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Study on the Phenomena of Potential Well in the View Point of Natural Statistical Physics

Yoshifumi Ito

Published Sep 1, 2012• 0 citations
OpenAlex

Abstract

In this paper, we study on the phenomena of potential well in the view point of natural statistical physics. We derive the Schrodinger equation of potential well by virtue of my new theory of natural statistical physics. This derivation is very new and natural and reasonable in the physical sense. We solve the eigenvalue problem for the Schrodinger equation of potential well. Thereby we obtain the new and true eigenfunctions of the Schrodinger operator of the potential well. By virtue of my new theory, we can understand the phenomena of potential well naturally and reasonably in the physical sense. At last, we study the meaning of the impact force naturally and reasonably in the physical sense. As for these results, we refer to Ito [1] [4]. Especially, as for the phenomena of potential well, we refer to Ito [2], [4]. 2010 Mathematics Subject Classi cation. Primary 81Q05 ; Secondary 81Q99, 81V45, 82D35.

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In this paper, we study the derivation and the solution of Schrodinger equation of the hydrogen atoms using the theory of Natural Statistical Physics. Using these results, we study the phenomena of the spectra of hydrogen atoms and the phenomena of the stability of hydrogen atoms. Here we consider the system of hydrogen atoms for which we need not to consider the in uence of the outer electro-magnetic eld. This is the case where there is no in uence of outer electro-magnetic eld or where we can neglect the in uence of the outer electro-magnetic eld. In this paper, we succeeded in deriving the Schrodinger equation in the natural and reasonable way by the method of variational calculus. Thereby we can obtain the complete understanding of the phenomena of the spectra of hydrogen atoms and the phenomena of the stability of hydrogen atoms. We remark that the model of the system of hydrogen atoms considered in this paper does not concern with the scattering state of hydrogen atoms. In this paper, Fourier's method plays the fundamental role. For the results of this paper, we refer to Ito[16], [19], [20], [22], [24], [25], [26]. 2010 Mathematics Subject Classi cation. Primary 82D99, 82B99, 81Q99.

•Sep 1, 2012•0 citations
In this paper, we study on the phenomena of potential well in the view point of natural statistical physics. We derive the Schrodinger equation of potential well by virtue of my new theory of natural statistical physics. This derivation is very new and natural and reasonable in the physical sense. We solve the eigenvalue problem for the Schrodinger equation of potential well. Thereby we obtain the new and true eigenfunctions of the Schrodinger operator of the potential well. By virtue of my new theory, we can understand the phenomena of potential well naturally and reasonably in the physical sense. At last, we study the meaning of the impact force naturally and reasonably in the physical sense. As for these results, we refer to Ito [1] [4]. Especially, as for the phenomena of potential well, we refer to Ito [2], [4]. 2010 Mathematics Subject Classi cation. Primary 81Q05 ; Secondary 81Q99, 81V45, 82D35.
In this paper, we study the phenomena of potential well of infinite depth on the view point of natural statistical physics. The mathematical model of this physical system is the system of micro-particles moving periodically with constant velocity in the interval D = [−a, a], (a > 0). Thereby we obtain the structure of this physical system at the stationary state. Thus we clarify that this physical system is the composed state of the proper physical systems at the stationary state and the ratio of their composition is given by the sequence {|an|2}n=0 ∪ {|bn|2}∞n=1, where an, (n ≥ 0) and bn, (n ≥ 1) are the Fourier type coefficients of the initial state ψ ∈ L2. Then we obtain the energy expectation of the total physical system Ē =∞Σn=0 n2π2ℏ2/2ma2 (|an|2 + |bn|2), where m denotes the mass of one micro-particle.
Abstract Using the simplest but fundamental example, the problem of the infinite potential well, this paper makes an ideological attempt (supported by rigorous mathematical proofs) to approach the issue of ≪understanding≫ the mechanism of quantum mechanics processes, despite the well-known examples of the EPR paradox type. The new exact solution of the Schrödinger equation is analyzed from the perspective of quantum mechanics in the phase space. It is the phase space, which has been extensively used recently in quantum computing, quantum informatics and communications, that is the bridge towards classical physics, where understanding of physical reality is still possible. In this paper, an interpretation of time-dependent processes of energy redistribution in a quantum system, probability waves, the temperature and entropy of a quantum system, and the transition to a time-independent ≪frozen state≫ is obtained, which is understandable from the point of view of classical physics. The material of the paper clearly illustrates the solution of the problem from the standpoint of continuum mechanics, statistical physics and, of course, quantum mechanics in the phase space.
People believe that the Schrödinger equation cannot be used to describe macroscopic objects like the Earth, and Newtonian mechanics cannot be used to describe microscopic systems. The old concept of the relationship between the existing laws of quantum mechanics and classical mechanics undoubtedly has a serious impact on people's understanding of the natural world, the development of physics theories, and the application of existing physics theories. The continuous development of physics theory requires constant changes to some incorrect old concepts. The Schrödinger equation that can describe planetary motion was successfully obtained by replacing the potential energy in the Hamiltonian operator from electromagnetic interaction potential energy to gravitational interaction potential energy. If the distance between the sun and the earth is approximated as a constant, the energy eigenvalues obtained by solving the Schrödinger equation for the Earth's revolution are completely consistent with the results obtained directly using classical mechanics. The direct significance of establishing and applying such equations is that they can simultaneously use classical mechanics and wave dynamics to describe all objects (no longer limited by the mass of the objects), simplifying the calculation process of quantum mechanics. It has been proven that classical mechanics and wave dynamics are compatible, and there is no insurmountable gap between them. This result has a huge positive impact on the theoretical updates and applications of quantum mechanics.
In this paper, we study the derivation and the solution of Schrodinger equation of the hydrogen atoms using the theory of Natural Statistical Physics. Using these results, we study the phenomena of the spectra of hydrogen atoms and the phenomena of the stability of hydrogen atoms. Here we consider the system of hydrogen atoms for which we need not to consider the in uence of the outer electro-magnetic eld. This is the case where there is no in uence of outer electro-magnetic eld or where we can neglect the in uence of the outer electro-magnetic eld. In this paper, we succeeded in deriving the Schrodinger equation in the natural and reasonable way by the method of variational calculus. Thereby we can obtain the complete understanding of the phenomena of the spectra of hydrogen atoms and the phenomena of the stability of hydrogen atoms. We remark that the model of the system of hydrogen atoms considered in this paper does not concern with the scattering state of hydrogen atoms. In this paper, Fourier's method plays the fundamental role. For the results of this paper, we refer to Ito[16], [19], [20], [22], [24], [25], [26]. 2010 Mathematics Subject Classi cation. Primary 82D99, 82B99, 81Q99.