Methods of Integration Common to Classical Mechanics and Wave Mechanics

Michele Marrocco

The Eherenfest theorem states that Schrodinger representation of quantum mechanics (wave mechanics) reproduces Newton laws of motion in terms of expectation values. Remarkably, the contrary is considered elusive and, indeed, many authors have tried to obtain wave mechanics starting from other alternative frameworks of classical mechanics (for instance, Hamilton-Jacobi theory). Despite this common opinion, we present here a simple method to make Newtonian dynamics develop naturally into Schrodinger representation. The proof is based on the assumption of matter waves and is laid out in three fundamental steps. First, the role of classical density functions is underlined in view of their use to define constants of the motion for massive particles. Thanks to this preparatory step, density functions generate wave functions whose spatial and time variables obey Newton laws of motion. The resulting wave equation is defined in dependence on a parameter that plays the identical role of the constant K introduced by Schrodinger in the original formulation of his theory. In the final step, the classical wave equation is treated under the hypothesis of conservative forces common to the Eherenfest theorem and, after some algebra, the Schrodinger equation emerges by means of the identification of the classical momentum with de Broglie momentum of matter waves.

R. Faure

Independent Researcher, Denmark, Knud Haugaard Sorensen

This paper explores the concept of osmotic momentum within quantum mechanics, offering a novel theoretical framework that integrates stochastic mechanics with generalized electrodynamics. By revisiting Edward Nelson’s interpretation of the Schrodinger equation, we propose that osmotic momentum arises from interactions with gauge wavesan extension to classical field components. Additionally, we outline a method for experimental detection of these waves using a “quantum lens”, a device designed to convert gauge waves into detectable photons. This work bridges gaps between quantum mechanics, gravity, and dark energy, suggesting that gauge waves could unify these phenomena under a common theoretical framework. Experimental validation of this model could redefine our understanding of quantum and relativistic systems. Keywords: Quantum mechanics; Quantum lens; Gravity; Gauge waves.

Yu. P. Pyt’ev

Frederik Jozef Belinfante

The fact that the Schr\"odinger and the Dirac equations for the wave function of an electron are differential equations of the first order with respect to time, while in classical theories differential equations of the second order are common, has necessitated a slightly different setup of the canonical theory in both fields. Under influence of the classical methods, second-order equations are also often used in the quantum theory of particles of integer spin, thus causing a difference between the treatment of Fermi-Dirac and of Einstein-Bose particles.It is shown here that conformity between classical and quantum-mechanical methods can be achieved easily by use of first-order equations throughout, thus avoiding a superfluous distinction between integral and half-odd-integral spin fields. The classical theory of a point charge in an electromagnetic field of force is set up here from this point of view.