Journal of Computational Methods in Sciences and Engineering

Quantum Theory with Statistical Interpretation The Hydrogen-Like Atom Problem

Е.П. Жидков, А. В. Зорин

Published Feb 1, 2002• 2 citations

Abstract

In the paper the energy spectrum of the hydrogen like atom in the quantum mechanics with the non negative quantum distribution function (QDF) is studied. The essential spectrum of the Hamiltonian has been calculated. As for the eigenvalues situated beyond the essential spectrum, a method for numeri cal evaluation of their numbers and values is proposed.

DOI: 10.3233/jcm-2002-21-240

Related papers

MAPLE program for modelling hydrogen-like atoms in quantum mechanics with non-negative distribution function

A.V. Zorin, N. P. Tretyakov

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.

Вестник Российского университета дружбы народов. Серия: Математика, информатика, физикаJan 1, 20183 citations

Hydrogen-like atom with nonnegative quantum distribution function

А. В. Зорин, Leonid A. Sevastianov

Among numerous approaches to probabilistic interpretation of conventional quantum mechanics (CQM), the closest to N. Bohr’s idea of the correspondence principle is the Blokhintzev-Terletsky approach of the quantum distribution function (QDF) on the coordinate-momentum ( q , p ) phase space. The detailed investigation of this approach has led to the correspondence rule of V.V. Kuryshkin parametrically dependent on a set of auxiliary functions. According to investigations of numerous authors, the existence and the explicit form of QDF depends on the correspondence rule between classical functions A ( q , p ) and quantum operator A . At the same time, the QDF corresponding to all known quantization rules turns out to be alternating in sign or overly complex valued. Finally nonexistence of nonnegative QDF in CQM was proved. On the other hand, from this follows the possibility to construct quantum mechanics where a nonnegative QDF exists. We consider a certain set of auxiliary functions to construct explicit expressions for operators O ( H ) for the hydrogen atom. Naturally, these operators differ from the related operator Ĥ in CQM, so that spherical coordinates are no longer separable for a hydrogen-like atom in quantum mechanics with nonnegative QDF.

Physics of Atomic NucleiApr 1, 20079 citations

Computer modeling of hydrogen-like atoms in quantum mechanics with nonnegative distribution function

А. В. Зорин, Leonid A. Sevastianov, N. P. Tretyakov

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.

Programming and Computer SoftwareMar 1, 20076 citations

The Hydrogen-Like Atom

Ahmad Amer

This chapter presents a detailed analysis of the hydrogen-like atom within the framework of quantum mechanics. The radial Schrödinger equation is solved for the Coulomb potential, yielding the quantized energy spectrum, energy eigenstates, and the associated degeneracy of levels. The expectation value of the electron's radius is calculated, and a comparison is drawn between quantum predictions and classical circular orbits. The quantum two-body problem is introduced to account for the motion of the nucleus, revealing its influence on the energy levels of the electron. This correction underpins the operation of the mass spectrometer and supports the spectroscopic identification of hydrogen isotopes such as deuterium and tritium. The analysis also explains how similar effects confirm the presence of helium in the solar spectrum through astrophysical observations. Metadata/Keywords: hydrogen-like atom, Coulomb potential, energy spectrum, quantum orbits, nucleus motion, two-body problem, mass spectrometer, hydrogen isotopes, helium, solar spectrum

CRC Press eBooksNov 5, 20250 citations

Calculation of Transition Probabilities in Quantum Mechanics with a Nonnegative Distribution Function in the Maple Computer Algebra System

A. V. Zorin, N. P. Tretyakov

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.

Computational Mathematics and Mathematical PhysicsJan 1, 20201 citation

The electron motion in the practically coulomb field

D. V. Kalinovsky, E. S. Savin

will be added. One can explain the nature of the term by examining the relativistic case. Below the formulas for the radial and angular parts of the wave function were evaluated. The term for the energetic levels of the hydrogen atom for specific quantum numbers was given. We calculated the values of several specific radial functions for different quantum numbers, as well as the values of the radial function at the origin of coordinates and wide apart. A number of mean values for different powers of r was calculated. For the continuous spectrum the radial eigenfunctions were calculated.

Apr 1, 20130 citations