Eigensolution and Thermodynamic Properties of Standard Coulombic Potential

The Combination of Coulomb potential with itself(standard Coulombic potential) is studied under the non-relativistic wave equation. The energy equation and its corresponding un-normalized redial wave are obtained using parametric Nikiforov-Uvarov method by applying a Green-Aldrich approximation scheme to the centrifugal term. The energy equation obtained was used to calculated the partition function from where the thermodynamic properties such as the mean energy, specific heat capacity, entropy and free energy are calculated. Numerical results are generated for the standard Coulombic potential and its special cases. The special cases are Coulomb potential with negative potential strength and the other is also Coulomb potential with positive potential strength. The study showed that the energy of the system is fully bounded. It is noted that the two special cases which are Coulomb-Coulomb potentials with positive and negative potential strengths are equal provided the strength are equal but opposite in sign. The thermodynamic properties aligned with those of the literature but has some unique behaviours.


Introduction
In the past decades, the solutions of the nonrelativistic wave equation for different potential systems were obtained using numerical techniques.Studies have shown that the numerical techniques are cumbersome, complicated and time-consuming methods.However, the approximate methods are simple and less time consuming.These methods admit wide range of potential functions.The different methods have their unique approach to solving problems but produce equivalent results.For instance, the solution of the generalized Morse potential function was obtained by Zhang et al. [1] using supersymmetry quantum mechanics in one of their articles.The same generalized Morse potential was studied by Onate et al. [2] using parametric Nikiforov-Uvarov method.The results of these two studies were compared by Onate et al. and the two results were found to be equivalent.Similarly, Falaye [3] studied hyperbolic potential in the frame work of asymptotic iteration method and compared his result with that of the Ikhdair et al. [4] that used the Nikiforov-Uvarov method.The two results were found to be equivalent.In ref. [5] and ref. [6], the solutions of the shifted Deng-Fan potential were obtained with different approximate methods and the results are found to be equivalent.These showed that the method to be used to obtained the solutions for any potential function do not affect the results but depends on the choice and complexity by the authors.Various potential models such as Manning-Rosen potential, Eckart potential, Rosen-Morse potential, Pὂschl-Teller type potential models, Hulthẻn potential, Hellmann potential, Yukawa potential to mention but a few, have received adequate attentions in the bound state via approximate methods.In the recent time, the study of bound state has been extended to the thermodynamic properties where the energy obtained in the bound state is used to calculate the partition function which in turn used to calculate the thermodynamic properties such as mean energy, free energy, entropy and heat capacity.This is due to the applications of thermal properties in sciences [7][8][9].Different potential models have been reported extensively.For instance, Dong and Cruz-Irisson [10] calculated the thermodynamic properties of the modified Rosen-Morse potential and examine both the temperature parameter and the maximum quantum state respectively as a function of the various thermodynamic properties.Ikot et al. [11] also studied the thermodynamic properties of the shifted Tietz-Wei potential where they reported that some thermodynamic properties decrease as the temperature parameter decreased while others increased as the temperature parameter increases.In the work of Khordad and Sedehi for double ring-shape potential [12], it was shown that the different thermodynamic properties have different variation with the temperature.Inyang et al. [13] calculated the various thermodynamic properties of Eckart-Hellman potential in one of the recent studies.Their result shows that the thermodynamic properties as a function of the maximum quantum state exhibit different features as the temperature parameter increases.Khordad [14] in one of his study examined the thermodynamical properties of triangular quantum wires, entropy, specific heat, and internal energy.Inyang et al. [15], studied masses and thermodynamic properties of a quarkonium system.Njoku et al. [16] in their study obtained approximate solutions of Schrodinger equation and thermodynamic properties with Hua potential.In ref. [17], the thermodynamic properties and mass spectra of a quarkonium system with ultra-generalized exponential-hyperbolic potential.Demirci and Sever [18] calculated the nonrelativistic thermal properties of Eckart plus class of Yukawa potential.Ramantswana et al. [19] in their own study, determined the thermodynamic properties of CrH, NiC and CuLi diatomic molecules with the linear combination of Hulthen-type potential plus Yukawa potential.The thermodynamic properties of some diatomic molecules confined by an-harmonic oscillating system was also studied by Oluwadare et al. [20].Wang et al. [21] in a special form, predicted the ideal-gas thermodynamic properties for water and deduced the average relative deviations of the predicted reduced molar Gibbs free energy and molar entropy of water from NIST data.Motivated by the interest in thermodynamic properties, the present study wants to examine the nonrelativistic solutions and thermodynamic properties of the standard Coulombic potential.The standard Coulombic potential in this work is a combination of Coulomb-Coulomb potentials with different potential strengths.The standard Coulombic potential in this case is given as The parameters 0 V and 1 V are potential strengths that determines the depth of the potential.In this situation, the two special cases of the standard Coulombic potential are the Coulomb potentials in which one has positive potential strength and the other Coulomb potential has negative potential strength.The excess of taking opposite sign for the potential strength is to examine which of the potential strength will have a higher energy eigenvalue.It is interested to know that the result for the positive Coulomb potential as a special case can reproduce the result of the Yukawa potential as a special case of the Hellmann potential.This study will also examine the effect of the negativity and positivity of the potential strength on the energy eigenvalues.The present study will adopt the methodology of parametric Nikiforov-Uvarov because, it is less cumbersome and gives an accurate result.


are polynomials of at most second degree and () s  is a first-degree polynomial.Defining ( ) where  and k are constants.Since the square root of the polynomial  in equation (3) must be square, then this defines the constant .
k Replacing k in equation (3), we define Since the derivative of  should be negative [23,24], ( ) 0 s   and ( ) 0. s   Then a choice of solution can be made.If  in equation ( 4) is redefined as The hypergeometric equation has a particular solution with degree .n The solution of equation ( 2) can be obtained with the product of two independent parts [25] ( ) ( ) ( ), s s y s   = (7) where ()  ys can be written as  should satisfy the condition [26]   The other factor is defined as The general form of the Schrödinger equation ( 2) for any physical potential model is transformed to the form [27] ( ) ( ) In other to use the parametric form of Nikiforov-Uvarov method, Tezcan and Sever [28] deduced the following by comparing equation (11) with equation ( 2), to obtain the following Now, substituting equation ( 12), into equation ( 3), we have The parameters in equation ( 13) are deduced as follow: Based on the rule of Nikiforov-Uvarov method, the function under the square root of equation (13), must be the square of the polynomial.This implies that ( ) where 9    is defined as follows: ( ) For the same , k we obtain the value for  using equations ( 5), ( 12) and ( 13) ) are obtained.Using equations ( 4), (17) and (19) we obtain the following condition to obtain the energy eigenvalue equation [29][30][31][32][33].

Thermodynamic Properties
In other to calculate the thermodynamic properties of standard Coulombic potential, the energy equation is written in a compact form as ) ( 1) After simplifying the energy equation, it becomes easier to determine the partition function from where the thermodynamic properties can be calculated.The partition function for any potential system in the bound state is defined as [38][39][40][41][42] , 0 ( ) , where  is the maximum quantum state and k  is a Botzmann constant and T is the temperature.To obtained a desired result, the sum is replaced with an integral and thus, equation (37) turns to the form where 2 .

T b c
 =   (40) Having obtained the partition function, the thermodynamic properties can be calculated as follows [43][44] (I): The Mean Energy (III): The Entropy (IV): The Free Energy

Special cases of the standard Coulombic potential
There are two special cases of the standard Coulombic potential.
Case I: The negative Coulomb potential.This is obtained when 1 0, V = and the potential (1) becomes Case II: The positive Coulomb potential.This is obtained when 0 0, V = and the potential (1) becomes

Discussion
Using equation (33) and the potential parameters given in Table 1, the spectra for the standard Coulombic potential are presented in Table 1 for various states.It is shown that the spectra for the − wave are higher than their counter pact for the s − wave.This shows that the inclusion of the approximation scheme has increased the energy of the system.It is also shown that the energy varies directly with the screening parameter for all the states.In Table 2, the spectra for the special cases of the standard Coulombic potential are presented.It is shown that the results with 0 0.3 V =− and 1 0, V = are equal to the results with 0 0 V = and 1 0.  2 also showed that the negative value of 0 V produces higher energy than its positive value.These results are reproduced only when 01 .VV = The comparison of the energy eigenvalue for the negative and positive Coulomb potential as special cases of the standard Coulombic potential and that of Coulomb potential with the Yukawa potential as special cases of the Hellmann potential are presented in Table 3.The present result and the result in re.[43] are perfectly in agreement for the two special cases.This shows that the results of the special cases of standard Coulombic potential are equal to the results of the special cases of Hellmann potential.It can be inferred that the numerical results of the standard Coulombic potential and that of the Hellmann potential are equivalent.The variation of the potential strengths 0 V and 1 V respectively against the energy are shown in Figure 1 and Figure 2. In each case, the energy of the system decreases as each of the potential strength increases respectively.The results in the two Tables together with Figures 1 and 2 showed that the energy of the standard Coulombic potential is fully bounded.The effects of the temperature parameter and maximum quantum state respectively on the partition function are shown in Figure 3.The partition function and the temperature parameter vary directly with one another from the origin.Thus, an increase in the partition function requires a decrease in temperature of the system.The same direct variation is observed between the partition function and the maximum quantum state for λ greater than or equal to two.However, for λ less than two, there is inverse variation.Figure 4 presented effects of the temperature parameter and maximum quantum state respectively on the mean energy.The mean energy and the temperature parameter vary indirectly with one another.
As the temperature parameter increases, the mean energy reduces for all maximum quantum states.This shows that an increase in mean energy requires temperature increase.This variation is seen to be contrary to the variation between the mean energy and the maximum quantum state for various temperature parameter.An increase in the maximum quantum state leads to increase in the mean energy.In Figure 5, the effects of the temperature parameter and maximum quantum state respectively on the heat capacity are shown.The heat capacity varies inversely with the temperature parameter while the maximum quantum state varies directly with the heat capacity.This shows that an increase in the heat capacity requires an increase in temperature.The variation of the heat capacity against the temperature parameter and the maximum quantum state is the same as the variation of mean energy against the temperature parameter and maximum quantum state as shown in Figure 4. Figure 6 is the effects of the temperature parameter and maximum quantum state respectively on the entropy.The entropy and the temperature parameter vary directly with one another from the origin.This physically means that the entropy of the system does not require an increase in temperature for its own increase.The same direct variation is observed between the entropy and the maximum quantum state for λ greater than or equal to two.However, for λ less than two, there is inverse variation.These variations are similar to the variation of partition function with temperature parameter and maximum quantum state shown in Figure 3.In Figure 7, the effects of the temperature parameter and maximum quantum state respectively on the free energy are presented.In this system, the free energy remains constant for all values of the temperature parameter.This indicates that there is no effect of temperature on free energy for this system.However, as the maximum quantum state increases, the free energy reduces for all values of the temperature parameter.In Figure 8, the partition function rises as the potential strength 1 V increases from 0.25 for different values of the other potential strength 0 .
V It is noted that the partition function for the lower value of 0 V has the highest value.In Figure 9, the mean energy reduces as the potential strength 1 V increases from 0.25 for different values of the other potential strength 0 .
V .The mean energy with the lower value of 0 V has the lowest value while the mean energy with the highest value of 0 V has the highest value.The heat capacity reduces as the potential strength 1 V increases for different values of the other potential strength 0 .
V The heat energy with the lower value of 0 V has the lowest value while the mean energy with the highest value of 0 V has the highest value as shown in Figure 10.The entropy rises as the potential strength 1 V increases from 0.25 for different values of the other potential strength 0 V as shown in Figure 11.It is noted that the partition function for the lower value of 0 V has the highest value.In Figure 12, the free energy reduces as the potential strength 1 V increases for different values of the other potential strength 0 .
V The free energy with the lower value of 0 V has the lowest value while the mean energy with the highest value of 0 V has the highest value.The potential strength 1 V has the same variation with partition function and entropy.Similarly, the effect of the potential strength 1 V on mean energy and free energy are the same.However, the variation of the potential strength 1 V on heat capacity is similar to that of the mean energy and free energy.Exception of the heat capacity, the variation of the potential strength 1 V with all the thermal properties has a turning point.

Conclusion
The recent work studied the nonrelativistic solutions of the standard Coulombic potential in the contest of bound states and thermodynamic properties.The energy of the standard Coulombic potential is fully bounded.The results of the two Coulomb subset potentials are not equal if the values of the potential strengths are equal even though the two potential strengths have the same effect on the energy of the system.However, with opposite numerical value, the two subset potentials reproduced the same numerical values.The thermodynamic properties showed different variation with the temperature parameter and maximum quantum state respectively.
Data availability: All the data used in this work are in the manuscript Funding: NA equation(1) and equation(27) into equation(26) and by defining a variable of the form ,

Figure 1 :Figure 2 :Figure 3 :
Figure 1: Energy of the standard Coulombic potential against the potential strength 0 V with

Figure 4 :
Figure 4: Effects of the temperature parameter and maximum quantum state respectively on the mean energy.

Figure 5 :
Figure 5: Effects of the temperature parameter and maximum quantum state respectively on the heat capacity.

Figure 6 :
Figure 6: Effects of the temperature parameter and maximum quantum state respectively on the entropy.

Figure 7 :
Figure 7: Effects of the temperature parameter and maximum quantum state respectively on the free energy.

Figure 8 :
Figure 8: Variation of partition function against the potential strength 1V .

3 V
= and 1 0, V = the results are different.It shows that the results of the special cases are The results in Table

Table 2 :
Energy eigenvalues of the special cases of the standard Coulombic potential with 1 ==for various quantum state, angular momentum and three values of the screening parameter.State 

Table 3 :
Comparison of energy eigenvalues of the Coulomb potential with 2