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International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT)

Application of the noncommutative theory of statistical decisions to the modeling of quantum communication channels

А. В. Зорин, Anton L. Sevastianov, Leonid A. Sevastianov

Published Nov 1, 2017• 5 citations
Publisher

Abstract

In this work the symbolic algorithm for the calculations of transition probabilities for hydrogen-like atoms in terms of quantum mechanics with non-negative probability distribution function is proposed. The problem was solved in terms of eigenvalues of the finite-approximated Ritz matrices. All the necessary functions, including wave functions, Sturmian functions and their Fourier-transforms, Clebsh-Gordan coefficients etc. were united in one single framework. The program is written using Maple. Results were compared with the data provided by NIST Atomic Spectra Database.

DOI
10.1109/icumt.2017.8255195

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In this work the symbolic algorithm for the calculations of transition probabilities for hydrogen-like atoms in terms of quantum mechanics with non-negative probability distribution function is proposed. The problem was solved in terms of eigenvalues of the finite-approximated Ritz matrices. All the necessary functions, including wave functions, Sturmian functions and their Fourier-transforms, Clebsh-Gordan coefficients etc. were united in one single framework. The program is written using Maple. Results were compared with the data provided by NIST Atomic Spectra Database.
The program is proposed for a realization of the symbolic algorithm based on the quantum mechanics with non-negative probability distribution function (QDF) and for calculations of energy levels for hydrogen-like atoms. The program is written up in the language MAPLE. In the framework of the algorithm an original Maple package for calculations of necessary functions, such as hydrogen wave functions, Sturmian functions and their Fourier-transforms, Clebsch-Gordan coefficients, etc. is proposed. Operators of observables are calculated on the basis of the QDF quantization rule. According to the Ritz method, eigenvalues of Ritz matrices represent spectral values of the quantity under investigation, i.e. energy. As an example, energy levels of hydrogen-like atoms are calculated and compared with experimental data retrieved from the NIST Atomic Spectra Database Levels Data. It turns out that this theory seems to be equivalent to the traditional quantum mechanics in regard to predictions of experimental values. However, the existence of a phase-space probabilistic quantum theory may be an important advance towards the explanation and interpretation of quantum mechanics.
The computer program is proposed for a realization of the symbolic algorithm based on the quantum mechanics with non-negative probability distribution function (QDF) and for calculations of energy levels for hydrogen-like atoms. In the framework of the algorithm an original Maple package for calculations of necessary functions, such as hydrogen wave functions, Sturmian functions and their Fourier-transforms, Clebsch- Gordan coefficients etc is proposed. Operators of observables are calculated on the basis of the QDF quantization rule.
Abstract In the Maple computer algebra system, an algorithm is implemented for symbolic and numerical computations for finding the transition probabilities for hydrogen-like atoms in quantum mechanics with a nonnegative quantum distribution function (QDF). Quantum mechanics with a nonnegative QDF is equivalent to the standard theory of quantum measurements. However, the presence in it of a probabilistic quantum theory in the phase space gives additional possibilities for calculating and interpreting the results of quantum measurements. The methods of computer algebra seem to be necessary for the relevant calculations. The calculation of the matrix elements of operators is necessary for determining the energy levels, oscillator strengths, and radiation transition parameters for atoms and ions with an open shell. Transition probabilities are calculated and compared with experimental data. They are calculated using the Galerkin method with the Sturm functions of the hydrogen atom as coordinate functions. The verification of the model showed good agreement between the calculated and experimentally measured transition probabilities.
An algorithm for describing operator structures and calculating energy levels for hydrogen-like atoms in the quantum mechanics with nonnegative distribution function (Kuryshkin’s quantum mechanics) is developed. The algorithm is implemented in Maple. An original library of necessary functions and quantities (Coulomb wave functions, Sturm functions, their Fourier images and kernels of integral transformations, Clebsch-Gordan coefficients, products of spherical harmonics, etc.) has been created. In accordance with the quantization rules in Kuryshkin’s quantum mechanics, operators of observable quantities are computed. Based on the Hamiltonian obtained, energy levels of a hydrogen-like atom are calculated by the Ritz method (in the basis of Sturm functions). Numerical values of model parameters are determined by comparing results obtained with experimental data for hydrogen and alkali metals taken from the NIST Atomic Spectra Database Levels Data library.