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SOLVING PROBLEMS OF QUANTUM MECHANICS BY METHODS OF INTEGER THEORY

M. Ye. Svavil’nyi

Published Mar 17, 2023• 0 citations
Publisher

Abstract

In the present review of the author's works, the calculation of the radiation spectrum of a hydrogen-like atom is described in detail using the principle of commensurability of conserved quantities, which is introduced by the author. The result obtained allows us to give a clear physical interpretation of Bohr's quantum hypothesis. It follows from the data obtained that the “resolved” states of quantum systems are the result of the need to fulfill the requirement of having a common measure of conserved quantities for all participants in physical interaction before and after interaction.

DOI
10.5281/zenodo.7747276

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In the present review of the author's works, the calculation of the radiation spectrum of a hydrogen-like atom is described in detail using the principle of commensurability of conserved quantities, which is introduced by the author. The result obtained allows us to give a clear physical interpretation of Bohr's quantum hypothesis. It follows from the data obtained that the “resolved” states of quantum systems are the result of the need to fulfill the requirement of having a common measure of conserved quantities for all participants in physical interaction before and after interaction.
The paper proposes a new method of describing physical reality using the mathematical apparatus of integer theory. Applying this apparatus becomes possible after introducing the principle of commensurability of conserved quantities into physics. It states that at the interaction of physical objects, only such exchanges of conserved quantities (energies, momentums, etc.) are possible so that these quantities have a common measure before and after the interaction of objects. Methods of integer theory solve two classical problems of physics: the radiation spectrum of a hydrogen-like atom and the scattering problem. The known results of such solutions are obtained, and new data are also obtained. It follows from the obtained that the physical reason for the ‘allowed’ states of atomic systems is the necessity of commensurability of the excited states of the atom with its ground state in all conserved quantities. This gives physical foundations to the quantum hypothesis of N. Bohr, which it has not had so far. The strict causal relation between stationary quantum states of atomic systems is also established, i.e. the principle of causality in the quantum world is restored.
Einstein extended the quantization to light itself with the photon hypothesis with the explanation of the photoelectric effect. Bohr used Planck's ideas to explain the line emission of hydrogen atoms which gave an indication that the newborn quantum mechanics was giving a better description of experiments. In this chapter, the authors not follow a rigorous mathematical description. The postulates come in classes: the description of states, the description of physical quantities, the description of the act of measurement, the effect of the measurements on the state and the time evolution of a quantum system. Sommerfeld modifications of Bohr's theory was seen favorably because it gave a solutions to a few anomalous effects like the Zeeman effect and the Stark effects. As in the classical case of a two body problem, positive values of the total energy E correspond to free particles while negative values of E correspond to bound states. In quantum mechanics, electrons are indistinguishable.
An attempt is made to present a consistent detailed theory of optical phenomena, based on the suggestion recently made by the author in collaboration with Bohr and Kramers. This suggestion is that atoms are radiating or absorbing continuously, during their stationary states, and that transitions influence radiation only in terminating the radiation characteristic of the old state, and substituting that of the new. The atoms, on the other hand, are supposed not to change their energy while radiating, but to change it discontinuously on transition. This necessitates giving up the detailed application of the conservation of energy in interaction between atoms and radiation. In the present paper, an additional suggestion is made, namely that resonance radiation is to be identified with the radiation carried out by the spherical wavelets which, by their interference with the external field, also produce absorption. This is practically a return to the classical picture of resonance radiation. Statement of the theory consists formally of two parts: the description of the behavior of atoms, and of radiation, when the two interact. The first involves the statement of the probabilities of transition of atoms, and is taken without change from Einstein. The probability of interruption of coherent vibrations is also discussed. The second requires the specification of the fields emitted by atoms in any steady state under the influence of any external field. These consist of spherical wavelets of the frequencies of the various quantum lines which the atom can emit or absorb in the stationary state in which it is. For emission frequencies, wavelets are emitted even in the absence of an external field; for absorption frequencies, external fields induce wavelets, similar to the wavelets of linearly bound electrons in electron theory, which on the one hand interfere with the external field to produce the phenomena of dispersion, etc., and on the other carry out the resonance radiation in all directions. Detailed descriptions of these fields are given. In a discussion of the theory, it is first shown that the assumptions which have been made about the field satisfy the correspondence principle, in that for large quantum numbers the field aproaches that which would be emitted classically on account of the interaction of an external field and a multiply periodic atom. On the other hand, it is shown that the field has a character essentially like that of classical electron theory, which is known to be in general agreement with experiments. Next, the energy and momentum relations at interaction of atoms and light are investigated, and it is shown that, although these quantities are not precisely conserved, still the assumptions as to atomic and radiation processes have been so made that they are conserved on an average over a great number of atomic processes. Finally the spectral resolution of emitted and absorbed radiation is considered; the theory gives a minimum breadth for the lines, depending on the finite length of wave trains resulting from the finite life in stationary states, and it is shown that Kirchhoff's law is obeyed, the emission and absorption lines being of the same breadth. By way of illustration, specific application of the theory is made to the simple examples of emission of light by bombardment of electrons at the resonance potential, resonance radiation, its quenching by presence of foreign gases, absorption, scattering, and disperison.
The first objective of this chapter is to briefly review some of the basic ideas about the structure of matter, in particular the concepts of microscopic physics, in order to recall the knowledge gained in previous physics (and chemistry) courses and make it more precise. Our review will be very concise, and most statements will be made without any proof or detailed discussion. A second objective is to give a brief description of some of the crucial stages in the early development of quantum physics. We shall not follow the strict historical order of this development or present the arguments used at the beginning of the last century by the founding fathers of quantum mechanics; rather, we shall stress the concepts which we shall find useful later on. Our last objective is to give an elementary introduction to some of the basic ideas, like those of a quantum particle or energy level, that will reappear throughout this text. We shall base our review on the Bohr theory, which provides a simple, though far from convincing, explanation of how energy levels are quantized and how the spectrum of the hydrogen atom arises. This chapter should be reread later on, once the basic ideas of quantum mechanics have been made explicit and illustrated by examples. From the practical point of view, it is possible to skip the general considerations of Sections 1.1 and 1.2 at the first reading and begin with Section 1.3, returning to those two sections later on as needed.
Quantum theory of measurements is an extremely important part of quantum mechanics. Currently perturbations by quantum measurements of observable quantities of atomic systems are rarely taken into account in computing algorithms and calculations. In the previous studies of the authors, constructive model of quantum measurements has been developed and implemented in the form of symbolic and numerical calculations for the hydrogen-like atoms. This work describes a generalization of these results to the alkali metal atoms.