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Journal of Shihezi University

Classical Oscillator Model of the Ground State of Hydrogen Atom

YU Chang-feng

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

According to the prophesy made by quantum mechanics that the angular momentum of ground state hydrogen atom is zero,in the paper,the ground state hydrogen atom is treated as a one-dimensional classical oscillator,and by using a potential function with phase factors and repulse force,the ground state energy of the hydrogen atom as well as an analytic expression of the vibrational orbit of electron are given.

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According to the prophesy made by quantum mechanics that the angular momentum of ground state hydrogen atom is zero,in the paper,the ground state hydrogen atom is treated as a one-dimensional classical oscillator,and by using a potential function with phase factors and repulse force,the ground state energy of the hydrogen atom as well as an analytic expression of the vibrational orbit of electron are given.
The hydrogen atom is the Rosetta stone of the early twentieth century atomic physics. The attempt to decipher its structure and properties led to the development of quantum mechanics and the unraveling of many of the mysteries of atomic, molecular, and solid state physics, and a good deal of chemistry and modern biology. Unlike the various one-dimensional model problems that we have been studying in the previous chapters, the hydrogen atom is a real physical system in three dimensions. It consists of an electron moving in a spherically symmetric potential well due to the Coulomb attraction of the positively charged nucleus. In three dimensions, the electron is not constrained to move linearly. It can execute orbital motions and, thus, has angular momentum. Not only is the total energy of the electron in the atom quantized, its angular momentum also has interesting and unexpected quantized properties that cannot possibly be understood on the basis of classical mechanics and electrodynamics. They are, however, the natural and necessary consequences of the basic postulates of quantum mechanics, as will be shown in this chapter. According to classical mechanics and electrodynamics, it is not possible to have a stable structure consisting of a small positively charged nucleus at the center of an electrically neutral atom with an electron sitting in its vicinity. For the electron not to be attracted into the positive charge, it must be orbiting around the nucleus so that the centrifugal force will counter the Coulomb attraction of the nucleus and maintain a constant electron orbit.
In this chapter, the authors solve the Schrodinger equation for a harmonic oscillator using two methods, analytical method and algebraic method. They discuss the quantum properties of the harmonic oscillator. A harmonic oscillator is a system that exhibits simple harmonic motion or periodic motion, a motion that repeats itself after equal intervals of time. Many systems such as spring, simple pendulum, vibrating string and molecular vibration can be approximated as simple harmonic oscillators. The authors consider a simple harmonic oscillator, a spring whose spring constant is "k" attached to a mass "m" as a simple harmonic oscillator. They describe the Schrodinger equation in spherical coordinates. The authors also solve the three-dimensional Schrodinger equation for the hydrogen atom. For the hydrogen atom, the potential energy function is dependent only on radial distance and is independent of angular variables. The hydrogen atom consists of a positively charged proton surrounded by an electron that orbits around it.
The problem of the motion of a particle attracted by two fixed centers of force according to the Coulomb force law can be treated by classical mechanics and has been used in quantum theory by Pauli and Messen for a theory of the hydrogen molecule ion.1 In the quantum mechanics, where the energy levels are determined as the “eigenwerte” of Schrodinger’s equation, the variables are separable and the boundary value problem is easily set up. But thus far a satisfactory treatment of the differential equations involved is lacking. Burrau2 has recently carried out a numerical integration of the problem for the lowest energy level of an electron moving under the influence of two fixed centers of Coulomb attraction as a function of the distance apart of these centers. In this paper, Burrau’s data are used to give a semi-quantitative discussion of the neutral hydrogen molecule. His values are: Nuclear separation 1.0 1.3 1.6 1.8 2.0 2.2 2.4 2.95 Electronic energy 2.896 2.648 2.436 2.309 2.204 2.109 2.025 1.836 The unit of separation is the Bohr l1 orbit radius of hydrogen atom, that of energy is the ionization potential of atomic hydrogen.
In Part one of this Paper a hypothesis is forwarded of the electron charge in an atom existing in a distributed form. To check it by methods of electrodynamics and mechanics (without invoking the formalism of quantum mechanics and the concepts of the wave function and of the operators), the potential, kinetic, and total energies were calculated for three states of the hydrogen atom, which were found to agree closely with the available experimental data. The Part two of the Paper offers additional assumptions concerning various scenarios of motion of elements of the distributed electron charge which obey fully the laws of theoretical mechanics. The angular momentum of the ground-state hydrogen atom calculated in the frame of theoretical mechanics is shown to coincide with the spin which is ℏ/2\hbar/2ℏ/2.