The subject of this article is a problem: substantiation of the periodic system of isotopes using quantum mechanical calculations, search for unification of strong and electromagnetic interactions, and general search for fundamental cause of periodicity. This article is a continuation of and a theoretical section of the following article: The periodic change of properties at the level of nuclei and the vertical symmetry of subgroups of isotopes were discovered in the “periodic system of isotopes,” which was validated against ten types of experimental data. A specific algorithm, the idea of multilayer periodicity of the atom, from the electrons to the nucleus, was used to create the periodic system of isotopes. This study proposes a unified system of quantum numbers that may be used to characterise both electron and nucleon shells as a description of multilevel periodicity (binomial probabilistic interpretation). The problem of a particle in a one-dimensional potential well was solved using the binomial interpretation; quantum-mechanical calculations for the probability functions of the orbitals and periods of both electrons and nucleons were performed – characteristic equations were obtained, electronic orbital projections were reproduced, and the binomial interpretation was shown to correspond to the family of spherical harmonics. Calculation and analysis of Schrödinger equation solutions for the binomial interpretation of quantum numbers have been conducted for electron orbitals. For this interpretation, the spatial nature of quantum numbers is shown in the form of degrees of freedom. Expressions are obtained and planar projections of nucleon nucleon orbitals are built using the multilevel periodicity approach. Two alternative spherical coordinate systems were proposed, for which Lame coefficients were calculated and Laplace equations were derived, after a critical analysis of the modern spherical coordinate system was performed, possible errors in the construction of electron orbitals were shown, and, taking into account the drawbacks, two alternative spherical coordinate systems were proposed, taking into account the drawbacks. A spatial model with changing degrees of freedom 0-n is presented as a search for the fundamental cause of multilevel periodicity; its manifestation in nature (crystal forms) is discovered; a number of experiments are proposed; and predictions about the applicability of the multilevel periodicity principle in quark theory are made.

**Author(S) Details**

**A. S. Magula
**Department of Applied Physics, University of Ukraine, Ukraine.

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