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Study on Systems of Hydrogen Atoms in the View Point of Natural Statistical Physics

Yoshifumi Ito

Published Sep 1, 2012• 0 citations
OpenAlex

Abstract

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.

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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.
In this paper, the classical Bohr’s model of the hydrogen atom has been revisited. Two values of fundamental physical properties of an electron in the hydrogen atom has been identified. These physical properties ( & ) are constant in nature. The aim for the review was to contribute to the solution of disagreement between the Bohr’s wavelength ( ) and the Balmer’s experimental observation ( ) for the emission spectrum of hydrogen atom. There are two other constants and that were identified in the Bohr’s equation of the hydrogen atom. The four fundamental physical constants are intrinsic properties of an electron and can be applied to multi-electron system. They can also be obtained from Schrodinger’s equation for hydrogen atom at steady state. These constants may be subjected to scrutiny for their determination for better understanding. Also, since Bohr’s model of hydrogen atom is based on classical mechanics, this paper has provided an alternate method of solving simple problems in atomic physics under Bohr’s model to aid good mental picture of hydrogen atom to scientists.
This chapter begins with the simple solution of the Schrödinger equation for hydrogen atom for which the energy value has been derived. The Bohr radius of H atom has also been computed. After this, this chapter provides the generalised solution of the Schrödinger equation for H atom/hydrogen-like species in which F equation, R equation, and T equations have been derived in a simple and lucid way. This chapter also deals with the Laguerre differential equation, Laguerre polynomials, Rodrigues formula for the Laguerre polynomials, Laguerre associated equation and its solution, associated Laguerre polynomials, Rodrigues formula for the associated Laguerre polynomials, and solution of the radial equation. It also provides normalisation of the radial wave function and complete wave function; for the hydrogen atom, hydrogenic atomic orbital; electron density of 1 s electron; mathematical background of shape of orbitals; the most probable distance of electron from nucleus of hydrogen atom, and average distance of electron from the nucleus of hydrogen atom. The end of this chapter describes the references and adequate computational problems (solved problems and question on concepts).
As the most basic and abundant atom in the universe, hydrogen atoms have long attracted the interest of scientists and provide the best subject for the exploration of quantum mechanics. This article provides a comprehensive study of the multifaceted characteristics and implications of the hydrogen atom. First, the research deeply studied the basic derivation of the Schrödinger equation of the hydrogen atom and provided the necessary theoretical framework. On this basis, modern computing tools are used to visualize the electron cloud of hydrogen atoms and reveal their shapes and characteristics under different orbits. In addition, the study investigated atomic transitions and associated spectra, elucidating basic quantum transition principles and serving as a benchmark for studying more complex atomic systems. Eventually, the research focus shifted to a specific direction, studying the Stark effect in hydrogen atoms, especially in the n=2 state. This research provides a novel perspective on understanding the fundamental properties of the hydrogen atom and its implications in modern physics research.
This chapter has provided an alternate method of solving simple problems in atomic physics under Bohr’s model to aid good mental picture of hydrogen atom to scientists. Two values of fundamental physical properties of an electron in the hydrogen atom has been identified. By limiting the orbiting electrons to a series of circular orbits having discrete radii, Bohr could account for the series of discrete wavelengths in the emission spectrum of hydrogen. Light, he proposed, radiated from hydrogen atoms only when an electron made a transition from an outer orbit to one closer to the nucleus. These physical properties (Ve = 174090.938ms-1& Pe = 1.585860888 × 10-25kgms-1) are constant in nature. The aim for the review was to contribute to the solution of disagreement between the Bohr’s wavelength (λ\lambdaλ0 = 91.12nm) and the Balmer’s experimental observation (λ\lambdaλ0 = 91.18nm) for the emission spectrum of hydrogen atom. There are two other constants δ\deltaδ = 7.199822 × 10-10 eVm and ζ\zetaζ = 1503796277m-1 that were identified in the Bohr’s equation of the hydrogen atom. The four fundamental physical constants can be applied to multi-electron systems because they are inherent characteristics of an electron. They can also be derived from the steady state hydrogen atom Schrodinger's equation. For a deeper understanding, these constants may be scrutinised for their determination. The study recommends that current quantum atomic models and equations governing quantum physics should be revisited for better applications and understanding of the physical world.
A hydrogen atom in a constant magnetic field is a problem whose Schrödinger equation has been considered to be analytically unsolvable. However, it has been found for the case of two-dimensional space some specific values of the magnetic field for which the problem has analytical solutions. In this paper, we give a full theory for these analytical solutions. First, we construct an equation to calculate of exact-solution-points, the values of magnetic field which allow exact solutions for the Schrödinger equation. The explicit forms of the wave functions corresponding to these analytical solutions are also given. Second, we analyse the exact-solution-points and discover some important rules. (i) There exist (n − 1)2 different values of the magnetic field so that the Schrödinger equation of a two-dimensional hydrogen atom in such field has exact analytical solutions with the principal quantum number n. (ii) For more details, among these (n − 1)2 exact analytical solutions, there are nr=n−|m|−1 solutions with the same magnetic quantum number m. Both properties are also proved mathematically in this paper. For illustration, some concrete values are shown for the cases of quantum number n up to 8.