In this paper we study a system which consists in an elastic medium carrying transverse waves and one punctual high elastic medium density, called concretion. We compute the equation of motion for the concretion as well as the wave equation of this classical and theoretical system. Afterwards we always consider the case in which the concretion is not the wave source any more. Then, the velocity of the concretion obeys an equivalent de Broglie guidance formula and moves as if exists an equivalent Bohm quantum potential. A strictly equivalent free Schr\odinger equation is retrieved; and we analyse this system in a linear or spherical cavity. We compute the energy (and momentum) of the concretion, naturally defined from the energy (and momentum) density of the vibrating elastic medium. Provided one condition about the amplitude of oscillation, it strikingly appears that the energy and momentum of the concretion not only write in the same form as in quantum mechanics, but also encapsulate equivalent relativistic formulas.
In this paper we study a system which consists in an elastic medium carrying transverse waves and one punctual high elastic medium density, called concretion. We compute the equation of motion for the concretion as well as the wave equation of this classical and theoretical system. Afterwards we always consider the case in which the concretion is not the wave source any more. Then, the velocity of the concretion obeys an equivalent de Broglie guidance formula and moves as if exists an equivalent Bohm quantum potential. A strictly equivalent free Schr\odinger equation is retrieved; and we analyse this system in a linear or spherical cavity. We compute the energy (and momentum) of the concretion, naturally defined from the energy (and momentum) density of the vibrating elastic medium. Provided one condition about the amplitude of oscillation, it strikingly appears that the energy and momentum of the concretion not only write in the same form as in quantum mechanics, but also encapsulate equivalent relativistic formulas.
We introduce a dual wave-particle macroscopic system, where a bead oscillator oscillates in an elastic medium which obeys the Klein-Gordon equation. This theoretical system is mostly inspired by bouncing droplets experiments and bead sliding on a vibrating string experiments. This system is studied using a common and simple mathematical formalism. We compute the motion equation of the bead as well as the wave equation of the system. We introduce the effective velocity of the bead with respect to the elastic medium and the wave ψ, created by the bead, which modulates the natural wave of the medium. Provided some conditions, ψ obeys an equation analogous to the free Schrödinger equation. In the case of linear and spherical cavities, the particle-like characteristics of the bead, expressed with its effective velocity, are proportional to the corresponding wave-like characteristics of the system. This paper is a translation of Dualité onde-corpuscule formée par une masselotte oscillante dans un milieu élastique : étude théorique et similitudes quantiques.
In this paper we suggest a macroscopic toy system in which a potential-like energy is generated by a non-uniform pulsation of the medium (i.e. pulsation of transverse standing oscillations that the elastic medium of the system tends to support at each point). This system is inspired by walking droplets experiments with submerged barriers. We first show that a Poincare-Lorentz covariant formalization of the system causes inconsistency and contradiction. The contradiction is solved by using a general covariant formulation and by assuming a relation between the metric associated with the elastic medium and the pulsation of the medium. (Calculations are performed in a Newtonian-like metric, constant in time). We find (i) an effective Schrodinger equation with external potential, (ii) an effective de Broglie-Bohm guidance formula and (iii) an energy of the `particle' which has a direct counterpart in general relativity as well as in quantum mechanics. We analyze the wave and the `particle' in an effective free fall and with a harmonic potential. This potential-like energy is an effective gravitational potential, rooted in the pulsation of the medium at each point. The latter, also conceivable as a natural clock, makes easy to understand why proper time varies from place to place.
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.
Since the interest in the quantum phenomena started, there have been two main approaches to the quantum theory, the first approach investigates the motion of a wave-particle object by assuming the wave has the characteristics of a moving particle, momentum and energy, this approach was developed by physicists Max Planck, Albert Einstein, and others. The second approach investigates the motion of a particle-wave object by assuming the particle has the characteristics of a moving wave, wavelength, wave vector, and frequency, this approach was developed by physicists De Broglie, Schrodinger and others which led to the presence of quantum mechanics. This article focuses on the second approach and merging it with classical mechanics. An early mathematical formulation for the quantum theory was proposed by De Broglie’s relations and later on, another formulation was given by Schrodinger’s equation. A successful attempt to bridge the gap between classical mechanics with quantum theory implemented via a mathematical derivation for De Broglie’s relations and Schrodinger formulation, The derivation process overcame a main obstacle was the convergence of the deterministic nature of classical mechanics with the probabilistic nature of the quantum theory by adopting an innovative method based upon replacing constant values in a deterministic relation with probabilistic variables in the same relation, and this led to switch the relation from deterministic representation to probabilistic representation. Both classical and quantum explanation are given for the deduced formulas. An energy approach is instrumented. Since classical mechanics formulas are not sufficient to explain the quantum phenomena, a new concept of complementary energy introduced in order to connect the parameters of a moving particle with the parameters of an associated wave, the value of the complementary energy for each particle adjusted the total energy equation of N moving particles without changing the magnitude of the total energy of a system of N particles. In special cases, the value of the complementary energy for some particles equals the value of the total energy for the particles. The derivation process indicates that complementary energy corresponds to the potential energy on the surface of the material which prevents the particles to escape from the boundary of the matter to the outer space and maintains its coherence, complementary energy concept also manifest the impact of the moving particles on a single particle due to the energy conveyed during direct contact in case of collision or by the remote influence through an existing field in the space, so it represents an energy of dual nature for both matter and radiation wave combined. The complementary energy is not introduced as mathematical assumption in order to manipulate the total energy equation, it’s actually exists in the total energy equation and its validity has not been applied before. A general format for any potential energy is obtained regardless the nature of field it represents (a universal format for potential energy) that is by using mathematical manipulation for the equation of total energy of N moving particles. The derivation process led to a particle-wave formula. The substitution into the particle-wave formula with the speed of light, De Broglie’s relations, revealed. Some additional applications for the particle-wave formula are suggested, also as a well known energy quantization formula obtained by applying all of De Broglie’s relation, a general energy, and speed formulas resulting during the derivation process. Matching time independent equation called Schrodinger’s amplitude equation with classical mechanics formulation demonstrated by deducing both the wave function and the magnitude of the wave vector from the classical formulation after converting it to probabilistic formulation. A dedicated section for wave function solutions for Schrodinger’s time dependent equation was given in terms of the wave functions and the magnitude of the wave vectors obtained from the matching process of Schrodinger’s amplitude equation with classical mechanics.
A hypothesis is put forward according to which the formulas of Relativity are a consequence of the laws of quantum physics and may be applied only to quantum objects. According to the wave - particle duality quantum bodies are simultaneously the waves and considered in the present research with a use of the wave equation and requirement of its Lorentz‘s invariance. The property of invariance with respect to the Lorentz’s transform is analysed in hyperbolic coordinates in an area of complex numbers. An examination showed that formulas of relativistic kinematics can be obtained based on the condition of equality of the amplitudes and phases of the wave, observed from different inertial frames. There is no need to introduce the concept of a unified entity of space-time, but the return to the idea of aether and matter waves de Broglie occurs. The physical model becomes borrowed partly from quantum field theory. It is assumed that the moving particle binds to a wave of the aether disturbance, and only this wave acts on the other particle like an electromagnetic field pushes an electrical charge. The model of colliding elastic balls becomes inapplicable, but equations of relativity mechanics remain the same, however, with a new interpretation. The mass of a particle in all coordinate systems is constant, in exchange the expressions of energy and momentum alter relative the classical case, include the property of aether – coefficient c of the wave equation, and acquire relativistic forms.
In this paper we suggest a macroscopic toy system in which a potential-like energy is generated by a non-uniform pulsation of the medium (i.e. pulsation of transverse standing oscillations that the elastic medium of the system tends to support at each point). This system is inspired by walking droplets experiments with submerged barriers. We first show that a Poincaré-Lorentz covariant formalization of the system causes inconsistency and contradiction. The contradiction is solved by using a general covariant formulation and by assuming a relation between the metric associated with the elastic medium and the pulsation of the medium. (Calculations are performed in a Newtonian-like metric, constant in time). We find (i) an effective Schrödinger equation with external potential, (ii) an effective de Broglie-Bohm guidance formula and (iii) an energy of the `particle' which has a direct counterpart in general relativity as well as in quantum mechanics. We analyze the wave and the `particle' in an effective free fall and with a harmonic potential. This potential-like energy is an effective gravitational potential, rooted in the pulsation of the medium at each point. The latter, also conceivable as a natural clock, makes easy to understand why proper time varies from place to place.