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Journal of Hengyang Normal University

Survey on Schrodinger Equation of Hydrogen Atom System

Junmin Zhang

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

This article analysised the characteristics of the hydrogen atom system,simplified the hydrogen atom as two bodies questions,and infered the steady state Schrodinger equation of the hydrogen atom system.We obtained that the quality of steady state Schrodinger equation is not the electronic quality,and is not the atomic quality,but is the systematic reduced mass.

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This article analysised the characteristics of the hydrogen atom system,simplified the hydrogen atom as two bodies questions,and infered the steady state Schrodinger equation of the hydrogen atom system.We obtained that the quality of steady state Schrodinger equation is not the electronic quality,and is not the atomic quality,but is the systematic reduced mass.
The hydrogen atom is the simplest of all atoms consisting of proton of charge e and the electron of charge -e. It is a two particle system. A two-body problem can always be reduced to a one-body problem by choosing the centre of mass coordinate and the relative coordinates. The existence of degenerate energy eigenvalues implies that linear combination of the corresponding eigenfunctions are solutions of the wave equation with the same energy. The isotopes of atomic hydrogen, deuterium and tritium also provide examples of hydrogenic systems. Since the mass of deuterium is md˜2mp and that of tritium is mt˜3mp, the reduced masses of them will be slightly different from that of hydrogen. In addition to deuterium and tritium, there exists other hydrogenic systems in which the role of proton is played by another positive particle.
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.
In the scientific community dominates the conviction that the hydrogen atom is the simplest, and existing theories based on equations of Schrodinger and Dirac fully describe all the subtleties of behavior of an electron in a hydrogen atom. In this paper and in the following articles will be shown that in addition to the known state-based solutions of the Schrodinger and Dirac equations, there are states which were not studied and they were not known, or often simply were discarded with the phrase have no physical meaning. Further it will be shown that the existing equations of quantum mechanics, namely the Dirac equation, Klein-Gordon equation and especially the Schrodinger equation, can not accurately describe all the subtleties of behavior of an electron in a hydrogen atom. As a result there will be given a reasonable basis for the feasibility of an experimental search or targeted synthesis of new unknown states of the hydrogen atom.
This study conducted a detailed study on the wave function characteristics of the hydrogen atom model and explored its potential applications in quantum computing, atomic spectral analysis and quantum information processing based on the Schrödinger equation. By solving the Schrödinger equation of the hydrogen atom, the form of the wave function was clearly obtained, and the distribution law of electrons at various energy levels was demonstrated. The study constructed a quantum theoretical model of the hydrogen atom, derived the regulatory effect of quantum numbers on the wave function, and solved the Schrödinger equation with the help of numerical analysis technology to ensure the accuracy and robustness of the wave function evaluation. The study conducted a detailed analysis of the energy level, wave function profile and electron probability density distribution of the hydrogen atom. The simulation results show that with the increase of energy level, the oscillation frequency of the wave function increases significantly, and the distribution space of electrons also expands accordingly. In addition, this study also explored the practical application of the hydrogen atom model. In the field of quantum computing, the wave function characteristics provide theoretical support for the initialization and manipulation of quantum bits; in atomic spectral analysis, accurately calculated energy levels and wave functions help to improve the accuracy of spectral measurement; in quantum information processing, the simulation of quantum entangled networks based on the hydrogen atom model provides new ideas for quantum communication and quantum computing. These applications demonstrate that the hydrogen atom model is not only of great significance in theoretical physics, but also has broad application prospects in emerging technology fields.
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 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).