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.
This study delves into the realm of Schrödinger equations and wave mechanics, aiming to see if they could provide a foundation comparable in generality to that formulated in the preceding chapter, and to bridge the gap between classical and quantum physics. It begins by deriving a family of Schrödinger-like equations. The study introduces an intuitive approach for deriving an entire spectrum of Schrödinger-like equations, encompassing classical and novel variants, through the utilization of the complex wave function and its conjugate, along with the assumptionsA01, A08. Examining the concepts of the complex wave function
ψ and the multi-vector wave function
Ψ reveals their equal generality, suggesting the applicability of Schrödinger-like equations in the context of multi-vectors, including both
ψ and the phase
Φ . To explore the correspondence principle between quantum and classical mechanics, a well-studied example is dissected in Section 2.4, necessitating the development of a suitable wave function (2.50). The use of the real function [1, (1.50)] to form the complex test function (2.50) is a logical step, and its substitution into the Schrödinger equations aims to yield an identity. The substitution of (2.50) reveals a discrepancy between the left and right sides of the Schrödinger equations (2.25, 2.26), highlighting their inadequacy in describing the classical mechanics example accurately. This observation underscores the limitations of classical Schrödinger equations and their departure from the correspondence principle. Consequently, a more comprehensive set of Schrödinger-like equations (2.58-2.68) is derived, rendering the previously considered equations (2.25, 2.26) irrelevant. Substituting the test function (2.50) into equations (2.58-2.68) demonstrates their soundness and suitability for the example, establishing the correspondence principle between classical and quantum mechanics. Our analysis indicates that the classical set [1, (1.4-1.6)] exhibits a higher degree of generality than the quantum one (2.58-2.68), thereby deferring the definitive validation of this assertion to forthcoming research endeavours. These findings underscore the potential significance of our framework in bridging the gap between quantum and classical mechanics, promising further exploration and validation in future studies.
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.
This paper presents a pedagogical framework for understanding quantum mechanics through classical and wave-based intuition. Starting from algebraic derivations rooted in Newtonian mechanics and wave properties, and incorporating de Broglie's matter-wave duality, we re-derive Planck's relation E=hf. Using the same classical-wave analogy, we extend the approach to foundational quantum models such as the particle in a box, the quantum harmonic oscillator, and other related systems. The goal is to provide an intuitive, visual way helping learners grasp the physical meaning behind key models without relying solely on abstract formalism. Note: This is an updated version of a previously published paper, with refined derivations and an expanded conceptual framework.
Despite its successes, quantum mechanics (QM) has remained mysterious to all who have encountered it. Starting with Bohr and progressing into the present, the departure from intuitive, physical reality has widened. The connection between QM and reality is more than just a “philosophical” issue. It reveals that QM is not a correct or complete theory of the physical world and that inescapable internal inconsistencies and incongruities arise when attempts are made to treat it as physical as opposed to a purely mathematical “tool.” Some of these issues are discussed in a review by F. Laloe [Am. J. Phys. 69, 655 (2001)]. In an attempt to provide some physical insight into atomic problems and starting with the same essential physics as Bohr of e moving in the Coulombic field of the proton and the wave equation as modified by Schrodinger, a classical approach is explored that yields a remarkably accurate model and provides insight into physics on the atomic level. The proverbial view, deeply seated in the wave-particle duality notion, that there is no large-scale physical counterpart to the nature of the electron may not be correct. Physical laws and intuition may be restored when dealing with the wave equation and quantummechanical problems. Specifically, a theory of classical quantum mechanics (CQM) is derived from first principles that successfully applies physical laws on all scales. Rather than using the postulated Schrodinger boundary condition “Ψ → 0 as r → ∞,” which leads to a purely mathematical model of the electron, the constraint is based on experimental observation. Using Maxwell’s equations, the classical wave equation is solved with the constraint that the bound (n = 1)state electron cannot radiate energy. By further application of Maxwell’s equations to electromagnetic and gravitational fields at particle production, the Schwarzschild metric is derived from the classical wave equation, which modifies general relativity to include conservation of spacetime in addition to momentum and matter/energy. The result gives a natural relationship among Maxwell’s equations, special relativity, and general relativity. CQM holds over a scale of spacetime of 85 orders of magnitude — it correctly predicts the nature of the universe from the scale of the quarks to that of the cosmos. A review is given by G. Landvogt [Internat. J. Hydrogen Energy 28, 1155 (2003)].