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International journal of engineering inventions.

The Core of Generalized Wave Mechamics: Quantum Mechanics Equation with Explicit Force

Runsheng Tu

Published Sep 12, 2026• 0 citations
Publisher

Abstract

Classical mechanics and quantum mechanics have long been framed as binary opposites—but this dichotomy has yielded no real benefits, only persistent confusion. The fundamental equations of quantum mechanics cannot logically demarcate a clear boundary between the quantum and classical realms; in fact, the massm and potential energy V in the Schrödinger equation seamlessly traverse the divide between the microscopic and macroscopic. The electromagnetic potential energy in the Schrödinger equation can be replaced with the potential energy of interactive forces, which can further be converted into force itself using the relation F = V/r. Derivations have now produced a force operator and quantum mechanical equations that explicitly incorporate force—including a quantum formulation of Newton’s second law. These results provide direct evidence that quantum and classical mechanics are not incompatible, but complementary. By resolving the historical absence of "force" as a formal concept in quantum mechanics, we bridge the conceptual gap between macroscopic classical frameworks and microscopic quantum theory, challenging long-held perceptions of their irreconcilability. This unified approach explains why nanoscale particles exhibit quantum properties, simplifies calculations of quantum forces, and lays the groundwork for a generalized wave mechanics that unifies mechanical descriptions across all scales.

DOI
10.5281/zenodo.22719653

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Classical mechanics and quantum mechanics have long been framed as binary opposites—but this dichotomy has yielded no real benefits, only persistent confusion. The fundamental equations of quantum mechanics cannot logically demarcate a clear boundary between the quantum and classical realms; in fact, the massm and potential energy V in the Schrödinger equation seamlessly traverse the divide between the microscopic and macroscopic. The electromagnetic potential energy in the Schrödinger equation can be replaced with the potential energy of interactive forces, which can further be converted into force itself using the relation F = V/r. Derivations have now produced a force operator and quantum mechanical equations that explicitly incorporate force—including a quantum formulation of Newton’s second law. These results provide direct evidence that quantum and classical mechanics are not incompatible, but complementary. By resolving the historical absence of "force" as a formal concept in quantum mechanics, we bridge the conceptual gap between macroscopic classical frameworks and microscopic quantum theory, challenging long-held perceptions of their irreconcilability. This unified approach explains why nanoscale particles exhibit quantum properties, simplifies calculations of quantum forces, and lays the groundwork for a generalized wave mechanics that unifies mechanical descriptions across all scales.
Classical mechanics and quantum mechanics have long been framed as binary opposites—but this dichotomy has yielded no real benefits, only persistent confusion. The fundamental equations of quantum mechanics cannot logically demarcate a clear boundary between the quantum and classical realms; in fact, the mass m and potential energy V in the Schrödinger equation seamlessly traverse the divide between the microscopic and macroscopic. The electromagnetic potential energy in the Schrödinger equation can be replaced with the potential energy of interactive forces, which can further be converted into force itself using the relation F = V/r. Derivations have now produced a force operator and quantum mechanical equations that explicitly incorporate force—including a quantum formulation of Newton’s second law. These results provide direct evidence that quantum and classical mechanics are not incompatible, but complementary. By resolving the historical absence of “force” as a formal concept in quantum mechanics, we bridge the conceptual gap between macroscopic classical frameworks and microscopic quantum theory, challenging long-held perceptions of their irreconcilability. This unified approach explains why nanoscale particles exhibit quantum properties, simplifies calculations of quantum forces, and lays the groundwork for a generalized wave mechanics that unifies mechanical descriptions across all scales.
The explanation of quantum mechanics has never been satisfactory. The successful application of quantum mechanics is generally attributed to the mathematical system of quantum mechanics. The application research of quantum encryption communication and quantum computers using quantum entanglement is very difficult. The binary opposition between quantum and classical has hindered human scientific thinking. The widespread quest to quantize gravity mirrors humanity's enduring drive to unify quantum and macroscopic physical frameworks. This research initiates its inquiry by integrating quantum mechanics with classical mechanics, yielding novel analytical perspectives, and introduces the Schrödinger-Tu equation—a wave function that incorporates gravitational potential energy and is capable of describing macroscopic objects. Validating the practical viability of merging quantum and classical approaches, several computational cases involving atoms and molecules yield successful outcomes. Additionally, this study derives a wavemechanical formulation of classical mechanical laws, furnishing direct empirical evidence that quantum mechanics and classical mechanics are not mutually exclusive but rather compatible and complementary. Further substantiation of their compatibility encompasses the following key points: The mass term m in the Schrödinger equation (SE) can be scaled to sufficiently large values to characterize macroscopic entities; The steady-state SE (Tψ+Vψ=Eψ) inherently integrates the wave function ψ with the classical energy conservation relation T+V=E; Potential energy functions within the SE can be derived from macroscopic force fields; Newton's second law F=ma and the SE are mutually derivable, establishing a fundamental theoretical linkage; Electron diffraction experiments concurrently demonstrate wave-particle duality, a phenomenon that bridges both mechanical frameworks. This paradigm shift—recasting the relationship between quantum mechanics and classical mechanics from one of incompatibility to compatibility—represents a pivotal theoretical advancement, with the potential to catalyze the development of a generalized wave mechanics.
The widespread quest to quantize gravity mirrors humanity’s enduring drive to unify quantum and macroscopic physical frameworks. This research initiates its inquiry by integrating quantum mechanics with classical mechanics, yielding novel analytical perspectives, and introduces the Schrödinger-Tu equation—a wave function that incorporates gravitational potential energy and is capable of describing macroscopic objects. Validating the practical viability of merging quantum and classical approaches, several computational cases involving atoms and molecules yield successful outcomes. Additionally, this study derives a wave-mechanical formulation of classical mechanical laws, furnishing direct empirical evidence that quantum mechanics and classical mechanics are not mutually exclusive but rather compatible and complementary. Further substantiation of their compatibility encompasses the following. key points: • The mass term m in the Schrödinger equation (SE) can be scaled to sufficiently large values to characterize macroscopic entities • The steady-state SE (Tψ +Vψ=Eψ) inherently integrates the wave function ψ with the classical energy conservation relation T+V=E; • Potential energy functions within the SE can be derived from macroscopic force fields; • Newton’s second law F=ma and the SE are mutually derivable, establishing a fundamental theoretical linkage; 5.Electron diffraction experiments concurrently demonstrate wave-particle duality, a phenomenon that bridges both mechanical frameworks.This paradigm shift—recasting the relationship between quantum and classical mechanics from one of incompatibility to ompatibility—represents a pivotal theoretical advancement, with the potential to catalyze the development of a generalized wave echanics.
In reality, micro objects and macro objects coexist harmoniously, and each macro object depends on micro particles to exist (macro is a collection of micro objects, indicating that they have a very extensive relationship and are symbiotic). However, classical mechanics and quantum mechanics have long been regarded as binary oppositions. This is a serious disconnect between theory and phenomena, requiring the establishment of new theories and methods that are compatible between quantum and classical. Prior to this, many reasons and methods for combining quantum theory with classical theory have been found (providing numerous computational examples). The wave mechanics equations of classical mechanics laws have been written. The technical method used is still the conversion rule of the Hermitian operator in the fundamental postulates of quantum mechanics. There are no new assumptions. The key is to use the relationship between force and energy in the planetary model to obtain wave mechanics equations that include force, Newton's second law, and the law of conservation of momentum. Thus, quantum mechanics, which is compatible with classical mechanics, truly becomes a form of 'mechanics'. This kind of mechanics is generalized wave mechanics. Quantum and classical are coordinated and compatible, resulting in no clear boundary between quantum and classical. It will seriously affect the credibility of quantum decoherence and wave function collapse.
From the fact that many people are dedicated to the quantization of gravity, it can be seen that people crave the integration of quantum theory and macroscopic theory. Let's first combine quantum mechanics with classical mechanics to gain insightful experiences. The citation introduces the Schrödinger-Tu equation, which has gravitational potential energy and can describe macroscopic objects. Several successful computational examples related to atoms and molecules using the combination of quantum and classical methods for use were listed. The wave mechanics representation method of classical mechanics laws has been derived. This indicates that quantum mechanics and classical mechanics can be compatible and coexist. There is still much evidence to suggest that the Schrödinger equation (SE) cannot exclude classical mechanical laws: The mass m in the SE can be large enough to bring the described object into the macroscopic range; Steady state SE is the Tψ+Vψ=Eψ, a combination of the wave function ψ and the expression T+V=E in classical mechanics; The potential energy function in the SE can originate from macroscopic force fields; F=ma and the SE can be converted to each other; Completed electron diffraction experiments that exhibit both wave and particle characteristics simultaneously. Important conceptual changes can drive the renewal of physics. The shift from quantum and classical incompatibility to compatibility is an important conceptual change that can lead to the birth of general wave mechanics.
Replacing the potential energy of electromagnetic interactions in the original Schrödinger equation with the potential energy of gravitational interactions can lead to the Schr ö dinger equation of gravitational potential energy. It is a product of the combination of classical mechanics and quantum mechanics, suitable for describing macroscopic and microscopic systems. A quantum chemistry method that combines classical mechanics and quantum mechanics can be established. Multiple computational examples have been provided for applying this method. The established basic particle structure configuration of wave elements can explain the source of electron spin magnetic moments. These three facts constitute a new quantum mechanics theoretical system — a localized realism quantum mechanics system that combines classical mechanics with quantum forces. This has a positive impact on promoting the development of material structure theory and quantum theory