematical formulation for modeling energy transfer processes in many-electron quantum systems using differential forms. The proposed integral structure captures the time-dependent nature of energy dynamics and orbital coupling via an exterior product (wedge), suggesting compatibility with both Hodge theory and quantum chemical multi-body problems. The formulation is numerically verified and theoretically aligned with state-of-the-art quantum simulations. Originality and AI-use statement: This work is an original research output by Begüm Yıldırım. AI tools, if used, were limited to language refinement, grammar correction, formatting, translation assistance, and clarity improvement. The conceptual framework, research direction, interpretation, models, and conclusions belong to the author. External sources, datasets, or prior works are cited where applicable.

DOI
10.5281/zenodo.20281389

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This article proposes and verifies a novel integral formulation for describing the time-dependent energy distribution in many-electron quantum systems. The integral is designed using differential geometry concepts and is shown to be consistent with both relativistic and non-relativistic formulations of energy density. Recent developments in quantum energy density theory, specifically those based on Dirac’s equation and the Terletsky-Margenau-Hill quasiprobability, serve as theoretical confirmation. In addition, numerical simulations demonstrate the consistency and integrability of the proposed model. Originality and AI-use statement: This work is an original research output by Begüm Yıldırım. AI tools, if used, were limited to language refinement, grammar correction, formatting, translation assistance, and clarity improvement. The conceptual framework, research direction, interpretation, models, and conclusions belong to the author. External sources, datasets, or prior works are cited where applicable.
We present a systematic analysis of the introduction of correlation in the many-electron time-dependent problem with an accurate formulation based on geometric algebra notation. This provides a systematic definition of the configuration space, of the external potentials, of the one-electron operators for a many-electron system, and of the electron–electron interaction terms. We arrive both at a formal equation for the total energy and at the equation for the time-evolution of the wavefunction. From this, using the new geometric notation and the indistinguishability and equivalence of the electrons and the fact that we are interested either in the ground state or in states near the ground state, we formulate a variational problem from which a set of tractable equations, which self-consistently define the many-electron wavefunction and density, is obtained. The main emphasis is on the electron–electron correlation.
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