Application of the Shannon Entropy in the Construction of a Paraconsistent Model of the Atom

Abstract

In this work, we present a model of the atom that is based on nonclassical logic called paraconsistent logic (PL), which has the main property of accepting the contradiction in logical interpretations without the conclusions being annulled. The model proposed in this work is constructed with the extension of PL called paraconsistent annotated logic with annotation of two values (PAL2v) that is associated with an interlaced bilattice of four vertices. We used the logarithmic function of the Shannon entropy H(s) with the inclusion of the normalized Planck constant ħ to construct the paraconsistent equations. Through the analyses of the interlaced bilattice, comparative values are obtained for some of the phenomena and effects of quantum mechanics, such as superposition of states, quantum entanglement, wave functions, and equations that determine the energy levels of the layers of the atom. At the end of this article, we use the hydrogen atom as the basis of the representation of the PAL2v model, where the values of the energy levels in six orbital layers are obtained. As an example, we present a possible method of applying the PAL2v model to the use of Raman spectroscopy signals in quality detection of lubricating mineral oil.With respect to the logic applied to quantum mechanics among various studies of quantum probabilistic logic formalism, one of the most important was developed by von Neumann in 1932 [11,12]. In his work, von Neumann assumed that each physical system is associated with a Hilbert space H (separable), with its unit vectors corresponding to possible physical states of the system. Each real "observable" random quantity is represented by a self-regulated operator A in H whose spectrum is the set of possible values of A [13][14][15][16].According to these previous works, mathematics in quantum mechanics can be considered a nonclassical probability calculation, which is supported by a nonclassical propositional logic [17,18].Nonclassical logics are created with the purpose of opposing the binary principles of classical logic, thus providing better conditions for the construction of physical-mathematical models with more approximate results. Currently, there are several types of nonclassical logics, and in general, we can consider that only those logics that are indestructible in the presence of the contradiction receive the denomination of paraconsistent logic (PL). Therefore, PL is a nonclassical logic that has, as its fundamental characteristic, the opposition to the principle of noncontradiction [19][20][21].The fundamental theory of PL has been developed in the area of philosophical logic [23], and a formal framework for inconsistent theories was proposed by da Costa [20]. Further details of the logical formalization of PL, the mathematical implications, and their theorems can be found in [10], [22], and [23]. Blair and Subrahmanian [24] presented applications of PL to logical programming and extended the formalization of three-valued semantics. With this initial work, a theory was developed for ...