In this work we propose a new approach to the explanation of the nature of electron based on the corpuscular-wave monism using the further development of the optical-mechanical analogy to describe the physical reality. In this theory the motion of an electron is considered to occur along a trajectory the presence of which is a reflection of the existence of a particle, as well as it is assumed that any motion is defined by a wave V(x,t). It is assumed that there is an explicit relationship between the trajectory and wave equations of the electron, which are established on the basis of the local variational principle. In this approach, an electron wave propagating in free space takes along the electron trajectory. We used this theory to describe the electron motion in a hydrogen-like atom and found its stationary states. The energies of these states coincide with the known quantum mechanics solutions for the stationary energies of the hydrogen-like atom, however, in our approach the spatial trajectories of the electron have the form of the surfaces, which are formed in the region of nodes of the standing electron wave. These surfaces have the form of spheres for the spherical symmetrical electron states and the radii of these spheres coincide with the radii of the Bohr orbits of these states. Thus, in this approach the trajectory and wave measurements of the electron get a consistent spatial description that is inherent to the picture of the corpuscular-wave monism. We discuss the considerable correspondence of the proposed theory with the quantum mechanics results describing the stationary and non-stationary motions of the electron in the atom and their difference. We believe that measuring of the spatial configuration of the observed electron trajectory surfaces in an atom could be a deep examination of the standard quantum theory.
In this work we propose a new approach to the explanation of the nature of electron based on the corpuscular-wave monism using the further development of the optical-mechanical analogy to describe the physical reality. In this theory the motion of an electron is considered to occur along a trajectory the presence of which is a reflection of the existence of a particle, as well as it is assumed that any motion is defined by a wave V(x,t). It is assumed that there is an explicit relationship between the trajectory and wave equations of the electron, which are established on the basis of the local variational principle. In this approach, an electron wave propagating in free space takes along the electron trajectory. We used this theory to describe the electron motion in a hydrogen-like atom and found its stationary states. The energies of these states coincide with the known quantum mechanics solutions for the stationary energies of the hydrogen-like atom, however, in our approach the spatial trajectories of the electron have the form of the surfaces, which are formed in the region of nodes of the standing electron wave. These surfaces have the form of spheres for the spherical symmetrical electron states and the radii of these spheres coincide with the radii of the Bohr orbits of these states. Thus, in this approach the trajectory and wave measurements of the electron get a consistent spatial description that is inherent to the picture of the corpuscular-wave monism. We discuss the considerable correspondence of the proposed theory with the quantum mechanics results describing the stationary and non-stationary motions of the electron in the atom and their difference. We believe that measuring of the spatial configuration of the observed electron trajectory surfaces in an atom could be a deep examination of the standard quantum theory.
Based on the method of V-function, a continuation of the optical-mechanical analogy is attained. In contrast to classic quantum mechanics, a trajectory-wave motion of the particle is explored. We highlight the presence of energy quantization of the particle and the availability of solution without a particle in the case of rectilinear uniform motion at constant speed. A solution to the direct and inverse problems of dynamics is searched for in a new statement for a hydrogen-like atom. When solving a direct problem, we find a stationary wave function of the electron in a hydrogen-like atom, with its properties investigated. When searching for a final solution to the stationary wave equation, we take into account a solution to the inverse problem of dynamics for the electron. A linear dependence between two particular solutions is shown. The second linearly independent solution is found, decaying exponentially to zero. We present charts of the stationary solution for a wave of the particle (electron) for three lower stationary states. Energy levels of a hydrogen-like atom are determined as a solution to the inverse problem of dynamics, which fully coincide with the classical results by Schrödinger and Bohr. A wave function is regarded as a physical reality, which makes it possible to open up new possibilities in order to study the structure of the microcosm
Based on the method of V-function, a continuation of the optical-mechanical analogy is attained. In contrast to classic quantum mechanics, a trajectory-wave motion of the particle is explored. We highlight the presence of energy quantization of the particle and the availability of solution without a particle in the case of rectilinear uniform motion at constant speed. A solution to the direct and inverse problems of dynamics is searched for in a new statement for a hydrogen-like atom. When solving a direct problem, we find a stationary wave function of the electron in a hydrogen-like atom, with its properties investigated. When searching for a final solution to the stationary wave equation, we take into account a solution to the inverse problem of dynamics for the electron. A linear dependence between two particular solutions is shown. The second linearly independent solution is found, decaying exponentially to zero. We present charts of the stationary solution for a wave of the particle (electron) for three lower stationary states. Energy levels of a hydrogen-like atom are determined as a solution to the inverse problem of dynamics, which fully coincide with the classical results by Schrödinger and Bohr. A wave function is regarded as a physical reality, which makes it possible to open up new possibilities in order to study the structure of the microcosm
The paper introduces a method of V-function, from which it follows that the trajectory motion of an object is connected with the wave motion. We formulate a local variational principle (LVP) and the basic theorems of the method of V-function, which allow a new statement of the direct and inverse dynamics problems. Based on the method of V-function, we consider the optical-mechanic analogy, which is presented in a new extended version and investigate the properties of the wave nature of electron motion in a hydrogen-like atom and the motion of an object (particle) for a harmonic oscillator. We demonstrate that in a physical sense, the wave V(x,t) characterizes the properties of an action showing itself in the motion of an object (particle). By its node, the wave is connected with the position of the particle and so leads it. The particle (trajectory) in its turn generates a wave propagating along it. We establish that the energy levels of a hydrogen-like atom, obtained using the method of V-function, correspond entirely to the classical results of quantum physics, while the energy levels of a harmonic oscillator do not contradict the known Schrodinger's results.
This paper presents a geometric and wave interpretation of energy quantization in the hydrogen atom, based on the de Broglie closure condition of the electron wave in a circular orbit. In this concept, energy quantization is a secondary phenomenon resulting from the fact that the electron wave in each orbit consists of exactly n full periods, and the transition to level n+1 corresponds to the addition of one full period. Combining the wave condition with the classical equilibrium of the Coulomb and centripetal forces leads to values of the orbital radii and a discrete energy spectrum consistent with solutions of the Schrödinger equation for the hydrogen atom. It is shown that energy quantization can be interpreted as a consequence of the resonant nature of the electron's wave nature and the condition of phase uniqueness after a complete orbit around the nucleus.1 The model under consideration is semiclassical in nature and serves as an intuitive representation of known results from quantum mechanics. It is assumed that the allowable states correspond to configurations in which the de Broglie wave forms a standing wave containing an integer number of full periods around the orbital circumference. This condition leads directly to the quantization of angular momentum according to the Bohr model. The waveband model provides a one-dimensional analogy of the full quantum description and can serve as a teaching tool to facilitate understanding the geometric aspects of energy quantization in the hydrogen atom. It demonstrates that energy quantization is a natural consequence of the standing wave geometry and the addition of successive full periods along the orbit as the system transitions to the next energy eigenstate after activation. The electron is a stable eigenstate of a quantum field whose behavior in bound systems can be geometrically interpreted as the self-resonance of a de Broglie wave satisfying the condition of single-valent phase, rather than as a local particle with a classical trajectory. Self-resonance of a wave means that the condition of single-valent phase of the wave function after a complete orbit around the nucleus is satisfied, i.e., the requirement that the phase change be an integer multiple of 2π; this is equivalent to the Bohr–de Broglie condition for the closure of the de Broglie wave.
This chapter considers Schrodinger's model of hydrogen atom and analyzes in details the quantum wave mechanics approach to the motion of the electron in the hydrogen atom. The stationary Schrodinger equation is not easy to solve as it stands, and the whole problem looks rather complicated when expressed in terms of Cartesian coordinates. An essential difference between Bohr's and Schrodinger's models is that there are three quantum numbers, n , l , m , rather than the one number n describing a state of the electron. Moreover, in Bohr's model the atom is, at each instant, in one and only one energy state, that is., at each instant, the state of the atom is definite, but according to Schrodinger's model, the atom can exist in a superposition of states. In other words, the atom can be simultaneously in more than one energy state at a given instant. This is clearly incompatible with Bohr's model.
The atom model of Niels Bohr is commonly considered as antiquated even if it describes the atomic spectrum of hydrogen quite accurately. The later published relation of Louis De Broglie could arithmetically be implemented into Bohr’s formulation, leading to the concept of standing waves as the existence cause of excited electron states. However, at that time it was not possible to find well defined electron trajectories being classically describable. As a consequence, the »quantum mechanics« were developed by several authors, particularly by Schrodinger and by Heisenberg, but delivering a hardly under¬standable formalism where the classical physical laws appear being abrogated while abstract terms replace concrete and imaginable ones, abandoning the particle perception of mass points. In particular, Heisenberg’s »uncertainty principle« and the assumption of state probabilities seem to be in a striking variance to the idea of standing waves. In contrast, a formulation is given here which exactly describes the electron trajectories in the exited states solely by applying classical physical laws. Firstly, the original Bohr-model - in combination with De Broglie’s relation - is rolled up. From this formula system a vibration frequency - corresponding to the De-Broglie frequency – is deduced which is n-times larger than the rotation frequency of the Bohr-model. Furthermore, a direct coherence between that vibration-frequency of the electron and the frequency of the involved light is evident being explainable as a resonance effect. Then, a three-dimensional model is proposed where the electron oscillates and pulses perpendicularly to a virtual rotation plane i.e. rotating around a vertical axis, accompanied by a perpetual energy exchange between potential and kinetic energy. This leads at the excited, metastable energy states to well-defined, three-dimensional and wavy trajectories winding up on a surface similar to the one of a hyperboloid, whereas at the ground state the trajectory is planar and stable.