Bound state solutions and thermal properties of the N-dimensional Schrödinger equation with Varshni plus Woods-Saxon potential via Nikiforov-Uvarov method
Subject Areas : Journal of Theoretical and Applied Physics
Ridha Horchani
1
(Department of Physics, College of Science, Sultan Qaboos University, Al-Khod, Muscat, Sultanate of Oman.)
Safa Al-Shafii
2
(Department of Physics, College of Science, Sultan Qaboos University, Al-Khod, Muscat, Sultanate of Oman.)
Noora Al-Hashimi
3
(Department of Physics, College of Science, Sultan Qaboos University, Al-Khod, Muscat, Sultanate of Oman.)
Akpan Ikot
4
(Theoretical physics Group, Department of Physics, University of Port Harcourt, Nigeria.)
Ituen Okon
5
(Department of Physics, University of Uyo, Uyo. Nigeria.)
Uduakobong Okorie
6
(Department of Physics, Akwa Ibom State University, Uyo. Nigeria.)
Carlos Duque
7
(Grupo de Materia Condensada‑UdeA, Instituto de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Antioquia UdeA, Medellín, Colombia.)
Enoch Oladimeji
8
(Department of Physics, Federal university Lokoja (FULOKOJA), Lokoja, Nigeria.)
Keywords:
Abstract :
Bound state solutions and thermal properties of the N-dimensional Schrödinger equation with Varshni plus Woods-Saxon potential via Nikiforov-Uvarov method
Ridha horchani1, Safa Al shafii1, noora al hashimi1, Akpan N. IKOT3, ITUEN. B. Okon4, UDUAKOBONG. S. OKORIE5*, CARLOS. A. DUQUE6 and ENOCK O. Oladimeji7, 8.
1Department of Physics, College of Science, Sultan Qaboos University, P.O. Box 36, P. C. 123, Al-Khod,
Muscat, Sultanate of Oman.
2Institut Polytechnique des Sciences Avancées Ecole d'Ingénieurs de l'Air et de l'Espace, Paris, France
3Department of Physics, Theoretical Physics Group, University of Port Harcourt, Choba, Nigeria
4 Department of Physics,University of Uyo, Nigeria
5Department of Physics, Akwa Ibom State University, Ikot Akpaden, Nigeria
6Grupo de Materia Condensada-UdeA, Instituto de Física, Facultad de Ciencias Exactas y Naturales,
Universidad de Antioquia UdeA, Calle 70 No. 52-21, Medellín, Colombia
7Institute of Physical Research and Technology, Peoples’ Friendship University of Russia, Moscow
(RUDN), Russia
8Department of Physics, Federal university Lokoja (FULOKOJA), Lokoja, Nigeria.
Abstract: We have solved the Schrödinger equation for Varshni plus Woods-Saxon potential in N-dimensions within the framework of Nikiforov-Uvarov method by using Greene-Aldrich approximation scheme to the centrifugal barrier term. We obtained the numerical bound state energies for both physical parameters and some diatomic molecules for various values of screening parameter which characterizes the strength of the potential. We obtained the energy eigen equation in a closed and compact form and applied it to study partition function and other thermodynamic properties as applied to four selected diatomic molecules namely: Nitrogen (N2), Carbon (II) Oxide (CO), Nitrogen Oxide (NO) and Hydrogen (H2) molecules, respectively using experimentally determined spectroscopic parameter. The numerical energy eigenvalues obtained both for physical parameters and for selected diatomic molecules at various dimensions (
) reveals that constant degeneracies occurs for S and P quantum state. The result also shows that 1S-quantum state has the highest bound state energies which are experimentally verified because of its proximity to the nucleus of an atom. To ascertain the accuracy of our work, the thermodynamic spectral diagram produces an excellent curves as compared to work of an existence literature. This research has application in the field of molecular spectroscopy.
Keywords: N-dimensional Schrödinger equation; Nikiforov-Uvarov method; eigenvalues; Varshni plus Woods-Saxon potential.
Correspondence Authors Email: uduakobongokorie@aksu.edu.ng
1. Introduction
From the early days of quantum mechanics, the study of exactly solvable problems has attracted a considerable attention in many areas of physics, particularly in atomic physics, information theory, nuclear physics, particle physics, and molecular physics [1, 2]. Many studies have addressed problems involving Schrödinger equation for spherically symmetric potentials in N-dimensional space by using different analytical procedures. For instance, Oyewumi et al. [3] investigated the exact solutions of the Schrödinger equation in the N-dimensional spaces for the pseudoharmonic oscillator potential by means of the ansatz method. Ntibi et al. [4] studied the analytical solution of the relativistic Klein-Gordon equation with Yukawa potential via the Nikiforov-Uvarov method. Gönül and Koçak [5] presented explicit solutions for N-dimensional Schrödinger equations with position-dependent mass. Oyewumi [6] discussed the analytical solution of the Kratzer–Fues potential in N-dimensional space. Ikhdair and Sever [7] investigated the exact solution of the Klein–Gordon equation in N-dimensions in the presence of the equal scalar and vector pseudoharmonic potential plus the ring-shaped potential using the Nikiforov–Uvarov method. Recently, with the experimental proof of the Schrödinger wave equation, researchers have made great effort to solve the Schrödinger equation by the combination of two or more potentials, which plays an important role in many areas of Physics [8]. For instance, Inyang et al. [9] presented analytical solutions of the N-dimensional Schrödinger equation for Varshni–Hulthén potential within the framework of Nikiforov–Uvarov method and by using Greene–Aldrich approximation scheme to the centrifugal barrier. Edet et al. [10] obtained bound state solutions of the Schrödinger equation for the modified Kratzer potential plus screened Coulomb potential model, within the framework of Nikiforov–Uvarov method. Later, Edet et al. [11] obtained any l-state solutions of the Schrödinger equation interacting with the Hellmann–Kratzer potential. William et al. [12] obtained bound state solutions of the radial Schrödinger equation by the superposition of Hulthén plus Hellmann potential within the framework of Nikiforov–Uvarov (NU) method for an arbitrary l-state. Inyang et al. [13] studied the approximate analytical solutions of the Schrödinger equation with a class of Yukawa plus Eckart potentials, with the Greene-Aldrich approximation in the centrifugal term.
The essence of combining two or more physical potential models is to have a wider range of applications. In addition, a comparative study of the potential functions by Steele et al. [14] and by Varshni [15] showed that the larger the number of parameters in the analytical potential energy function, better the fits with experimental data will it be. Hence, motivated by the success of the combination of exponential-type potentials, we seek to investigate the bound state solutions of the Schrödinger equation with combined Varshni and Woods-Saxon potential (VWSP) of the form:
, (1)
Where 
is the Varshni potential which is greatly important with applications, cutting across nuclear physics, particle physics, and molecular physics [16, 17]:
, (2)
where 
and 
are potential strength parameters, 
is the screening parameter, and 
the inter-nuclear separation. The Varshni potential is a short-range repulsive potential energy. Additionally, 
is the Wood–Saxon potential which is one of the important short-range potentials in physics. Its relevance to diverse areas of physics including nuclear and particle physics, atomic physics, molecular physics, condensed matter, and chemical physics has been of great interest and concern to researchers in recent times [18]:
, (3)
where is the screening parameter and 
is the potential strength constant.
The plot of the combine potential with the screening parameter is presented in Fig. 1(a). It must be noted that the exact solution of relativistic and non-relativistic equation for most of these physical potentials creates a problem where the application of the approximation is indispensable [19]. For instance, in the case of the Schrödinger equation, when the angular momentum quantum number is present, one can resort to solve the non-relativistic equation approximately via a suitable approximation scheme [20]. Some of such approximations, yielding good results, consist of the conventional approximation scheme proposed by Greene and Aldrich [21], the improved approximation scheme by Jia et al. [22], the elegant approximation scheme [23], the Pekeris approximation [24], the improved approximation scheme by Yazarloo et al. [25], improved approximation scheme in Refs. [26-28] and in Refs. [29, 30]. Therefore, to obtain approximate solutions, we employ a Greene–Aldrich approximation [21]
. (4)
Fig. 1(b) shows a comparison between the centrifugal term and the Greene–Aldrich approximation. It can perfectly model the centrifugal term around the equilibrium distance.
The paper is organized as follows: In section 2 we present the N-dimensional Schrödinger equation. In section 3, we derive the bound states solutions of the Schrödinger equation with VWSP using NU method. In section 4, we present the results and discussion. Conclusions are given in Sect. 5. In addition, a review of the Nikiforov-Uvarov (NU) method [31–34] is presented in Appendix A.
Figure 1: (a) Plot of the Varshni plus Woods-Saxon potential as a function of inter-nuclear distance for different screening parameter. We choose 
, and 
. In (b) is depicted the comparison between centrifugal term and Pekeris-type approximation.
2. N-Dimensional Schrödinger Equation
The Schrodinger equation for a spherically symmetric potential in N-dimensions [35] is written as:
, (5)
where the Laplacian operator is defined as
(6) .
Also, 
is the potential,
is the reduced mass, 
is the reduced Planck constant, 
is the energy spectrum, and 
represents the angular coordinate. The hyperspherical harmonic functions are the eigenfunction of the operator 
. Thus, we write
. (7)
In above equation, 
is the hyperspherical harmonics and 
is the hyper radial wave function. It is well-known that 
is a generalization of the centrifugal barrier for the N-dimensional space and involves the angular coordinate () and the eigenvalues of the hyperspherical harmonic functions are given by
(8) ,
in which 
is the arbitrary angular momentum quantum number. We choose a common ansatz for the wave function in the form
. (9)
The Schrodinger equation in N-dimensions can be written as
. (10)
3. Bound state solutions of the Schrödinger equation with Varshni plus Woods-Saxon potentials
In this study, we adopt the Nikiforov-Uvarov (NU) method, which is based on solving the second-order differential equation of the hypergeometric type. The details are given in the appendix A. Substituting Eqs. (1) and (4) into Eq. (10), we obtain
. (11)
By using the coordinate transformation 
, we obtain:
, (12)
where
, (13)
, (14)
and
. (15)
The Eq. (12) has the same shape as Eq. (A1) in the appendix A (see below). We have the following parameters:
, (16a)
, (16b)
and
. (16c)
Substituting Eqs. (16) into Eq. (A6) of the appendix A, we obtain:
. (17)
To find the constant 
, the discriminant of the expression under the square root of Eq. (17) must be equal to zero. As such, we have that
. (18)
Substituting Eq. (18) into Eq. (17) yields 
as
. (19)
According to Eq. (A5) of the appendix A, 
can be written as
(20)
and
(21)
Referring to Eq. (A7) of the appendix A, we define the constant l as,
(22)
and taking the derivative of s(x) with respect to x from equation (16b), we have
. (23)
Substituting Eqs. (23) and (20) into Eq. (A8) of the appendix A, we obtain
. (24)
By comparing Eqs. (24) and (22), using Eq. (16) yields the energy eigenvalues equation of the VWS potential as a function of 
and as
(25) ,
where
(26) .
Special cases:
If we set 
and 
in Eq. (25), we obtain the energy eigenvalue equation for the s-wave of VHP as
. (27)
When we set 
in the previous equation, we obtain the energy eigenvalues for Varshni potential
, (28)
where
. (29)
If we set 
in Eq. (25), we obtain the energy eigenvalue equation for Woods-Saxon potential
, (30)
where
(31) .
4. Thermodynamic Properties
In this section, we present the thermodynamic properties for the potential model. Generally, the thermodynamic properties of quantum systems can be obtained from the exact partition function given by
(32) ,
where 
is an upper bound of the vibrational quantum number obtained from the numerical solution of 
given as 
, 
where 
and 
are Boltzmann constant and absolute temperature respectively. In the classical limit, the summation in Eq. (32) can be replaced with an integral:
(33) . In order to obtain the partition function, then Eq. (25) can be presented in a close and compact form as
(34) ,
where
(35a) ,
(35b)
and
(35c) .
Let
. Then Eq. (34) reduces to
(36) .
Using Eq. (32), the partition function equation is then expressed as
(37) .
Using mathematica software 10.0 version, Eq. (37) is evaluated as
(38) .
Using Eq. (38), other thermodynamic properties can be obtained as follows:
(a) Vibrational mean energy: This is given by the equation
(39) ,
where
(40) .
(b) Vibrational heat capacity: This is given by the equation
(41) ,
where
(42) .
(c) Vibrational entropy: This is given by the equation
(43) .
(d) Vibrational Free energy: This is given by the equation
(44) .
5. Discussion
Table 1 is the spectroscopic diatomic molecular potential parameter adopted from Ref. [37] for the numerical computation of energy eigenvalues for the selected diatomic molecules namely, Nitrogen (N2), Carbon (II) oxide (CO), Nitrogen oxide (NO) and Hydrogen (H2) molecules, respectively. In Table 2, we numerically reported energy eigenvalues (in eV) for the 1s, 2s, 2p, 3s, 3p, 3d, 4s, and 4p states with different values of the potential strength and for 
, 
, and 
. As the quantum number increases, the energy eigenvalues of the combined potential become more bounded. We observed that the energy increases as the screening parameter 
, and the number of dimensional space 
increase.
Table 3 is the numerical bound state energies for the selected diatomic molecules for different dimensions using spectroscopic screening parameter 
. It is observed that the numerical results for all the diatomic molecules have degeneracy at quantum state 2s at 
and 2p at 
dimension, respectively. The same trend is observed at quantum state 3s at and 3p at dimension, respectively.
From Table 3, we also observed that 1s has the highest bound state energy for all the diatomic molecules. This totally agrees with theoretical and experimental investigations. The electrons in the S-orbital are close to the nucleus than other shells. Hence, it will possess greater ionization energy. Generally, the bound state energies of these molecules tend to decrease with an increasing quantum state especially at higher quantum state. Correspondingly, we also observed that the energy increases as the screening parameter , and the number of dimensional space increases. The variation of the energy eigenvalues with the screening parameter, potential strength, and the reduced mass for 1s, 2p, 3p, and 3d states is shown in Figs. 2-6. In Fig. 2, we show the variation of energy as a function of the screening parameter for various numbers of dimensional space and observed that the energy increases as d increases until it reaches a maximum value and then it drops. The variation of the energy eigenvalues as function of the reduced mass has a similar behavior. The plot of the energy eigenvalues with the reduced mass for the states 1s, 2p, 3p, and 3d for various values of the screening parameter with , and 
are displayed in Fig.3. From Figs. 4 to 6, we plotted the variation of the energy eigenvalues of the different states as a function of the VWS strengths in three dimensions. We observed that there is an increase in energy as the potential strengths, and increase. However, the energy decreases as the potential strength increases. Fig. 7(a-c) are the plot of partition function against the inverse temperature parameter (
) for the selected diatomic molecule for (
) dimension, the plot of partition function against the inverse temperature parameter () for the selected diatomic molecule for (
) dimension and the plot of partition function against the upper bound vibrational quantum number (), for fixed values of for () dimension respectively. In Figs. 7(a) and (b), the partition function increases exponentially with an increase in the inverse temperature parameter () for and respectively. But in Fig. 7(c), the partition function which all converges together from the vertical axis slowly start diverging at 
and at the same time increases exponentially with an increase in the upper bound quantum number .
Fig. 8 (a-c) are the plot of vibrational mean energy against the inverse temperature parameter () for the selected diatomic molecule for () dimension, the plot of vibrational mean energy against the inverse temperature parameter () for the selected diatomic molecule for () dimension and the plot of vibrational mean energy against the upper bound vibrational quantum number (), for fixed values of for () dimension, respectively. In Fig. 8(a), the vibrational mean energy increases monotonically from origin before diverging linearly with slight maximum turning point at 
for () dimension. The same is applicable to Fig. 8(b), the slight turning point occurs at 
for () dimension. In figure 8(c), the variation of vibrational mean energy increases monotonically with an increase in upper bound vibrational quantum number for fixed value of inverse temperature parameter.
Fig. 9 (a-c) are the plot of specific heat capacity against the inverse temperature parameter () for the selected diatomic molecule for () dimension, the plot of specific heat capacity against the inverse temperature parameter () for the selected diatomic molecule for () dimension and the plot of specific heat capacity against the upper bound vibrational quantum number (), for fixed values of for () dimension respectively. Fig. 9(a) is a sinusoidal curve that has maximum turning point at 
and minimum turning point at for () dimension. The same is applicable to figure 9(b) where the maximum turning point occurs at 
for 
dimension as applied to all the diatomic molecules. Fig.s 9(a) and 9(b) show symmetrical characteristics for both dimensions, though the curve spacing is more conspicuous at higher dimension . In Fig. 9(c), the variation of heat capacity against the maximum upper bound vibrational quantum number shows a parabolic curve with maximum turning point at 
for fixed value of.
Figs. 10 (a-c) are the plot of vibrational entropy against the inverse temperature parameter () for the selected diatomic molecule for () dimension, the plot of vibrational entropy against the inverse temperature parameter () for the selected diatomic molecule for () dimension and the plot of vibrational entropy against the upper bound vibrational quantum number (), for fixed values of for () dimension respectively. In Figs. 10 (a) and (b), the variation of entropy against show parabolic curves that concave upward with various minimum turning points as applied to all the diatomic molecules for fixed value of vibrational quantum number. In Fig. 10(c), the vibrational entropy increases exponentially with an increase in for fixed values of for all the diatomic molecules.
Figs. 11(a-c) are the plot of vibrational free energy against the inverse temperature parameter () for the selected diatomic molecule for () dimension, the plot of vibrational free energy against the inverse temperature parameter () for the selected diatomic molecule for () dimension and the plot of vibrational free energy against the upper bound vibrational quantum number (), for fixed values of for () dimension respectively. In Fig. 11(a), the vibrational free energy increases monotonically from the origin before spreading out linearly in a quantized manner with an increase value of for () dimension. Fig. 11(b) is a parabolic curve that slowly decreases from the maximum in an increasing value of for () dimension as applied to all the diatomic molecules. In Fig. 11(c), the variation of free energy against the upper bound vibrational quantum number shows a unique symmetrical quantized curves that increases monotonically, but converges at 
. The trend of our thermodynamic curves agrees excellently with some work of an existing literature as can been seen in Refs. [38-40].
Table 1: Model parameters for selected diatomic molecules in this study [37].
Molecule |
|
|
| 7.00335 | 2.69860 |
| 6.86059 | 2.29940 |
| 7.46844 0.50391 | 2.75340 1.94260 |
|
|
|
Table 2: Energy eigenvalues (in eV) of the Varshni plus Woods-Saxon potential for different states. We choose 
, and 
.
State | N |
|
|
|
|
1s | 2 | -3.144822531 | -1.863734568 | -1.585268347 | -1.481234568 |
3 | -2.239511111 | -1.528044444 | -1.381649383 | -1.334469444 | |
4 | -1.825000111 | -1.378400444 | -1.294472605 | -1.275068444 | |
2s | 2 | -1.815032111 | -1.367328444 | -1.283160605 | -1.264380444 |
3 | -1.588629938 | -1.285630864 | -1.237024726 | -1.235764198 | |
4 | -1.454628628 | -1.239739002 | -1.213632962 | -1.224194104 | |
2p | 2 | -1.454628628 | -1.239739002 | -1.213632962 | -1.224194104 |
3 | -1.369284028 | -1.212136111 | -1.201318596 | -1.220367361 | |
4 | -1.311886454 | -1.19470631 | -1.194839693 | -1.220265569 | |
3s | 2 | -1.449763322 | -1.234971655 | -1.209845207 | -1.222267574 |
3 | -1.362002778 | -1.205511111 | -1.197037346 | -1.220117361 | |
4 | -1.303456824 | -1.187654458 | -1.191565619 | -1.223169273 | |
3p | 2 | -1.303456824 | -1.187654458 | -1.191565619 | -1.223169273 |
3 | -1.262738778 | -1.176555111 | -1.190016901 | -1.228587111 | |
4 | -1.233469444 | -1.169580257 | -1.19061996 | -1.234987695 | |
3d | 2 | -1.233469444 | -1.169580257 | -1.19061996 | -1.234987695 |
3 | -1.211859568 | -1.165216049 | -1.192408265 | -1.241674383 | |
4 | -1.195549129 | -1.162551545 | -1.194834409 | -1.248292965 | |
4s | 2 | -1.300691392 | -1.185481619 | -1.190874261 | -1.224848285 |
3 | -1.258402778 | -1.173611111 | -1.190192901 | -1.233611111 | |
4 | -1.22827275 | -1.16664472 | -1.192444753 | -1.244071993 | |
4p | 2 | -1.22827275 | -1.16664472 | -1.192444753 | -1.244071993 |
3 | -1.206237346 | -1.16272716 | -1.196252709 | -1.25505216 | |
4 | -1.189773981 | -1.160752728 | -1.200846243 | -1.26594977 | |
4d | 2 | -1.189773981 | -1.160752728 | -1.200846243 | -1.26594977 |
3 | -1.177254478 | -1.160038322 | -1.205784133 | -1.276462812 | |
4 | -1.167592901 | -1.160149383 | -1.210810734 | -1.286449383 |
State | N |
| CO molecule | NO molecule |
|
1s | 2 | -3.627134168 | -3.313056789 | -3.702990275 | -1.626741087 |
3 | -3.931335591 | -3.609236041 | -4.026345636 | -4.937948246 | |
4 | -3.742178623 | -3.329433039 | -3.790289993 | -7.211445685 | |
2s | 2 | -3.768797705 | -3.304912845 | -3.799465994 | -9.308201167 |
3 | -3.820667899 | -3.358728412 | -3.855886360 | -9.099935006 | |
4 | -4.226552582 | -3.620004834 | -4.237631309 | -12.93900696 | |
2p | 2 | -4.226552582 | -3.620004834 | -4.237631309 | -12.93900696 |
3 | -4.492124066 | -3.809420964 | -4.494309293 | -14.81922035 | |
4 | -4.723410398 | -3.978818673 | -4.719508119 | -16.30494777 | |
3s | 2 | -4.443598836 | -3.761619921 | -4.442160003 | -15.38911589 |
3 | -4.446738608 | -3.769729378 | -4.447417627 | -15.16872798 | |
4 | -5.158396853 | -4.281600958 | -5.136475219 | -20.59996679 | |
3p | 2 | -5.158396853 | -4.281600958 | -5.136475219 | -20.59996679 |
3 | -5.566906548 | -4.582290969 | -5.534612588 | -23.43103226 | |
4 | -5.917996763 | -4.843089904 | -5.877687922 | -25.76504992 | |
3d | 2 | -5.917996763 | -4.843089904 | -5.877687922 | -25.76504992 |
3 | -6.228068294 | -5.074804751 | -6.181206131 | -27.76851221 | |
4 | -6.504514349 | -5.282284892 | -6.452148144 | -29.51739950 | |
4s | 2 | -5.468715816 | -4.501019422 | -5.435355107 | -23.34423329 |
3 | -5.455674087 | -4.494254190 | -5.423730323 | -23.11864665 | |
4 | -6.417174661 | -5.199566787 | -6.359715059 | -30.15498880 | |
4p | 2 | -6.417174661 | -5.199566787 | -6.359715059 | -30.15498880 |
3 | -6.963055189 | -5.603895759 | -6.892598492 | -33.96879576 | |
4 | -7.436187030 | -5.955921309 | -7.355069609 | -37.20090230 | |
4d | 2 | -7.436187030 | -5.955921309 | -7.355069609 | -37.20090230 |
3 | -7.859076840 | -6.271569634 | -7.768814707 | -40.04312740 | |
4 | -8.241000462 | -6.557331028 | -8.142743437 | -42.57794080 |
Figure 2: The plot of the energy spectra for 1s, 2p, 3p, and 3d states as a function of the screening parameter, , for various numbers of dimensional space. The results are for and .
Figure 3: Energy eigenvalues variation with the reduced mass for the states 1s, 2p, 3p, and 3d for various values of the screening parameter. We choose , and .
Figure 4: The plot of the energy spectra for 1s, 2p, 3p, and 3d states as a function of . We choose 
, and 
.
Figure 5: The plot of the energy spectra for 1s, 2p, 3p, and 3d states as a function of . We choose 
, and .
Figure 6: The plot of the energy spectra for 1s, 2p, 3p and 3d states as a function of . We choose and .
(a) (b)
(c)
Figure 7: (a) The plot of partition function against the inverse temperature parameter () for the selected diatomic molecule for () dimension, (b) The plot of partition function against the inverse temperature parameter () for the selected diatomic molecule for () dimension, (c) The plot of partition function against the upper bound vibrational quantum number (), for fixed values of for () dimension.
(a) (b)
(c)
Figure 8: (a) The plot of vibrational mean energy against the inverse temperature parameter () for the selected diatomic molecule for () dimension, (b) The plot of vibrational mean energy against the inverse temperature parameter () for the selected diatomic molecule for () dimension, (c) The plot of vibrational mean energy against the upper bound vibrational quantum number (), for fixed values of for () dimension.
(a) (b)
(c)
Figure 9: The plot of specific heat capacity against the inverse temperature parameter () for the selected diatomic molecule for () dimension, (b) The plot of specific heat capacity against the inverse temperature parameter () for the selected diatomic molecule for () dimension, (c) The plot of specific heat capacity against the upper bound vibrational quantum number (), for fixed values of for () dimension.
(a) (b)
(c)
Figure 10: The plot of vibrational entropy against the inverse temperature parameter () for the selected diatomic molecule for () dimension, (b) The plot of vibrational entropy against the inverse temperature parameter () for the selected diatomic molecule for () dimension, (c) The plot of vibrational entropy against the upper bound vibrational quantum number (), for fixed values of for () dimension.
(a) (b)
(c)
Figure 11: The plot of vibrational free energy against the inverse temperature parameter () for the selected diatomic molecule for () dimension, (b) The plot of vibrational free energy against the inverse temperature parameter () for the selected diatomic molecule for () dimension, (c) The plot of vibrational free energy against the upper bound vibrational quantum number (), for fixed values of for () dimension.
6. Conclusions
The analytical solutions of the N-dimensional Schrödinger equation for the combined Varshni plus Woods-Saxon potential are obtained via NU method by using Greene-Aldrich approximation scheme to the centrifugal barrier. The numerical energy eigenvalues are obtained for various values of 
and . The energy equation has been presented in a compact form in order to obtain the partition function and other thermodynamic properties as applied to four selected diatomic molecules. The thermodynamic studies reproduce excellent curves in agreement to work of existing literatures. The numerical bound state energies show degeneracies in numerical values for some quantum states. The numerical bound state energies decrease with an increase in quantum state for various screening parameters and experimentally determined spectroscopic parameters. The thermodynamic plots also reveal that higher dimensions portrayed similar characteristics. The behaviors of the energy eigenvalues for different states are illustrated graphically. Therefore, studying of analytical solution of the N-dimensional Schrödinger equation for Varshni plus Woods-Saxon potential could provide valuable information on the quantum mechanics dynamics at atomic and molecular physics and opens new window.
Appendix A
In this section, we give a brief description of the conventional Nikiforov-Uvarov (NU) method. A more detailed description of the method can be obtained from Ref. [36].With an appropriate transformation 
, the one-dimensional Schrödinger equation can be reduced to a generalized equation of hypergeometric type, which can be written as follows:
, (A1)
where 
and 
are polynomials, at most second-degree, and 
is at most a first-order polynomial. To find particular solution of Eq. (A1) by separation of variables, if one deals with
(A2)
then Eq. (A1) becomes
, (A3)
where
, (A4)
, (A5)
, (A6)
and
. (A7)
The polynomial 
with the parameter 
and prime factors show the differentials at first degree be negative. However, determination of parameter is the essential point in the calculation of 
. It is simply defined by setting the discriminate of the square root to zero [36]. Therefore, one gets a general quadratic equation for . The values of can be used for calculation of energy eigenvalues using the following equation
. (A8)
Furthermore, the other part 
of the wave function in Eq. (A2) is the hypergeometric-type function whose polynomial solutions are given by Rodrigues relation:
, (A9)
where 
is a normalizing constant and the weight function 
must satisfy the condition [36]
. (A10)
Acknowledgments and Funding Declaration
CAD is grateful to the Colombian Agencies: CODI-Universidad de Antioquia (Estrategia de Sostenibilidad de la Universidad de Antioquia and projects "Efectos de capas delta dopadas en pozos cuánticos como fotodetectores en el infrarrojo", "Propiedades magneto-ópticas y óptica no lineal en superredes de Grafeno", "Efectos ópticos intersubbanda, no lineales de segundo orden y dispersión Raman, en sistemas asimétricos de pozos cuánticos acoplados", and "Estudio de propiedades ópticas en sistemas semiconductores de dimensiones nanoscópicas"), and Facultad de Ciencias Exactas y Naturales-Universidad de Antioquia (CAD exclusive dedication project 2020-2021). CAD also acknowledges the financial support from El Patrimonio Autónomo Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación Francisco José de Caldas. (project: CD 111580863338, CT FP80740-173-2019).
Author’s Contributions Statement
All the authors contributed to this work. R. Horchani conceptualized the idea. S.A. Shafii and N.A. Hashimi wrote the first draft. A.N. Ikot carried out the literature analysis. I.B. Okon computed the thermal properties. E.O. Oladimeji carried out the numerical computations. C.A. Duque carried out the discussion of the results. U.S. Okorie proof-read the work, making sure that all corrections were made in accordance with the journal guidelines.
Competing Interest Statement
We humbly state here that we have no conflicts of interest to disclose amongst authors.
Data Availability Statement
All data enclosed in this manuscript are generated using Maple software and are available on request.
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