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Journal of Xi'an University of Technology

A Visualization Analysis of Hydrogen Electronic Cloud Distribution

Wei Shi

Published Jan 1, 2007• 1 citation
OpenAlex

Abstract

With an aim at the abstract conception of wave functions for hydrogen atom in quantum mechanics,this paper suggests the radial distribution,angular distribution and spatial distribution function of hydrogen atom electronic cloud,and also,Matlab software is used to plot their corresponding distribution charts.The analytical results indicate that the distribution patterns of hydrogen atom electronic clouds are co-determined by the radial and angular probability distribution,and that the electronic cloud distribution density is so uneven that the orbit trace can be observed.Accordingly,this method can reveal the radial,angular and spatial probability distribution laws of hydrogen atom electronic clouds intuitionally,whereby a new thinking way for analyzing the abstract wave function visualization in quantum mechanics is provided in this paper.

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With an aim at the abstract conception of wave functions for hydrogen atom in quantum mechanics,this paper suggests the radial distribution,angular distribution and spatial distribution function of hydrogen atom electronic cloud,and also,Matlab software is used to plot their corresponding distribution charts.The analytical results indicate that the distribution patterns of hydrogen atom electronic clouds are co-determined by the radial and angular probability distribution,and that the electronic cloud distribution density is so uneven that the orbit trace can be observed.Accordingly,this method can reveal the radial,angular and spatial probability distribution laws of hydrogen atom electronic clouds intuitionally,whereby a new thinking way for analyzing the abstract wave function visualization in quantum mechanics is provided in this paper.
Electron spin and orbital interaction outside the nucleus is a classical and important problem in quantum mechanics. In this paper, a simplified model is constructed for the phenomenon of atomic energy level splitting in weak magnetic field in quantum mechanics, and the wave function of the fine structure of hydrogen-like atoms under perturbation is derived. According to the probability interpretation of quantum mechanics, we obtain the probability density function of the fine structure electron cloud of hydrogen-like atoms. The Monte Carlo selection method is used to sample, and the electron cloud images under different quantum numbers are drawn by Mathematica software. The experimental results are explained regularly, so that people can intuitively understand the distribution of electron cloud in three-dimensional space.
While in the process of studying hydrogen atom wave function in quantum mechanics course,some frequently-asked questions about the radial probability distribution,tree-dimensional angular distribution,and movement trace are raised by beginners.The visualized interpretations for those questions are made by the three dimension model set up in computer,which can not only help students to establish the correct quantum images about the movement of the particles,but also provide a reference for visualization about other quantum phenomena.
Previous attempts to represent graphically the Schrodinger electron cloud for hydrogen-like atoms have been so successful that similar graphical representations have been looked for on the Dirac theory. With a single electron specified on Dirac's theory by a set of four wave functions instead of one, the probability density $\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}={\ensuremath{\psi}}_{I}{{\ensuremath{\psi}}_{I}}^{*}+{\ensuremath{\psi}}_{2}{{\ensuremath{\psi}}_{2}}^{*}+{\ensuremath{\psi}}_{3}{{\ensuremath{\psi}}_{3}}^{*}+{\ensuremath{\psi}}_{4}{{\ensuremath{\psi}}_{4}}^{*}$ is treated by considering, as on the Schrodinger theory, the independent angular and radial factors separately. Tables and graphs are given for $\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}$ as a function of $\ensuremath{\theta}$ alone. The graphs are compared with and shown to correspond closely to the space quantization of the classical electron orbits. Graphs representative of $\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}$ as a function of rrr alone are given and compared with the electronnucleus distances of the corresponding electron orbits. The radial and angular factors are brought together by means of a mechanical device described previously, which when photographed represents closely the electron-cloud $\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}$. Photographs are given for the states 12S121^{2}S_{\frac{1}{2}}12S21​​, 22P12,322^{2}P_{\frac{1}{2}, \frac{3}{2}}22P21​,23​​, 32D32,523^{2}D_{\frac{3}{2}, \frac{5}{2}}32D23​,25​​ 42F52,724^{2}F_{\frac{5}{2}, \frac{7}{2}}42F25​,27​​, and 52G725^{2}G_{\frac{7}{2}}52G27​​, with $m=\ifmmode\pm\else\textpm\fi{}\frac{1}{2},\ifmmode\pm\else\textpm\fi{}\frac{3}{2},\ensuremath{\cdots}\ifmmode\pm\else\textpm\fi{}j$
The simplest atom consisting of a proton and an electron is a hydrogen atom. The energy level and the probability of finding an electron in a hydrogen atom can be determined based on the function of radial wave. The function of radial wave of a hydrogen atom is influenced by the principal quantum number ( n ) and the orbital quantum number ( l ). Completion of the function of radial wave of a hydrogen atom in the principal quantum numbers( n ) 4 and 5 uses the time-independent Schrodinger equation approach in spherical coordinates, variable separation method and uses the associated Laguerre function. The normalized of the function of radial wave of a hydrogen atom R nl ( r ) at n = 4 produces four wave functions:R 40 ( r ); R 41 ( r ); R 42 ( r ); R 43 ( r ), whereas at n = 5 produces five wave functions:R 50 ( r ); R 51 ( r ); R 52 ( r ); R 53 ( r ); R 54 ( r ). The results of this research in the form of the functions of radial wave of hydrogen atom can be used to determine the behavior of electron in a hydrogen atom.
Quantum mechanics holds a very important position as the theoretical foundation of the micro world. Harmonic oscillator system is a very classical model in quantum mechanics. In this paper, the wave function of a three-dimensional linear harmonic oscillator is derived based on the linear harmonic oscillator model in quantum mechanics. According to the probabilistic interpretation of quantum mechanics, we obtain the probability density function of the harmonic oscillator electron cloud. In this paper, Monte Carlo selection method is used for sampling, and Mathematica software is used to draw electron cloud images with different quantum number, so that people can intuitively see the distribution of electron cloud in three-dimensional space.