Shannon entropy for Feinberg–Horodecki equation and thermal properties of improved Wei potential model

1 Introduction

The wave equations studied in quantum mechanics are basically categorized into relativistic and non-relativistic wave equations. The solutions of the Schrödinger equation occupy the main position in the non-relativistic quantum mechanics, while the Dirac equation and Klein−Gordon equation occupy the main positions in the relativistic quantum mechanics. The Schrödinger equation describes non-relativistic spinless particles, the Klein−Gordon equation describes the relativistic spin-zero particles, the Duffin–Kemmer–Petiau (DKP) equation describes the relativistic spin-zero and spin-one particles, and the Dirac equation focuses on the relativistic spin-half particles. These equations were solved by adopting different methodologies such as asymptotic iteration method [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18], Nikiforov−Uvarov (NU) method [19,20,21,22,23,24,25,26,27,28,29,30], supersymmetry quantum mechanics [31,32,33,34,35,36,37,38,39,40,41,42,43,44], factorization method [45], exact and proper quantization rule [46,47,48,49], shifted 1/N expansion method [50], and others. Over the years, many authors have dedicated their time to study the coordinate-like counterpart (time-independent part) of the Schrӧdinger wave equation with different physical potential models such as the Kratzer potential, Morse potential, Rosen−Morse potential, Manning−Rosen potential, Deng−Fan potential, and Pӧschl−Teller potential among others [51,52,53,54,55,56]. However, the time-dependent counterpart has not been given much attention. This could be probably because it was assumed that the time-dependent counterpart could not be applied to wider studies, unlike the time-independent counterpart. However, the report on Feinberg–Horodecki for some vector potentials was given in refs. [57,58]. In this study, we aim to obtain a solution to the time-dependent counterpart of the Schrӧdinger wave equation and apply it to study Shannon entropy, which has not been studied yet to the best of our knowledge. Entropy, generally, is the measure of a system’s thermal energy per unit temperature that is unavailable for doing useful work. The amount of entropy is a measure of molecular disorder or randomness since the work obtained is ordered from molecular motion. Shannon entropy has been given attention in statistical mechanics and mathematics. In 1948, Shannon [59] used Shannon entropy to study fundamental limits on signal processing operations. Entropic systems in the past few years have been deeply studied in statistical mechanics. The entropic systems measure the spread of the electron density. Shannon entropy precisely is the probabilistic measure of uncertainty that is used to determine the stability of a system. The computations of these entropies are based on the probability density function that is obtained from the wave function. In the present work, the Feinberg–Horodecki equation will be studied with a new expression for molecular Wei potential energy function. In 1990, Wei H. [60] proposed a four-parameter diatomic molecular potential function of the form

where D e is the dissociation parameter, t e is the equilibrium time point, b is the screening parameter that characterizes the range of the potential, and h is the potential parameter. This potential function has been studied very well and has been used to investigate the rotational vibrating levels of diatomic molecules [61,62,63,64,65,66]. The Wei potential given in equation (1) has been modified by Jia et al. [67] in one of their works as

The potentials given in equations (1) and (2) are time-dependent vector potential. These potentials are obtained by replacing r with t in the time-independent potential. The time-dependent vector potential has been used in the probability distribution for the scintillations caused by an electron hitting a screen after diffraction. The vector potential function given in equation (2) is a good molecular potential as it possesses molecular spectroscopic parameters. This potential has not received much attention due to its complexity when it comes to applications. This work is organized as follows: Section 2 is a brief review of the parametric Nikiforov–Uvarov method. Section 3 deals with the solutions of the Feinberg–Horodecki equation with the new expression for molecular Wei energy potential function while in Section 4, the computation of the entropic system is presented. In Section 5, we present the thermodynamic properties. In Section 6, we discuss the results of this work and give the concluding remarks in Section 7.

2 Parametric Nikiforov–Uvarov method

The parametric Nikiforov–Uvarov method derived by Tezcan and Sever [26] is the simplification of the original Nikiforov–Uvarov method. Without any derivation in this work, the general equation of the form [26] is given by

The conditions for the energy equation and the corresponding wave function of the above equation are

The parameters in equations (4) and (5) are obtained as

3 Feinberg–Horodecki equation and the Wei energy potential function

In this section, we calculate the quantized momentum and the wave function. The Feinberg–Horodecki equation with the time-dependent vector potential V ( t ) function is given by

where c is the speed of light, m is the mass, ℓ is the angular momentum quantum number, ψ n ( t ) is the wave function, and p n is the quantized momentum.

To approximate the term ℓ ( ℓ + 1 ) t 2 , in equation (7), we use the formula [66]

where

Substituting equations (2) and (8) into equation (7), we have

where

and e b r e is replaced by λ . To obtain the solutions of equation (12) with the parametric Nikiforov–Uvarov method, we define a variable of the form y = λ h e − b t . On using this transformation into equation (12), we have another differential equation of the form

where

Comparing equation (15) with equation (3), we have the parametric constant values in equation (6) as follows

Substituting the values of the parametric constants into equation (19) into equations (4) and (5) respectively, we have the quantized momentum and its corresponding wave function as

and N n is the normalization constant. Using the normalization condition, we can easily determine the normalization constant. Thus

Using another transformation s = 1 − 2 y in equation (26), equation (26) then becomes

where

Using the integral given in the appendix, we have the normalization constant as

4 Shannon entropy

In the concept of this work, Shannon entropy will be studied for the time point and momentum space using the probability density function ρ ( t ) = ∣ ψ n ( t ) ∣ 2 . The Shannon entropies for the time point and momentum space, respectively, are given by refs. [68,69,70,71,72,73,74,75,76,77]

To calculate the Shannon entropy for the time point, we define a new variable of the form s = 1 − y , which now changes equation (31) to the form

where we have defined

Equation (33) is the final simplified form by using equation (30), the formula, and the integral in the appendix.

To obtain the Shannon entropy for the momentum space, we further define x = 1 − 2 y , and use this in equation (32) to obtain

In equation (36), we have also defined 1 − x = 1 − 1 − x 2 . The Shannon entropy for the momentum space is simplified using equations (30) and (36), a relation, and integral given in the appendix.

5 Thermodynamic properties

In this section, we studied the improved Wei molecular energy potential function in the domain of thermodynamic properties. The energy equation used for the computations is obtained from the quantized momentum obtained in equation (20). Some simple transformations c → 1 and p n → E n used in equation (20) result in

To solve the thermodynamic properties, we consider the pure vibrational states of the molecules by deriving the eigenstates as

where

In equation (38), we have introduced ϖ = − τ + Λ 1 ± Λ 1 − Λ 2 for mathematical simplicity so that [ ϖ ] is the largest integer inferior to ϖ .

The vibrational partition function for the molecule is obtained as

where k is the Boltzmann constant. On substituting equation (38) into equation (40), we obtain

In the classical limit, the summation in equation (41) can be replaced by an integral

so that

where

Using equation (43), other thermodynamic systems for the diatomic molecules will be computed

I. Vibrational mean energy U :

where

II . Vibrational specific heat capacity C :

where

III. Vibrational mean free energy F :

And, finally,

IV. Vibrational entropy S :

6 Discussion of results

Table 1 presents the quantized momentum for various n and ℓ with four values of h . For h = − 1 , the quantized momentum varies inversely with both n and ℓ . For h = 0 , the quantized momentum varies directly with both n and ℓ . The spectroscopic parameters for four molecules studied here are given in Table 2. In Table 3, we presented the quantized momentum for three values of b with different states of CO , NO, CH , and ScH molecules. The effect observed in Table 1 was also observed in Table 3. The Shannon entropy for the time point, momentum space, and their sum for various values of the dissociation energy are presented in Table 4. The two entropies vary inversely with each other. The results in Table 4 confirmed and satisfied Bialynicki–Birula and Mycielski inequalities. The minimum bound for the sum of the entropies is 6.663738148, which is greater than 1 + log π (1.497206180) for a one-dimensional system. This is the entropic uncertainty relation, i.e., S(ρ) + S(γ) ≥ D(1 + log π), which is the major test of accuracy for a calculated Shannon entropy of any physical potential model. The results in Table 4 revealed that a diffused density distribution in the momentum space is associated with a localized density distribution at the time point. In Tables 5 and 6, we numerically presented the Shannon entropies for four molecules. In Table 5, where computation is done by varying the value of b , the entropies vary inversely with each other that obeys Heisenberg’s uncertainty relation. However, in Table 6 where the computation is done by varying the value of h , the entropies vary in the same direction.

We have computed the thermodynamic properties of the improved Wei molecular energy potential function for four molecules, i.e., CO, NO, CH, and ScH. The computations were done using the spectroscopic parameters in Table 2. The variation of each of the thermodynamic properties against the temperature (β) is examined. In Figure 1(a–d), we plotted the partition function Z(β) against β for CO, NO, CH, and ScH, respectively. The partition function increases as the temperature increases. The increase in temperature is highly noticeable after some temperature increase from the origin (−K). In the case of NO (Figure 1b), the partition function remains almost constant from –K to −0.3K. In Figure 2(a–d), we have plotted the vibrational mean energy U against β for the same molecules. It is noted that the mean energy decreases gradually from –K to −2K before a sharp decrease occurred. In each case, the vibrational mean energy becomes zero at 273°C. At 273°C, the mean energy for various quantum states is equal. In Figure 3(a–d), we have plotted vibrational specific heat C against β. As it can be seen, the vibrational specific heat increases monotonically as the temperature changes positively for all the molecules. At a temperature of about −2.85 K, the specific heat at different quantum states converged for the four molecules studied. The variation of vibrational mean free energy F against β is shown in Figure 4(a–d). As we can see from the figures, the vibrational mean free energy increases gradually from –K to −0.5K. A more gradual increase is seen between 0.5 to 0.25K before a very sharp increase that occurred for the four molecules studied. In Figure 5(a–d), we examined the behaviour of vibrational entropy against β. It is clearly observed that the variation of entropy with β has similar behaviour for CO and NO molecules (Figure 5a and b) where entropy increases as β increases. However, for CH and ScH molecules (Figure 5c and d), the behaviour of the entropy is similar but different from those of CO and NO. In CH and ScH, the vibrational entropy decreases as the temperature increases. The differences in the variation of entropy with β for these molecules probably result due to a significant difference in the mass of the molecules. CO and NO with higher mass values have the same behaviour while CH and ScH with lower mass values have the same behaviour.

Figure 1

Partition function against β for CO, NO, CH, and ScH at different vibrational quantum states for various values of largest integer inferior to ϖ: (a) CO, (b) NO, (c) CH, and (d) ScH.

Figure 2

The vibrational mean energy (U) against β for CO, NO, CH, and ScH at different vibrational quantum states for various values of largest integer inferior to ϖ: (a) CO, (b) NO, (c) CH, and (d) ScH.

Figure 3

The vibrational specific heat (C) against β for CO, NO, CH, and ScH at different vibrational quantum states for various values of largest integer inferior to ϖ: (a) CO, (b) NO, (c) CH, and (d) ScH.

Figure 4

The vibrational mean free energy (F) against β for CO, NO, CH, and ScH at different vibrational quantum states for various values of largest integer inferior to ϖ: (a) CO, (b) NO, (c) CH, and (d) ScH.

Figure 5

The vibrational entropy (S) against β for CO, NO, CH, and ScH at different vibrational quantum states for various values of largest integer inferior to ϖ: (a) CO, (b) NO, (c) CH, and (d) ScH.

7 Conclusions

In this study, we have solved the Feinberg–Horodecki equation for a given molecular energy potential function. The result of the Feinberg–Horodecki equation was used to study Shannon entropy as a theoretic quantity and the results were found to obey the Bialynicki–Birula and Mycielski inequalities, which is in agreement with the study under the time-independent Schrödinger equation. The thermodynamic properties of the molecular energy potential function were calculated for CO, NO, CH, and ScH molecules using their molecular spectroscopic parameters. Each of the thermodynamic properties studied in this work has similar features for the four molecules except for the entropy. To the best of our knowledge, this appears to be the first time, Shannon entropy is studied under the time-dependent Feinberg–Horodecki equation.