Nardelli, Michele (2009) On the mathematical connections between some equations concerning the calculation of all the eigenfunctions of atoms with the Thomas-Fermi method, some sectors of Number Theory, the modes corresponding to the physical vibrations of superstrings, p-Adic and Adelic free relativistic particle and p-Adic strings. Dip.Sc.Terra-Dip.Matem.Unina. (Unpublished)
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Abstract
According to quantum mechanics, the properties of an atom can be calculated easily if we known the eigenfunctions and eigenvalues of quantum states in which the atom can be found. The eigenfunctions depend, in general, by the coordinates of all the electrons. However, a diagram effective and enough in many cases, we can get considering the individual eigenfunctions for individual electrons, imagining that each of them is isolated in an appropriate potential field that represent the action of the nucleus and of other electrons. From these individual eigenfunctions we can to obtain the eigenfunction of the quantum state of the atom, forming the antisymmetrical products of eigenfunctions of the individual quantum states involved in the configuration considered. The problem, with this diagram, is the calculation of the eigenfunctions and eigenvalues of individual electrons of each atomic species. To solve this problem we must find solutions to the Schroedinger’s equation where explicitly there is the potential acting on the electron in question, due to the action of the nucleus and of all the other electrons of the atom. To research of potential it is possible proceed with varying degrees of approximation: a first degree is obtained by the statistical method of Thomas-Fermi in which electrons are considered as a degenerate gas in balance as a result of nuclear attraction. This method has the advantage of a great simplicity as that, through a single function numerically calculated once and for all, it is possible to represent the behaviour of all atoms. In this work (Sections 1 and 2) we give the preference to the statistical method, because in any case it provides the basis for more approximate numerical calculations. Furthermore, we describe the mathematical connections that we have obtained between certain solutions concerning the calculation of any eigenfunctions of atoms with this method, the Aurea ratio, the Fibonacci’s numbers, the Ramanujan modular equations, the modes corresponding to the physical vibrations of strings, the p-adic and Adelic free relativistic particle and p-adic and adelic strings (Sections 3 and 4).
| Item Type: | Article |
|---|---|
| Subjects: | 500 Scienze naturali e Matematica > 510 Matematica |
| Depositing User: | Michele Nardelli |
| Date Deposited: | 23 Jan 2009 |
| Last Modified: | 20 May 2010 12:01 |
| URI: | http://eprints.bice.rm.cnr.it/id/eprint/731 |
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