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Reinterpretation of the Schrödinger Equation via Photon and Matter Energy A Quantum-Relativistic Expression of the Hamiltonian

Hakan ÜNAL

Published Sep 15, 2026• 0 citations
Publisher
DOI
10.5281/zenodo.22766814

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By positing energy as intrinsic and independent—with momentum, time, and spatial properties emerging as functions of boost energy (either directly or indirectly)—E-Theory achieves a fully covariant, background-independent formulation for quantum mechanics. This paper demonstrates that by anchoring special relativity in the invariant energy–momentum relationship and reinterpreting energy as an independent variable rather than a dependent property, we transform the traditional four-vector—typically expressed via complex differential relationships—into three simple algebraic functions for momentum, time, and space. We can then transform the standard Time-Dependent Schrödinger Equation (TDSE) into a covariant, background-independent version simply by scaling the time derivative by a universal factor derived from the boost energy.This shift from the time domain (time-dependent) to the energy domain (energy-dependent) has profound consequences: a covariant TDSE emerges; both relativistic and non-relativistic dispersion are recovered from the standard non-relativistic Hamiltonian; inverse relativistic effects appear at ultra-low energies; and a single wave equation governs quantum evolution across all energy scales and particle spins. Most importantly, we free quantum mechanics from its reliance on a fixed temporal background and place it on equal footing with general relativity, thus laying the foundation for their ultimate unification.
The proposed paper presents the unobserved inadequacies in de Broglie's concepts of wave-particle duality and matter waves in the year 1923. The commonly admitted quantum energy or frequency expression hν=γmc2 is shown to be inappropriate for matter waves and is acceptable only for photons. The superluminal phase velocity expression c2/\u{psion}, for matter waves, is investigated in detail and is also reported to be inadequate. The rectifications in the inadequate concepts of de Broglie's theory and refinements in the analogy implementation between light waves and matter waves are presented, which provides the modified frequency and phase velocity expression for matter waves. Mathematical proofs for the proposed modified frequency and phase velocity expression are also presented. In accordance with the proposed concepts, a wave-particle duality picture is presented which elucidates the questions coupled with the wave-particle duality, existing in the literature. Consequently, particle type nature is shown to be a characteristic of waves only, independent from the presence of matter. The modifications introduced in the frequency expression for matter waves leads to variation in the wave function expression for a freely moving particle and its energy operators, with appropriate justifications. A new relation between the Kinetic energy and Momentum of the moving body is proposed and is subsequently applied to introduce novel General and Relativistic Quantum Mechanical Wave Equations. Applications of these equations in bound state quantum mechanical systems provide the information regarding particle's general and relativistic behavior in such systems. Moreover, the proposed wave equations can also be transformed into Schrödinger's and Dirac's equations. The interrelation of Schrödinger's, Dirac's and proposed equations with the universal wave equation is also presented.