The relationship between classical and quantum mechanics is explored in an intuitive manner by the exercise of constructing a wave in association with a classical particle. Using special relativity, the time coordinate in the frame of reference of a moving particle is expressed in terms of the coordinates in the laboratory frame of reference in order to provide an initial spatiotemporal function to work from in initiating the development of a quantum wave. When temporal periodicity is ascribed to the particle, a provisional spatiotemporal function for a particle travelling at constant velocity manifests itself as an running wave characterized by parameters associated with the moving particle. A wave description for bidirectional motion is generated based on an average time coordinate for a combination of oppositely directed elementary running waves, and the resulting spatiotemporal function exhibits wave behavior characteristic of a standing wave. Ascribing directional orientation to the intrinsic periodicity of the particle introduces directional sub-states; variations in the relative number of sub-states as a function of angle in combined states lead to spatially varying magnitudes for the associated waves. Further analysis leads to full mathematical expression for all waves representing free particle motion. A generalization for particles subject to force fields enables us to develop a governing differential equation identical in form to the Schroedinger equation.
The relationship between classical and quantum mechanics is explored in an intuitive manner by the exercise of constructing a wave in association with a classical particle. Using special relativity, the time coordinate in the frame of reference of a moving particle is expressed in terms of the coordinates in the laboratory frame of reference in order to provide an initial spatiotemporal function to work from in initiating the development of a quantum wave. When temporal periodicity is ascribed to the particle, a provisional spatiotemporal function for a particle travelling at constant velocity manifests itself as an running wave characterized by parameters associated with the moving particle. A wave description for bidirectional motion is generated based on an average time coordinate for a combination of oppositely directed elementary running waves, and the resulting spatiotemporal function exhibits wave behavior characteristic of a standing wave. Ascribing directional orientation to the intrinsic periodicity of the particle introduces directional sub-states; variations in the relative number of sub-states as a function of angle in combined states lead to spatially varying magnitudes for the associated waves. Further analysis leads to full mathematical expression for all waves representing free particle motion. A generalization for particles subject to force fields enables us to develop a governing differential equation identical in form to the Schroedinger equation.
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.
In a previous paper, we showed that the Hamilton equations of motion of a quantum particle are obtained as group velocities of the wave packet describing this particle only if the time dependent phase of a particle wave is proportional to the Lagrangian, not to the Hamiltonian as in a solution of the conventional Schrodinger equation. When a Lagrangian of relativistic form is considered, the wave packet of a quantum particle takes a physical form, with a finite spectrum of a cut off velocity (c). Based on a relativistic quantum principle, asserting the invariance of the time dependent phase for an arbitrary change of coordinates, we obtained the relativistic kinematics and dynamics, the electromagnetic field equations, the spin and the electromagnetic and gravitational interactions. When the Lagrangian is considered as a function of the Hamiltonian, we obtain a Schrodinger type equation with an additional term depending on the velocity and the momentum operator. Based on this equation, we investigate the dynamics of a relativistic quantum particle. In this framework, such a particle is described as a continuous distribution of conservative matter, according to the general theory of relativity. In an electromagnetic field, any time dependent phase variation is modified with a term proportional to a vector potential conjugated to the spatial coordinates and a scalar potential conjugated to time. However, this classical description tells us nothing about the structure of the physical world. Only Quantum Mechanics tells us something about the structure of this world. Namely, that this world is composed of species of identical quantum particles. On one hand, experimentally, it has been found that these particles are of a wavy nature. On the other hand, one could find that the simplest way to define a quantum particle is by a wave packet, with momenta conjugated to coordinates and energy conjugated to time Consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrodinger equation. We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves. We consider a quantum particle as a distribution of conservative matter propagating according to the General Theory of Relativity. We obtain the dynamics of this matter in a gravitational field, the propagation in plane waves perpendicular to geodesic tracks, and equations of conservation. We describe the matter dynamics as a positively defined density and show that, according to the general theory of relativity, such a distribution can be conceive only as of a fragment of matter with a finite mass equal to a mass , as a characteristic of the matter dynamics, – the matter quantization. The group velocities of the Fourier conjugate representations in the coordinate and momentum spaces describe the dynamics of a quantum particle in agreement with the Hamiltonian equations. Under the action of an external (non-gravitational) field, the acceleration of the quantum matter has two components: 1) A component perpendicular to the velocity, given by the relativistic mechanical component of the time dependent phase, and 2) A component given by the additional field terms of this phase. A free quantum particle is described by a non-dispersing wave function , contrary to the solution of the Schrodinger equation. A coherent electromagnetic field, in resonance with a system of active quantum particles in a Fabry-Perot cavity, has a wave vector approximately proportional to the metric elements, as the resonance frequency is approximately constant – a gravitational wave can be detected by the transmission characteristics of an active Fabry-Perot cavity. In a constant gravitational field, a quantum particle undertakes a velocity and an acceleration, which, at the boundary of a black hole are null – absorption and evaporation processes at the boundary of a black hole arise only by gravitational perturbations. Generally, a quantum particle is described by a time-space volume, called graviton, with a spin 2, and a distribution of a specific matter in this volume, with a half-integer spin for Fermions and an integer spin for Bosons. A graviton Lagrangian is obtained as a curvature integral on a graviton volume, and a Hamiltonian tensor is obtained for the gravitational coordinates and velocities. In a gravitational field, the time space coordinates are deformed. In such a field, any plane wave remains perpendicular geodesic, while an additional acceleration is possible in the wave plane.
The physics of special relativity is applied to a wave-packet model of a particle without the usual de Broglie-like assumption of Planck's rule, E = hv. The velocity of the particle is equated to the group-velocity of the wave-packet to associate the particle with its wave-packet. With such an identification it is demonstrated using special relativity that the momentum four-vector, P = [p, E/c2] of the particle is parallel and proportional to a central wave-number four-vector, K0 = [k0. ω0/c2], of the particle's wave-packet. The constant of proportionality, relating these two four-vectors, has the dimension of action and by diffraction experiments it has the value h/2π where h is Planck's constant. This refinement of the de Broglie wave-like nature of a particle suggests that special relativity underlies quantum physics more deeply than heretofore believed. The above quantum rules are used to find the wave-packet function in the one-dimensional case of a non-relativistic free particle and a particle in a conservative force field of potential V (x). This wave-packet function is shown to satisfy an integral time-evolution equation which relates the wave function at time t0 to the function at a slightly later time t. The resulting integral equation is equivalent to the propagator equation or path-integral hypothesised by Feynman. This fact also demonstrates, using the method of Feynman, that the wave-packet function satisfies Schrödinger's partial differential equation. The wave-packet function of a particle in a force field is directly shown to satisfy a linear integral equation which relates the wave-packet at one instant of time to its values at all later times. The kernel or propagator of this integral equation is also equivalent to Feynman's path integral. A computation demonstrates that the position of a moving particle in a simple quadratic potential well does not become more uncertain with the passage of time. This contrasts with the behaviour of a moving free particle which as a function of time becomes more and more difficult to accurately locate.
Special relativity combined with the stochastic vacuum flux impact model lead to an explicit interpretation of many of the phenomena of elementary quantum mechanics. We examine characteristics of a repetitively impacted submicroscopic particle in conjunction with examination of the ways in which effects associated with the particle's behavior appear in moving frames of reference. As seen from relatively moving frames of reference, the time and location of impacts and recoils automatically exhibit wave behavior. This model leads to free particle waves with frequencies proportional to the energy and wavelengths inversely proportional to the momentum. As seen from relatively moving frames of reference, impacts and their associated recoils can appear to an observer to take place simultaneously at multiple locations in space. For superposed free particle waves corresponding to bidirectional motion, an amplitude that varies sinusoidally with distance results. A governing equation identical in form to the Schroedinger equation is developed that describes the behavior of the impacts and their associated recoils. This approach permits many features of quantum mechanics to be examined within an intuitively visualizable framework.
In this paper the relationship of classical physics and quantum physics is studied by introducing a partial differential equation, which describes classical mechanics, but looks very similar to the Schrodinger wave equation of quantum mechanics. This work is largely based on David Bohm's causal interpretation of quantum mechanics, but is in some sense complementary to it. In Bohm's theory the Schrodinger wave equation is used to derive classical looking equations of motion for quantum physics. Here exactly the opposite is done. The equations of classical physics are put into a form resembling the Schrodinger equation of quantum physics.