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Experimental Verification of a Differential Energy Integral in Quantum Systems

begum yildirim

Published May 19, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.20285904

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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 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.
Quantum mechanics does not provide any ready recipe for defining energy density in space, since the energy and coordinate do not commute. To find a well-motivated energy density, we start from a possibly fundamental, relativistic description for a spin- 12 particle: Dirac's equation. Employing its energy-momentum tensor and going to the non-relativistic limit we find a locally conserved non-relativistic energy density that is defined via the Terletsky-Margenau-Hill quasiprobability (which is hence selected among other options). It coincides with the weak value of energy, and also with the hydrodynamic energy in the Madelung representation of quantum dynamics, which includes the quantum potential. Moreover, we find a new form of spin-related energy that is finite in the non-relativistic limit, emerges from the rest energy, and is (separately) locally conserved, though it does not contribute to the global energy budget. This form of energy has a holographic character, i.e., its value for a given volume is expressed via the surface of this volume. Our results apply to situations where local energy representation is essential; e.g. we show that the energy transfer velocity for a large class of free wave-packets (including Gaussian and Airy wave-packets) is larger than its group (i.e. coordinate-transfer) velocity.
In order to describe the time evolution of energy states we choose to abandon the non-relativistic Hamiltonian method, which has been the standard for nearly a century, in favor of a more fundamental, relativistically correct Lagrangian method based on theories originally proposed by Dirac and Einstein. Integral equations of motion for the absorption and emission of radiation are derived that underlie and anticipate the differentially motivated Schrödinger equation. This new interpretation applies to a large volume of experimental evidence of both classical and quantum mechanical origin. Among the examples discussed in support are Planck’s law describing black body radiation, the function of the simplest quantum mechanical system an electron cyclotron, atomic clocks, matrix mechanics, chaos theory, and evolutionary biology.
In order to describe the time evolution of energy states we choose to abandon the non-relativistic Hamiltonian method, which has been the standard for nearly a century, in favor of a more fundamental, relativistically correct Lagrangian method based on theories originally proposed by Dirac and Einstein. Integral equations of motion for the absorption and emission of radiation are derived that underlie and anticipate the differentially motivated Schrödinger equation. This new interpretation applies to a large volume of experimental evidence of both classical and quantum mechanical origin. Among the examples discussed in support are Planck’s law describing black body radiation, the function of the simplest quantum mechanical system an electron cyclotron, atomic clocks, matrix mechanics, chaos theory, and evolutionary biology.
In our investigation of the time evolution of energy states Einstein’s interpretation from more than a century ago is found to have greater internal consistency than that of non-relativistic quantum mechanics. We choose to align more closely with his methods by abandoning the non-relativistic Hamiltonian method, which has been the standard for nearly a century, in favor of a relativistically correct Lagrangian method. Integral equations of motion for the absorption and emission of radiation are derived that underlie and anticipate the differentially motivated Schrödinger equation. This new interpretation applies to a large volume of experimental evidence of both classical and quantum mechanical origin. Among the examples discussed in support are Planck’s law describing black body radiation, matrix mechanics, the function of the simplest quantum mechanical system an electron cyclotron, chaos theory, and evolutionary biology.
In order to describe the time evolution of energy states we choose to abandon the non-relativistic Hamiltonian method, which has been the standard for nearly a century, in favor of a more fundamental, relativistically correct Lagrangian method based on theories originally proposed by Dirac and Einstein. Integral equations of motion for the absorption and emission of radiation are derived that underlie and anticipate the differentially motivated Schrödinger equation. This new interpretation applies to a large volume of experimental evidence of both classical and quantum mechanical origin. Among the examples discussed in support are Planck’s law describing black body radiation, the function of the simplest quantum mechanical system an electron cyclotron, atomic clocks, matrix mechanics, chaos theory, and evolutionary biology.