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Advances in Theoretical & Computational Physics

The Premise and Result of Combining Quantum and Classical Theories: Towards Generalized Wave Mechanics

Runsheng Tu

Published Aug 25, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.33140/atcp.09.03.13

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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.
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.
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.
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.
The compatibility between quantum mechanics and classical mechanics is always worth discussing. Schrödinger equation (SE) can be derived from F=ma and T+V=E. Important classical physics formulas can be derived from the SE. The SE intuitively contains the mass m and the potential energy formula (this quality m can be large enough to enter the macro range and the potential energy comes from classical attraction). These three points determine that the SE is an organic combination of wave functions and classical mechanical laws. Based on this, it can be predicted that the SE, which can be used to describe macroscopic objects, can be established. It can be observed from the experiment of electron diffraction in a magnetic field that the volatility and property of moving electrons can be presented simultaneously. This experimental phenomenon, together with the mathematical analysis mentioned above, supports the conclusion that Newton's second law still applies in the microscopic world. According to the SE, it can be proven that "the energy of a moving particle described by the law of waves is equivalent to a multiple of its kinetic energy"; There is no absolute boundary between macro and micro (or classical and quantum), and they can be compatible.
Quantum mechanics and classical mechanics are incompatible, and the highly controversial concepts of "uncertainty" and "non realism" are deeply rooted old ideas. Their existence has seriously hindered the development of physics. It is the obligation of explorers to clarify the reasons, methods, application examples, and significance of combining quantum mechanics with classical mechanics. By analyzing the composition, structure, and function of the Schrödinger equation, it can be concluded that the equation itself is a product of the organic combination of classical mechanical laws and wave functions. Prove that the Schrödinger equation is an organic synthesis of the classical mechanical law T+V=E and the wave function (including its differential). Export the Schrödinger equation with gravitational potential energy that can describe macroscopic systems. Pure mathematical methods cannot limit the magnitude of the mass m in the Schrödinger equation. These three facts determine that classical mechanics and quantum mechanics can be used simultaneously to describe the same object that is not limited by mass (providing multiple successful computational examples of "simultaneous use"). These research conclusions can lead to conceptual innovation, theoretical innovation, methodological innovation, and knowledge renewal in physics.