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Theoretical and Natural Science

Exploring the hydrogen atom: From wave function to stark effect

Biyue Ren

Published Jan 23, 2024• 1 citation
Publisher

Abstract

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.

DOI
10.54254/2753-8818/30/20241104

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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.
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.
The unique properties of the hydrogen atom, especially those that result from the dynamic symmetry of the 1/r potential, are presented from both classical and quantal points of view. Using these properties as a starting point, the response of a classical hydrogen atom, an electron executing a Keplerian ellipse about a proton, to a weak electric field is described. Because the field is weak, the elliptical orbit may be treated as a dynamical element itself, variations in its shape and orientation depicting the classical Stark effect. The behavior of this orbit is then correlated with the quantum mechanical description of the first-order Stark effect in hydrogen, a subject that is included in most introductory courses in quantum mechanics.
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.
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.
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.