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Doklady BGUIR

Fundamentals of the Theory of a Hydrogen Atom for Problems of Synthesis of Quantum-Electronic Circuits

I. P. Kabiak

Published Feb 28, 2022• 0 citations
Publisher

Abstract

On the basis of the unconventional concept of the formation of the line spectrum of the hydrogen atom, an equation is obtained that is necessary for computer calculations of the energy emitted by an electron during the transition from a certain current energy level rλ,n to the receiver radius r0. The obtained solution of the energy equation made it possible to derive the formal relations necessary for calculating the values of the radii of the levels of the hydrogen atom and, accordingly, the velocities of the electron in these orbits. The applied aspect of the considered methodology consists in using the results of the performed calculations for the problems of synthesis and analysis of quantum-electronic circuits.

DOI
10.35596/1729-7648-2022-20-1-31-39

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On the basis of the unconventional concept of the formation of the line spectrum of the hydrogen atom, an equation is obtained that is necessary for computer calculations of the energy emitted by an electron during the transition from a certain current energy level rλ,n to the receiver radius r0. The obtained solution of the energy equation made it possible to derive the formal relations necessary for calculating the values of the radii of the levels of the hydrogen atom and, accordingly, the velocities of the electron in these orbits. The applied aspect of the considered methodology consists in using the results of the performed calculations for the problems of synthesis and analysis of quantum-electronic circuits.
The existence of line spectra can be explained bymeans of the first “quantum”model of the atom, developed by Bohr in 1913. Although the Bohr model of the hydrogen atom was eventually replaced, it yields the correct values for the observed spectral lines, and gives a substantial insight into the structure of atoms in general. The following derivation has the objective of obtaining the energy levels of the Bohr atom. If we can obtain the energy levels, we can reproduce the hydrogen atom spectra. The derivation proceeds with three major elements: first, use force balance to relate the velocity to the radius of the electron orbit, then use a quantum assumption to get the radius, then solve for the energies.
Transition of hydrogen electron through S-orbits has been studied.Energy levels and frequencies of the emitted spectral lines were computed through new simple atomic model.The equations assume that, the energy and frequency of the emitted lines of six known series are related to enclosed areas of successive orbits and proportionality constants.These constants are strongly related to all spectral lines in all series.The results found from this model will give abroad understanding of quantum mechanics principles.
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.
This chapter begins with the simple solution of the Schrödinger equation for hydrogen atom for which the energy value has been derived. The Bohr radius of H atom has also been computed. After this, this chapter provides the generalised solution of the Schrödinger equation for H atom/hydrogen-like species in which F equation, R equation, and T equations have been derived in a simple and lucid way. This chapter also deals with the Laguerre differential equation, Laguerre polynomials, Rodrigues formula for the Laguerre polynomials, Laguerre associated equation and its solution, associated Laguerre polynomials, Rodrigues formula for the associated Laguerre polynomials, and solution of the radial equation. It also provides normalisation of the radial wave function and complete wave function; for the hydrogen atom, hydrogenic atomic orbital; electron density of 1 s electron; mathematical background of shape of orbitals; the most probable distance of electron from nucleus of hydrogen atom, and average distance of electron from the nucleus of hydrogen atom. The end of this chapter describes the references and adequate computational problems (solved problems and question on concepts).
Definitions of the mechanical parameters entering the Bohr model of the hydrogen atom allowed us to calculate the time intervals connected with the electron transitions between the nearest-neighbouring energy levels in the atom. This is done in a strictly non-probabilistic way. The time results are compared with those derived earlier on the basis of the classical Joule-Lenz law for the energy emission adapted to the case of the electron transfer in the quantum systems. A similar formalism has been next applied to the harmonic oscillator and a particle moving in the one-dimensional potential box.
The basis for the hydrogen atom planetary model, which represents the development of the Rutherford-Bohr atom model, is presented. The measurements of the atom polarizability, radius, electric and magnetic moments, as well as the Stark and Zeeman effects are given as an argument. The central paragraph of the Article is a comparison of the binding energy obtained based on wave measurement data (the Rydberg constant) and the binding energy obtained by calculation according to the data of the electron charge and mass fundamental constants and applying the principle of momentum conservation. The substantiation of the atom stability is given proceeding from the atom planetary model. The nature and mechanism of the optical spectrum formation is discussed. The substantiation of the fine structure of the hydrogen atom spectral lines is presented, proceeding from the fundamental association between the electric field strength of the electron charge and the speed of its velocity. The origin of the optical spectrum hyperfine structure is discussed within the planetary model. The difficulty in constructing an acceptable theory of the electronic structure of atoms is primarily due to the complexity of obtaining direct information on the atoms internal structure. The point is that the outermost electron shell of an atom behaves itself like an effective screen for external probing electrical fields. This circumstance sharply limits the possibilities of experimental methods for studying the intra-atomic structure. On the other hand, it is imperative that the research methods being employed be non-destructive ones in order to obtain reliable information concerning the structure of atoms. This means that the action on the atom during measurement process must meet the condition of the electron shell perturbation smallness of the atom or ion being investigated. Concerns associated with the inaccessibility of the electronic structure of atoms served as a pretext for a conclusion of the limited possibilities of the empirical method of knowing the internal structure of atoms at the proper time.