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Quantum Mechanics of the Bound Isotropic Systems with Mathematica: Analysis and Visualization of Quantum States

Alexei V. Tikhonenko
Organization: Institute of Nuclear Power Engineering, Russia
Department: General Physics Department
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The aim of this electronic book is study Quantum Mechanics of the Bound Isotropic Systems with MATHEMATICA. Quantum Isotropic Rotator (Two-dimensional and Three- dimensional), Quantum Isotropic Oscillator (one, two and three- dimensional) and Hydrogen-like Atoms are fully considered with MATHEMATICA tools: - analysis of the solution of Schrodinger equations; - separation of variables in two and three- dimensional Schrodinger equations; - solution of differential equations; - special mathematical functions (Kummer confluent hypergeometric functions, Bessel functions, Hermite function, Laguerre function, Spherical Harmonics) and its properties; - asymptotic expanding and Taylor series, - two-dimensional plots (polar and filled plots); - three-dimensional plots (surface, density and parametric plots) It is analytically and graphically analyzed quantum states of Rotator, Oscillator and Hydrogen-like Atoms for given quantum numbers. Wave functions and probability densities are calculated in the Cartesian rectangular coordinates, polar coordinates and spherical coordinates and visualized by one, two and three-dimensional plots. It is analyzed structure of three-dimensional probability densities for different values of parameter. Comparative visualizations of the probability densities of Rotator, Oscillator and Hydrogen-like Atoms are presented.

Some important notes on running this electronic book

1. The most part of this electronic book is presentation of quantum states of Quantum Rotator, Quantum Oscillator and Hydrogen-like Atoms (two and three- dimensional plots of radial and angular functions, wave functions and corresponding probability densities). These plots require computer resources. Therefor it is necessary to look after parameter N3d (Plot Points for 3d Plots) before running files. Just parameter N3d defines final file size, time of evaluation and spending of computer resources. In order to have qualitative pictures it is recommended value N3d = 150 (and more). Just this value will provide detailed visualization of quantum states. But in this case it is necessary more computer resources. Acceptable quality level of pictures can be reached with N3d = 80. 2. Files and graphics are configured for 17 inch monitor with 1024 x 768 resolution. Some presentation graphics need extra rolling. 3. Most of files are relatively self-contained and can be running independently of one another. But sometimes it is necessary to clear data of previous files. 4. This electronic book is based on special style sheet "Classroom1.nb" (this file is attached). It is recommended to copy this style sheet file to StyleSheets directory: "...Wolfram Research\Mathematica\4.1 \SystemFiles\FrontEnd\StyleSheets". 5. Some files free from plots contain calculated Output cells.

*Applied Mathematics > Visualization
*Science > Physics > Quantum Physics

Physics, Quantum Mechanics, Visualization

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