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

In this paper, we present a model of the atom that is based on a nonclassical logic called paraconsistent logic (PL), which has the main property of accepting the contradiction in logical interpretations without the conclusions being annulled. The proposed model is constructed with an extension of PL called paraconsistent annotated logic with annotation of two values (PAL2v), which is associated with an interlaced bilattice of four vertices. We use the logarithmic function of the Shannon entropy H ( s ) to construct the paraconsistent equations and thus adopt a probabilistic model for representations in quantum physics. Through analyses of the interlaced bilattice, comparative values are obtained for some of the phenomena and effects of quantum mechanics, such as superposition of states, wave functions, and equations that determine the energy levels of the atomic shells of an atom. At the end of this article, we use the hydrogen atom as a basis for the representation of the PAL2v model, where the values of the energy levels in six orbital shells are obtained. As an example, we present a possible method of applying the PAL2v model to the use of Raman spectroscopy signals in the detection of lubricating mineral oil quality.


Introduction
The model of the atom was presented by Niels Bohr in 1913, where he proposed that electrons are particles with two kinds of motions in atoms. In the Bohr model, the electrons either move continuously around the nucleus in certain stationary orbits or discontinuously jump between these orbits [23] [29]. Subsequently, with the advances in quantum theory, new concepts, such as the ideas of superposition of states and quantum entanglement, have been proposed. Currently, the physical state of an electron is described by a wave function and in the foundations of quantum mechanics; the wave function is a description of the random discontinuous motion of particles. Moreover, the data on the physical properties of particles are uncertain, and all of the analyses are probabilistic [4] [18][35]. The square of the modulus of the wave function represents not only the probability of a particle being found at a certain location but also the probability of the particle being there [22][35] [28] [30] [31].
In 1926, Schrödinger proposed a partial differential equation for the wave functions of particles, such as electrons. The state of a system at a given time is described by a complex wave function, which is also referred to as the state vector in a complex vector space, and this abstract mathematical object enables the calculation of the probabilities of the outcomes of concrete experiments [7][36] [37]. In 1927, Heisenberg proposed the uncertainty principle, which shows the formal inequality relating the uncertainty of position  x and the uncertainty of momentum  p , as follows [9]: The electrons may be considered (to a certain probability) to be located somewhere within a given region of space. However, their exact positions are unknown [4] [7][18] [27][35]. The probability density is obtained using the square of the amplitude of the wave function, which usually involves a complex quantity [9]. Thus, its value is derived by multiplication with the conjugate complex, as follows [4] [7] [25][26 [27]: by the symbol  , and the second degree of evidence is unfavorable for the proposition P and is represented by the symbol  . These degrees of evidence are normalized, classified as a set of real numbers, and contained in the closed interval [0,1]. The annotation assigns a logical state to the proposition P. Thus, the information in PAL2v is a paraconsistent logical signal represented by the proposition P with the subscript of the annotation as ( ) ,  : ( , ) P  , where the annotation is composed of a pair of the degrees of favorable evidence ( )  and unfavorable evidence ( )  .  of PAL2v [13][14] [15].
As discussed in [12] [13], and [17], this representation of PAL2v has been recently investigated using an interlaced bilattice also known as the bilattice of Belnap [5][7] [14]. A bilattice is a structure B , , where B is a nonempty set, and elements. In studies of bilattice, the symbols, ⊗k and ⊕k are used to denote the meet and join operations that correspond to  k , respectively, and ⊗t and ⊕t are used to denote the meet and join operations that correspond to  t , respectively [5] [15]. The partial order  k is intended to represent the knowledge or information order, and  t is intended to represent the truth order [8].

Figure 1(b)
shows the interlaced bilattice of Belnap with the ordering  t and  k and the representations of the extreme logical states in their four vertices, t, f, ⊺, and ⊥, which denote truth, falsity, both, and none, respectively [5] [13]. The paraconsistent equations are obtained from mathematical transformations that map the values arranged in a unitary square on the Cartesian plane (USCP) to the associated bilattice of PAL2v [5] [12]. Initially, the degrees of evidence of PAL2v are considered on the USCP (which is also known as lattice κ), from where their values are mapped to lattice FOUR [13]. Given that, in the USCP, the values are allocated to the x-and yaxes, the USCP (lattice κ) is mapped to the associated lattice  of PAL2v by equating values with the degrees of evidence and implementing the following actions: value from the y-axis ( ) ( )

If
x is the value allocated to the x-axis of the USCP and y is the value allocated to the y-axis of the USCP, then = x  and = y  . The previously described actions create T1, T2, and T3 transformations, as described in [13] and [14], which results in the following: We denote the certainty degree (Dc) as X3 and the contradiction degree (Dct) as Y3 [13] [14]: Dc  → Certainty degree as a function of  and  :  The maximum negative value of the degree of certainty is −1 at the vertex of the extreme logical state "false" (f) and the maximum positive value is +1 at the vertex of the extreme logical state "true" (t). For these two conditions, the value of the degree of contradiction will always be 0 ( ) ( ) The maximum negative value of the degree of contradiction is −1 at the vertex of the extreme logical state "paracomplete" (⊥), and the maximum positive value is +1 at the vertex of the extreme logical state "inconsistent" (⊺). For these two conditions, the value of the degree of certainty will always be 0 ( ) [25,26]. Furthermore, PAL2v, when applied to quantum mechanics, is called paraquantum logic (PqL).
In the interlaced PqL bilattice, the values are represented by a universe of complex numbers, where the degree of contradiction lies in the imaginary axis and the degree of certainty lies in the real axis, with the origin at the point equidistant from the vertices of the bilattice; therefore, in this point, the degrees of certainty and contradiction are both equal to 0 [13] [16].
The paraconsistent logical state   , which defines the paraquantum logical state [31], is considered the point of intersection between the degrees of certainty

Dct
 are dependent on the μ and λ values, the distance between the logical state resulting from ( , )     and one of the extreme logical states t, f, ⊺, or ⊥, represented by the vertices of the PqL bilattice, is dependent on the values of μ and λ considered in the physical world. If we know the paraquantum logical state ετ in any region inside the PqL bilattice, then the values of the degrees of evidence can be calculated using the following equations [13] [17]: Non-commutation exists between the degrees of evidence of the PqL and is explained by the logical negation operation denoted by the symbol  .
The change of position of the degrees of evidence in the annotation negates proposition P. Therefore, given proposition P, its logical negation  P is represented by the exchange of the degrees of evidence in the annotation, as follows [18] [20]]: An interlaced bilattice [16] [17] in addition to the negation operation expressed in Eq (9) also enables the application of the complementation and conflation operations.
The logical complementation operation in the PqL, denoted by the symbol  , is an explicit complement to the unit of the degrees of evidence in the annotation. Given proposition P and its complement  P, we can express the complementation operation as follows: The logical conflation operation in the PqL, denoted by the symbol , is explained by the negation operation, followed by the complement to the unit of the degrees of evidence in the annotation [18]. Given proposition P and its conflation P, we can express the conflation operation as follows: For a logical-mathematical study, the interlaced bilattice associated with PqL can be divided into four quadrants [17]: (a) In Quadrant I, the degrees of certainty and contradiction are positive (there is no operator action on the annotation ( ) ,  ); (b) in Quadrant II, the degree of certainty is negative, while the degree of contradiction is positive (this is an action of the logical negation operator  over the annotation With the negation, complementation, and conflation operations only over the values of the degrees of certainty and contradiction obtained in Quadrant I, the results of the degrees of certainty and contradiction are obtained in the three other quadrants of the interlaced PqL bilattice. Therefore, given that we detect the paraquantum logical state in Quadrant I with the corresponding values of ( )  , The paraconsistent model of the atom will be presented and analyzed in this paper. We compare the use of probability in the Shannon entropy function, which will form the degrees of evidence, and the use of Bernoulli distribution to determine the probability value p of the paraconsistent analysis.
Shannon's work emphasizes a fundamental concept, that is, the entropy of the information, which has become well known as the Shannon entropy H(s). The Shannon entropy has complementary interpretations that can be either information quantity (after measurement) or uncertainty (before measurement) in a given probability distribution [33] [34]. To establish the current concept that H(s) is a function of entropy, similar to Boltzmann's H theorem, Shannon defined some statistical concepts through the equation where pi is the probability of a system being in cell i of its phase space, and k corresponds only to a certain unit of measure [33][34] [36]. The equation of entropy in the case of two variables, that is, p and q (where q = 1-p), is written as follows: where p is the probability, q is its complement (1-p), and the constant k depends on the variable used [36] [38].
As can be seen in [37] and [31], to obtain the maximum unit value of H(s) in Eq. (12), k is calculated as where  is the wavelength of the photon (wavenumber=1/wavelength); Z is the atomic number of the atom; 1 n is the principal quantum number of an energy level, for the atomic electron transition; 2 n is the principal quantum number of an energy level for the atomic electron transition, with The probability p is an outcome that generates the degrees of evidence for the analysis of proposition P for affirmation (true) or refutation (false). One form of representation whose results can be applied to the interlaced PqL bilattice is the Bernoulli trial process [18]. For this representation, we derive the random distribution of variable X, such that ( ) = E X p , and the variance of X is written as Var(X). In this case, the variance is a measure of how much the value of X varies from the expectation E(X) and is defined as 2 Var(X) =− pp . The standard deviation of the probability distribution is denoted by the symbol σ and is defined as the square root of the variance Var(X) [16]: A graph of Var(X) as a function of p ∈ [0,1] exhibits a parabola that opens downward [16].
In the first stage of the analysis, we show the trajectory of the logical states in the ground state and its main equations obtained in Quadrant I of the interlaced PqL bilattice. Moreover, the negation, complementation, and conflation operations are applied, and the model of the complete atom in the xy plane is formed in the perception of an observer in the vector base X. In the second stage of the analysis, the modeling equations of the complete atom are presented, and the trajectories of degenerate and nondegenerate quantum states are highlighted. In the third stage of the analysis, the modeling equations of the energy shells are derived from the mapping of the degrees of evidence that differs in terms of the direction of rotation, which is now done clockwise. In this manner, the paraconsistent model of an atom in the xy plane is formed in the perception of an observer in the vector base Y. Finally, the results of an example of the application of the paraconsistent model of an atom are correlated with the energy values extracted from the Rydberg formulas and presented based on the hydrogen atom. The results of the hydrogen atom show the curves obtained from the analysis of signals using only Quadrant I of the interlaced PqL bilattice.
A method of using Raman spectroscopy signals for the detection of lubricating mineral oil quality is presented with this final model.

1.Materials and Methods
In the construction of the paraconsistent model of the atom, the concepts and equations of PqL and the logarithmic function of the Shannon entropy H(s) are used. These fundamentals, equations, and concepts are applied to the in-depth analysis of the interlaced bilattice associated with PqL. In the proposed model, to represent the probabilistic functions according to the fundamentals of PqL, it is necessary to establish state vectors with unitary modules and that define the orbital paths and energy shells of the atom.
We consider an internal state vector with a unitary module (Pψint) and that has its origin located at the vertex of the true logical state (t) of the interlaced PqL bilattice. In this point, This expression is similar to the following function: where  and  are the degrees of evidence.
The external trajectory of the paraquantum logical states in Quadrant I can be completed for the variation of the inclination angle   of another vector, that is, PψCext.
With the same values of  and  , an external complementary vector (PψCextII) with a unitary module is created simultaneously, with its origin at the point The generated paraquantum logical state establishes the orbital trajectory of the state vector PψI2, whose inclination presents a variation of 45° to 90°, that is, an angular variation of In Quadrant I of the interlaced PqL bilattice,  and  are represented by probabilistic functions ( )  , which must present results that have their values varying simultaneously in the corresponding intervals, that is,

Representation of the Degrees of Evidence of PqL as Probabilistic Functions
First, we consider that and it is contained in the closed interval [0,1]. Therefore, p represents a probability value, and the normalized values X and X′ should be 86 adapted to the interlaced PqL bilattice. The two probability values must also be presented to form an annotation.
We also consider that as an initial condition, the two probabilistic sources 1 and 2 are out of phase at the angle Θ, such that in the amplitude variation of the probability value p, the probabilistic function of source 2 generates another function; that is, . These two probabilistic functions must have the following characteristics: a) when X is at its maximum unitary value, that is, () , the difference between X and X′ will be equal to ( ) when X is at half its maximum value, that is, the difference between X and X′ will be null From the reference probability value at source 1, which is considered a degree of favorable evidence under the previously mentioned conditions is derived as follows: From Eq (4), the degree of certainty of the interlaced PqL bilattice, which is now a probabilistic function, can be calculated as follows: In the same manner, the degree of contradiction shown in Eq (5) is also a probabilistic function, which can be calculated as follows: From Eq (6), the paraquantum logical state ψPqL that appears in the interlaced PqL bilattice will be represented by two probabilistic functions, as follows: In the representation of the functions, the paraquantum logical state In this case, the paraquantum logical state ψp that forms an external orbit trajectory will be constructed with two probabilistic functions, as follows: where ( ) p Dc is obtained using Eq (19), and ( ) p Dct is obtained using Eq (20).

Representation of Fundamental PqL-Equations
In this work, we will construct a paraconsistent model of the atom using the Shannon entropy to operate as a probabilistic function representative of the degrees of evidence ( ) ,  in the interlaced PqL bilattice. In this manner, the fundamental PqL-equations as well as degree of evidence equations will be probabilistic functions, that it will be inserted in the energy equations of the paraconsistent model of the atom. To apply the Shannon entropy function to the PqL equations, we will introduce an adjustment dimensionless value represented by the symbol h S , which will be called the Shannon normalization factor.

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The h S value is calculated as follows: Being that the Shannon entropy from Eq (12) where ( ) s PqL H is the Shannon entropy function presented in Eq (23).
According to Eq (18), with the inclusion of the Shannon entropy, the probabilistic function of the degree of unfavorable evidence can be expressed as follows: where ( ) From Eq (19), the probabilistic certainty degree of the ground state (level E1) can be calculated as follows: From Eq (20), the probabilistic contradiction degree of the ground state (level E1) can be calculated as follows: The values of the degrees of certainty and contradiction considered in the set of complex numbers , with their quantized probabilistic functions, represent the energy of the atom. In the proposed paraconsistent model of an atom, the ground state (level E1) is represented by the point of origin of the real and imaginary axes, which will be located at the point equidistant from the vertices of the interlaced PqL bilattice. In this representation, the paraconsistent logical state Pql  that defines the external orbital trajectory in the ground state, representing the complex numbers in Quadrant I, is expressed as follows: The probabilistic functions of the paraquantum logical state and the external orbital trajectory in the ground state of the model of an atom at the end of the state vector PψI1 with a unitary module; that is, where ( ) s PqL H is the probabilistic Shannon entropy function presented in Eq (23).
With these logical-mathematical considerations, some concepts of PqL can be compared with the concepts of quantum mechanics on the basis of the equations obtained in Quadrant I of the interlaced PqL bilattice. Therefore, in quantum mechanics, the quantum state is represented by and the vector norm is represented by The same paraquantum logical state in quantum mechanics will be achieved, with the following relations of equality: The quantum state of the quantum mechanics in the PqL is represented by the following well-known Dirac notation: In general, for n number of states related to En shells of energies: The representation of the degrees of certainty and contradiction and the ground state in level E1 will be unitary (E1 = 1) and is represented by where the potential energy of the ground state is ( ) ( )

PqL Energy Equations for the Observer in the Vector Base X
In Quadrant I of the interlaced PqL bilattice, the Shannon entropy functions simultaneously create the trajectories of the paraquantum logical states at the ends of two state vectors, thus establishing the ground state (level E1) of the quantum state of the particle.
The state vector PψCextI constructed with the complementary action, in relation to the original vector PψextI, has the same characteristics and differs only in terms of the angular variation. For the X observer, as defined in the mapping shown in Fig. 3, the projections of the real values in the x-axis, which represent the potential energy, and the imaginary values in the y-axis, which represent the kinetic energy, vary proportionally, indicating the equilibrium of values against the inherent probabilistic uncertainties of quantum mechanics.
The energies of the ground state are represented by the PqL equations, with the adapted function of the Shannon entropy having only the probability p as its variable. Using the logical operations of negation, complementation, and conflation, as well as the fundamentals of PqL, we will now define the n energy equations that form the n atomic shells of the paraconsistent model of the atom. Initially, through these operations, the ground-state energy equations of the three other quadrants of the interlaced PqL bilattice are obtained.
The negation operator applied to the functions of the paraquantum logical states that mark the orbital trajectory of the particle in the ground state of Quadrant I produces Quadrant II, as follows: In Quadrant III, the complementation operator applied to the functions of the paraquantum logical states that mark the orbital trajectory of the particle in the ground state of Quadrant I produces the following expressions: In Quadrant IV, the conflation operator applied to the functions of the paraquantum logical states that mark the orbital trajectory of the particle in the ground state of Quadrant I produces the following expressions: These PqL logical operations create the paraconsistent model of the atom, where the probabilistic trajectory of the particle in the ground state is a unit-radius circle composed of the Shannon entropy functions introduced in the degrees of certainty and contradiction equations.
These probabilistic trajectories related to the ground state are shown in the graphics of the results section.
In this search, we consider that the shells of the atom that relate to the degenerate states are represented by energy that is related to the fundamentally pure state but is not aligned to the x-axis of the real values.
In the interlaced PqL bilattice, the degenerate states have different values of contradiction degrees, which bring them close to the extreme logical state of inconsistency in Quadrants I and II and the extreme logical state of paracompleteness in Quadrants III and IV. In the same manner, the shells of the atom that relate to the nondegenerate states are represented by the energy that is related to the fundamentally pure state. In the interlaced PqL bilattice, the nondegenerate states are aligned to the x-axis of the real values and thus to the axis of the degrees of certainty.
The nondegenerate states have the same values of contradiction degrees and different values of certainty degrees, which bring them close to the extreme logical state of true (t) in Quadrants I and IV and the extreme logical state of false (f) in Quadrants II and III.
With these considerations, for the second atomic shell, the degree of favorable evidence μ is expressed in Eq. (24), and that of unfavorable evidence is expressed in Eq. (25). We can maintain a constant difference between the two degrees of evidence within a reasonable range of the probability variation p. For this, the degree of unfavorable evidence can be obtained by multiplication with the degree of favorable evidence, such that ( With these values of the degrees of evidence, the degree of certainty for the energy level E2 will have a constant value over a reasonable range of probability variation p. Therefore, Moreover, the degree of contradiction for the energy level E2 can be derived as follows: where ( ) s PqL H is the Shannon entropy function presented in Eq (23).
In the second atomic shell, the degenerate paraquantum logical state will be represented by the function: or by the Dirac notation for the X observer: Given the relation to the pure state of the ground state, in the second atomic shell, the pure or nondegenerate paraquantum logical state will be represented by the complement expressed in Eq (34) and the function expressed in Eq (27), as follows: These procedures can be continued for n atomic shells of the paraconsistent model of the atom. In Appendix A are the equations for the third, fourth, fifth and sixth energy shells used in the paraconsistent model of the atom.
The graphs resulting from the energy shell equations are shown in the results section.

PqL Energy Equations for the Observer in the Vector Base Y
For the interlaced PqL bilattice, the equations that translate this situation can be obtained using the same procedures performed to obtain the equations for calculating the degrees of certainty and contradiction. To derive the equations for the Y observer, we initially consider the same probabilistic function used for the X observer, with its values allocated to the same USCP (lattice κ). To obtain the degrees of certainty and contradiction for the Y observer, we will apply the actions that previously created the transformations that resulted in the degrees of certainty and contradiction, now considering the rotation of 45° to be clockwise.

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These actions are as follows: (a) expansion of 2 from the x-and y-axes ( ) ( ) For the Y observer, we denote the contradiction degree (Dct) as X3 and the certainty degree (Dc) as Y3: Therefore, from Eq (39) -transformation 3( ) p T , we obtain the certainty degree equation (Dc(μ(p),λ(p))), with its values projected on the y-axis of the lattice τ, and the contradiction degree equation (Dct(μ(p),λ(p))), with its values projected on the x-axis of the lattice τ. In Fig. 4(a), the mapping with the sequences of actions of the paraconsistent transformations for the Y observer is shown.
Given that the paraquantum logical state will have its values changed through the modification of the orbital trajectory, for the Y observer, its representation will be at the extremity of the internal state vector of the PqL, originating from the vertex where the inconsistent logical state is located.
For logical negation, the internal state vector will have its origin at the vertex, where the extreme logical state "paracomplete" is located.
Given that the paraquantum logical state ψτ is the point of intersection between the degree of certainty (Dc) and the degree of contradiction (Dct) located in the interlaced PqL bilattice, the representation of the Y observer in the form of a set of complex numbers will be expressed as follows:

A Representation of the Paraconsistent Model of the Atom
The mapping sequences of probabilistic evidence degrees with both the X and Y observers result in equations of superposed paraquantum logical states in Quadrant I of the interlaced PqL bilattice. Using the paraquantum equations, we can present the results of the superposed logical states as two bilattices comprising one superposed plane.
The energy equations for the Y observer are represented by the paraquantum logical states with a set of complex numbers, where the imaginary and real values will change depending on the observer. For the Y observer, the vector base will be orthogonal to the base X. This means that, for the Y observer, the imaginary values of the X observer will be their real values and the actual values of the X observer will be their imaginary values.
In the paraconsistent model, the equations form superposed paraquantum logical states located in the planes of the X and Y observers. Moreover, variations of the probability values are applied to the equations expressing the orbital trajectories of the particles in the two superposed planes as traces of energy in the overlapping shells of the atom.
In the next section, a paraconsistent model of the atom is constructed with PqL equations formalized with the Shannon entropy function. With the range of probability values (p), we will obtain the degrees of evidence and the degrees of certainty and contradiction forming the paraquantum logical states and energy values to show through the graphical results the individual behavior and its representations that simulate quantum phenomena.

Results and Discussion
The graphical results that will be presented are from simulations using the PqL equations. In the simulations with fundamental PqL-equations and for energy shells of the hydrogen atom, the number of sequential steps used was Nmax=100, therefore the probability values (p) are with intervals p in the order of 1/100. The value used as the Shannon normalization factor was 1.057402554 = h S , and the pi constant used was 3.14159265358 =  . Figure 6(a) shows the graphs of the results of the degree of favorable evidence μ(PqL) derived using Eq (24) and the degree of unfavorable evidence λ(PqL) derived using Eq (25).  (26) and (27), respectively, and the unitary module M(ψ)I1 of the quantized probabilistic function (Eq. 29) for a complete variation of probability. Figure 7 shows the simulation results with explications about the utilized equations, and the interlaced PqL bilattice circumscribed in the external orbital circumference.

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In this simulation, we used Eqs (24) to (27) and the logical operations of negation, complementation, and conflation. Figure 8 shows the results simulated with a group of Shannon entropy functions, which are obtained using the equations of the degrees of certainty in six atomic shells of the paraconsistent model of the atom for nondegenerate states.   In this simulation, we used Eqs. (24) to (76) and the adapted equations shown in the "PqL Equations for the Y Observer" section, with the logical operations of negation, complementation, and conflation. The wave function for each shell of the paraconsistent model of the atom is derived by multiplying the paraquantum logical state with the conjugate complex, according to Eq (2). For the ground-state energy, the wave function 2  is calculated using PqL E Dct is presented in Eq (27). For the other atomic shells, the axes of the references and the amplitudes of the degrees of certainty and contradiction derived by the equations will be considered. The graphs obtained by the applications of the equations that are related to the wave functions are shown in Fig. 10(a). Figure 10(b) shows the six wave functions represented in the paraconsistent model of the atom with orbital energy paths of shells E1, E2, and E3 for the X and Y observers.

Results Related to the Heisenberg Uncertainty Principle in the PqL
The reversibility characteristic of the PqL ensures that the degrees of evidence of probability can be obtained through Eqs. (7) and (8).
The certainty degrees at the x-axis With Eqs (48) and (49), the formal inequality of the Heisenberg uncertainty principle expressed in Eq (1) is represented in the PqL as follows: The graphical results of the simulations with the Heisenberg uncertainty principle equations are shown in Fig. 11(a).

Results Related to the Calculation of the Probability Value in the Paraconsistent Model
The PqL is a reversible logic, and in this manner, we can analyze the paraconsistent model of the atom from its inner part, considering the nucleus as the energy generator that spreads its values to its external part. With the equations considered in this manner, we can estimate the probable values of energy, as well as the probable location of the energy around the generating nucleus. The reversibility characteristic of the PqL ensures that the degrees of evidence of probability can be obtained through Eqs (7) and (8), and the result of its normalized value can be compared with Bernoulli's probabilistic function presented in Eq (14).
This comparison shows that the following equality is a good approximation: ( ) 2

Simulation Results with Energy Shell Values for the Hydrogen Atom
In this work, the paraconsistent model of the atom constructed with the probabilistic function of Shannon's entropy will be correlated to the hydrogen atom through the Rydberg formula. In this way, the energy values obtained in the paraconsistent atom model simulation are all correlated with the values defined in Eq (13); therefore, for application in the hydrogen atom, the Rydberg energy is the unit for energy and is calculated by ( ) As previously discussed, the energy values in each atomic shell will be obtained by multiplying each value of the degree of certainty obtained in each atomic shell of the paraconsistent model of the atom by the value of the ground-state energy of the hydrogen atom. For a better visualization of the results in the graphs, we will use in the equations the module of energy value 13.6 = Ryd E eV .

Energy Equations to Stabilize the Graphical Results in the Shells of Hydrogen Atom.
An adjustment factor adj F is considered from the observation of the extreme results From Eq (34), the adjustment factor of is applied to the degree of certainty to determine the energy of the second atomic shell with a constant value over the range defined by the maximum (p(max)) and minimum (p(min)) values of probability. Equation (34) in this adjusted format for a hydrogen atom becomes These procedures for PqL equations can be continued for n atomic shells of the paraconsistent model applied to the hydrogen atom.
The PqL equations for the third, fourth, fifth and sixth energy shells of the paraconsistent model of the atom hydrogen are showed in Appendix B. Figure 12(a) shows the simulation results for energy shells in the degenerate states of the hydrogen atom in the paraconsistent model of the atom by application of the application of equation (34) and the other equations of degrees of certainty that are in Appendix A. Figure 12(b) shows the results of the paraconsistent functions of the energies obtained from the hydrogen atom using multiples of the adjustment factor of

Results of the Paraconsistent Model of the Atom Applied to Raman Spectroscopy
Data from Raman spectroscopy are obtained through vibrational processes of molecules involving laser application and capturing responses in the form of energy pulses. The obtained Raman information is related to the spectral lines that are provided as frequency-dominated Raman shifts or wavenumbers expressed in cm −1 . In this research, we experimentally investigate how the paraconsistent model of the atom responds to variations in a particular wavelength range. This experimental simulation involves applying the normalized values of Raman intensity in the equation of the degree of favorable evidence presented in Eq (23). In this application, the values of probability p are replaced in the Shannon entropy function by the complement of the normalized values of Raman intensity (1-IRaman) throughout the spectrum of the sample.  the degree of evidence, which is probabilistic, is modified and becomes the degree of evidence of Raman intensity, with spectroscopy characteristics that interfere with the energy amplitude values in the hydrogen atom shells. Figure 13(a) shows two Raman spectroscopy signals in the 400 to 1,400 cm −1 range, which were recorded from a lubricating mineral oil sample. The first spectrum (Type 1) is related to the Raman data of normal lubricating mineral oil, and the second spectrum (Type 2) shows the Raman data of non-normal lubricating mineral oil. The Type-2 (non-normal) lubricating mineral oil had its temperature controlled and maintained at 101 approximately 127.5 °C for 8 h and cooled to room temperature to obtain the Raman sample. The two Raman data spectra were applied to the paraconsistent model of the hydrogen atom and analyzed through the representation of the energy levels of the shells presented in this work. Figure 13(b) shows the simulation results obtained for the energy levels of the six shells in the paraconsistent hydrogen atom. In a superficial analysis, it is verified that the spectrum of the normal lubricating mineral oil (Type 1) presents variations in the shells of the atom at several wavelengths.
Some parts of the spectrum of the non-normal lubricating mineral oil (Type 2) exhibit a few energy variations in the shells of the atom considering the investigated spectrum range.

Discussion
The equations presented in this work, as well as the method of obtaining them through interpretations of the interlaced PqL bilattice, follow the fundamentals of PAL2v, where the degree of favorable evidence μ must be accompanied by the degree of unfavorable evidence λ to form the annotation. In the same manner, the degree of certainty Dc must be accompanied by the degree of contradiction Dct to form the paraconsistent logical state.
In Fig. 6(a), the graphs of the results obtained by simulations with Shannon entropy show how the variation of probability p creates the path of logical states within the interlaced PqL bilattice. Notably, the correlation value of 0.5 between the curves of the degrees of evidence is the point that defines the boundaries (p(min) = 0.11 and p(max) = 0.89) between the evolution of the states in a balanced quantum system and the collapse of the wave function with the definition of a false or true final logical state. Figure 6(b) shows the unit value of the modulus of the internal vector Pψint that moves to create a geometric arc. The limits of its slope are defined by the equations of the Shannon entropy, and its movement with the two external vectors created by the complementarity of values indicates the uncertainty in the movement directions of these vectors.
The paraconsistent model of the atom presents the uncertainties that lead to incompleteness in the measurements, which is expected of a quantum system. This was demonstrated in the results of the simulations shown in Figs. 7(a) and 7(b), where the movements of the external vectors are antagonistic.
In Fig. 8, the results show the models separated by the reference of two observers. Figure 9 shows the simulation results for the complete paraconsistent model of the atom, as well as its projection of the state vectors in the imaginary y-axis of the contradiction and in the real x-axis in the case of an effected measurement. It was demonstrated how the actions of negation, complementation, and conflation operators applied to the logical states enabled the expansion of the probability values. From the analysis, the results obtained by the application of the operators to the logical states of Quadrant I show that the probabilistic trajectories of the particles in the atomic shells appear simultaneously in two directions, that is, clockwise and counterclockwise.
For a practical interpretation, the external and internal probabilistic trajectories of the paraconsistent model of the atom mean that the PqL operations (negation, complementation, and conflation) occur simultaneously, acting in four quadrants of the interlaced PqL bilattice. For an observer in X, in the first quadrant, each paraquantum logical state located at the end of the external state vector (Pψext) simultaneously generates a paraquantum logical state located at the end of the complementary external state vector (PψCext). This is also the same for a Y observer; thus, four paraquantum logical states are generated in Quadrant I. Therefore, each single logical state of the paraconsistent model of the atom is, in the reality, composed of eight paraquantum logical states. As each paraquantum logical state has one equation for the degree of certainty and one equation for the complement of the degree of contradiction, then 16 equations are used for this representation.
In the construction of the model, each of these PqL equations receives the probabilistic value p and its complement (1−p) so that through the Shannon entropy functions they can present the paraconsistent results. The variations of the probabilistic values and the frequency in which they are applied in the PqL equations result in the trajectories presented in the paraconsistent model of the atom. The countless PqL-equations of the paraquantum logical states obtained in the atom shells simultaneously generate all the internal trajectories that are presented in the paraconsistent model. In the paraconsistent model of the atom, the internal state vector with a unitary module (Pψint) has its variation only within the interlaced PqL bilattice. In this way, this internal vector, Pψint, supports all formalization of PqL through the paraquantum logical states located at its end. The external state vector (Pψext), in turn, exceeds the limits of the interlaced bilattice and represents energy values through its paraquantum logical states located at its end. In this way, the actions of these two vectors show through the probabilistic trajectories of the particles in the paraconsistent model of the atom a clear interface between the logical universe and the real/quantum world. Figures 10(a) and 10(b) show the simulation results of the wave functions for the ground-state energy shell and the wave functions for the other internal energy shells. These results verify the limits that define the evolution of the states. Figure 11(a) illustrates the validation of the results of the simulation with the Heisenberg uncertainty principle in the PqL, and Fig. 11(b) shows that it is possible to obtain the value of the probability p considering only the values of the degrees of certainty and contradiction.
With these simulations, quantum concepts, such as probability density and wave function, are wellestablished by the equations and probabilistic functions in the PqL. It is highlighted here that in the simulations performed to obtain the wave functions, the Heisenberg uncertainty principle equations, the probability values, and the values of the degrees of certainty and contradiction that make up the paraquantum logical states were used as input. The results demonstrate the application of the reversibility characteristic in the paraconsistent model of the atom.
In Fig. 12, the energies of the orbital shells represented by the pairs of values are well delineated in the representation of the hydrogen atom, with values close to those obtained by the Bohr model. On the basis of the results presented in Fig. 12, the experiments with Raman spectroscopy have been elaborated, which results are presented in Fig. 13. A visual analysis of the results shown in Fig. 13(b) indicates that with PqL equations, the differences in energy variations between a normal lubricating mineral oil (Type 1) and a nonnormal lubricating mineral oil (heated at 127.5 °C for 8 h; Type 2) can be verified. Therefore, this PqL-based technique can be useful for spectroscopic signal analysis and the verification of material properties at atomic levels.
In general, the simulation results define the probabilistic characteristics of the particle, whereby in quantum mechanics, before the measurement, any of the physical properties are always indefinite. In this manner, the configuration model exhibits the geometry of a sphere and can be described using equations that consider angular variables. A representation of the two planes can be made according to Fig. 9, where the orthogonality of the two planes forms an octahedron in which an analysis of the external and internal variables in the equations validates the variation of probability. With the probability representation obtained through the PqL equations in which the Shannon entropy is introduced, we can derive the equations of the degrees of certainty and contradiction by applying the conditions used for a paraconsistent Bloch sphere as an example.

Conclusions
The paraconsistent model of the atom proposed in this work is based on the foundations of the PAL and combines the concepts of the entropy of information theory with quantum mechanics. The proposed paraconsistent model of the atom, which applies the interlaced PqL bilattice, shows a convincing geometric aspect for the atomic particle. Thus, well-adjusted fundamentals for the phenomena of quantum physics have been demonstrated. Despite the impossibility of covering all quantum phenomena in this work, the equations obtained by the analyses provide the characteristics of symmetry, recurrence, and superposition of states. The quantum phenomenon of the superposition of states is explicitly expressed through the equations and the representation in which two observers are active. This procedure, which was conducted through the simultaneous use of equations and the analysis of an observer in the vector of the base X and an observer in the vector of the base Y, is similar to the quantum theory. The model presented in this work is innovative and opens a field of in-depth investigations of different conditions and the effects originating from interpretations of the model under diverse conditions and dimensions. All the equations feature good computability and ensure that all the procedures can be presented in matrix forms. Notably, in the presented paraconsistent model of the atom, no equation results in a defined value. In general, all the equations used in the simulations are probabilistic functions that result in indefinite trajectories of paraquantum logical states. Even the direction and orientation of the orbital trajectories shown in the result graphics are indefinite because they are always opposite to one another. In the simulations, the paraconsistent model of the atom was represented by equations that allowed showing the probabilistic trajectories of the paraquantum logical states that can be related to the probable orbits of the elementary particles. With the simulations carried out applying the PqLequations, the energy functions, represented by the degrees of certainty and contradiction, therefore, values generated in the quantum universe -Interlaced PqL Bilattice -, can express the probabilities to be measured in the real world. These equations have high significance for the beginning of new studies in computational applications of PqL. In its application to Raman spectroscopy data, the paraconsistent model showed versatility and a good representation of the energy variations at the atomic level of hydrogen. This work demonstrates that the use of the Shannon entropy in a paraconsistent model is an excellent method for modeling quantum systems. In the future, new simulation procedures will expand the application possibilities of the paraconsistent model of the atom in other areas, including computation, quantum logic gates, quantum systems for signal recognition, and quantum cryptography.

Supplementary Materials:
The data that support the plots within this paper and other findings of this study are available in additional material and from the corresponding author. In the third atomic shell, the degenerate paraquantum logical state will be represented by the function:

A2. Equations for Fourth atomic Shell of Energy
The degree of favorable evidence is equal to the degree of unfavorable evidence previously derived.  .

A3. Equations Fifth atomic Shell of Energy
The degree of favorable evidence is equal to the degree of unfavorable evidence previously derived. In the fifth atomic shell, the degenerate paraquantum logical state will be represented by the function:

A4. Equations for Sixth Atomic Shell of Energy
In the sixth atomic shell, the degree of favorable evidence is equal to the degree of unfavorable evidence previously derived. Therefore,  In the sixth atomic shell, the degenerate paraquantum logical state will be represented by the function: