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Latvian Journal of Physics and Technical Sciences

V -Function Method: Some Solutions of Direct and Inverse Dynamics Problems in A New Statement

N T Valishin, Fan T. Valishin

Published Feb 1, 2019• 2 citations
Publisher

Abstract

Abstract Based on the V -function method, the properties of wave nature of object motion are studied for object uniform motion with constant speed and for harmonic oscillator. It follows from the V -function method that object wave motion is inseparably linked with its trajectory motion. The V -function method consists of the principle of local variation and a new statement of the direct and inverse dynamics problems. The proposed approach made it possible to make the optico-mechanical analogy that obtained a new continuation. A comparison is made with the results obtained by Schrödinger for a harmonic oscillator.

DOI
10.2478/lpts-2019-0007

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Abstract Based on the V -function method, the properties of wave nature of object motion are studied for object uniform motion with constant speed and for harmonic oscillator. It follows from the V -function method that object wave motion is inseparably linked with its trajectory motion. The V -function method consists of the principle of local variation and a new statement of the direct and inverse dynamics problems. The proposed approach made it possible to make the optico-mechanical analogy that obtained a new continuation. A comparison is made with the results obtained by Schrödinger for a harmonic oscillator.
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.
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
Abstract On the basis of the V-function method, an optical-mechanical analogy is carried out; and further it receives a new continuation. In particular, the equality of the phase velocity of the wave and the velocity of the particle is shown. It is noted that according to its physical meaning the wave characterizes the properties of the action that manifests itself in the motion of the particle. The V-function method is based on the concept of a process-state and it consists of a local variational principle, a new formulation of the direct and inverse problem of dynamics. According to the V-function method, the trajectory motion of an object is associated with its wave motion. In this case it is assumed that there is an explicit connection between the trajectory and wave equations of the object. The application of the V-function method for the case of rectilinear motion of an object is shown.
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