Replace the potential energy in the Hamiltonian operator from electromagnetic interaction potential energy to gravitational interaction potential energy. Define that the kinetic energy of a moving object is equal to its de Broglie wave's hν. Successfully obtained the Schrödinger equation describing planetary motion. The Schrödinger equation itself is a combination of classical mechanics and quantum mechanics. Quantum mechanics based on it cannot completely exclude classical mechanics. Many examples have been introduced where classical mechanics and quantum theory can be combined to describe the same object (whether it is macroscopic or microscopic). Local realism and determinism still have a market in modern physics. Quantum mechanics and classical mechanics have a compatible and complementary relationship.
Replace the potential energy in the Hamiltonian operator from electromagnetic interaction potential energy to gravitational interaction potential energy. Define that the kinetic energy of a moving object is equal to its de Broglie wave’s hν. Successfully obtained the Schrödinger equation describing planetary motion. The Schrödinger equation itself is a combination of classical mechanics and quantum mechanics. Quantum mechanics based on it cannot completely exclude classical mechanics. Many examples have been introduced where classical mechanics and quantum theory can be combined to describe the same object (whether it is macroscopic or microscopic). Local realism and determinism still have a market in modern physics. Quantum mechanics and classical mechanics have a compatible and complementary relationship.
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.
Some recent experiments claim to show that any model in which a quantum state represents mere information about an underlying physical reality of the system must make predictions which contradict those of quantum theory. The present work undertakes to investigate the issue of reality, treading a more fundamental route from the Hamilton-Jacobi equation of classical mechanics to the Schrodinger equation of quantum mechanics. Motivation for this is a similar approach from the eikonal equation in geometrical optics to the wave equation in electromagnetic theory. We rewrite the classical Hamilton-Jacobi equation as a wave equation and seek to generalise de Broglie's wave particle duality by demanding that both particle and light waves have the freedom of being described by any square-integrable function. This generalisation, which allows superposition also for matter wave functions, helps us to obtain the Schrodinger equation, whose solution can be seen to be as much objective as the classical mechanics wave function. Several other equations which one writes in quantum mechanics, including the eigenvalue equations for observables, series expansion of energy states in terms of eigenstates of observables other than energy, etc., can be written in the classical case too. Absence of any collapse of the wave function, entanglement, etc. in the classical realm have their origin in the nonlinearity of the classical wave equation. These considerations indicate that many of the puzzles in quantum mechanics are present also in classical mechanics in a dormant form, which fact shall help to demystify quantum mechanics to a great extent.
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.
Striking parallelism between classical mechanics and quantum mechanics is evidenced for several bound system in a rather general form by introducing reasonably “quantum-mechanical momentum” from the Schrödinger equation. The following are proved: The “quantum-mechanical momentum” which is nearly equal to the corresponding classical momentum in “classical region” can be defined in the whole range of validity of the coordinate for the bound systems. In the same way as in classical mechanics, in quantum mechanics one can construct an associated wave accompanying a motion of a quantum particle and obeying the Schrödinger equation. The uniqueness condition on the associated wave throughout an orbit leads to the boundary condition on the wave function which is composed of the associated wave and to “quantum-mechanical quantization condition”, equivalently. In addition, all these quantum-mechanical quantities are approximately equal to the corresponding classical ones in the “classical region” of the coordinate. In particular it is demonstrated that even for a bound system of a singular (central) potential exists the ground state with the lowest energy eigenvalue. It is proved in one dimension that to any physical system obeying the Schrödinger equation always exists the corresponding classical system.