1 Introduction

The Schrödinger equation (SE) portrays numerous problems in various areas of Physics and Chemistry [1,2,3,4]. The exact solution of the Schrödinger equation (SE) with some solvable potential assumes an essential part in nuclei, atoms, molecules, and spectroscopy and in many fields of modern physics [5]. It is a laborious task to obtain analytic solutions to the radial SE with a given interaction potential without application of an approximation scheme [6,7,8,9]; also, there are a couple of potentials, for example, the harmonic oscillator and hydrogen atom, for which the SE can be solved with no approximation scheme [10, 11]. The applications of the solutions of the Schrödinger equation in some cases depend on the nature of the potential model [1, 12,13,14,15,16,17,18]. For many decades, researchers have been obtaining the solution of the SE via different methods which include point canonical transformation (PCT) [19, 20], the Nikiforov–Uvarov method [21,22,23,24,25], numerical methods [26,27,28], the asymptotic iterative method (AIM) [29, 31], WKB approximation method [32], supersymmetric quantum mechanics (SUSYQM) [33, 34], the factorization method [35, 36] and the Hill determinant method [37] and many more.

There are areas in which the solution to SE can be applied, for instance, in Ref. [48], the authors applied the solution of the SE with screened Kratzer potential in studying thermodynamics properties of the system. Other applications of the solution to SE include investigation of the mass spectra of quarkonium system, which is the main focus of this work. Potential models such as Martin, logarithmic and Cornell potentials have often been used which are commonly used in studying heavy quarkonium spectra [38,39,40,41,42,43]; in this work the screened Kratzer potential is used to investigate quark confinement. These potentials have two distinctive features, strong interaction-asymptotic freedom and confinement [38]. The screened Kratzer potential is one of the successful potential models for such systems because it produces its mass spectra in agreement with the experimental data [44]. The study of heavy quarkonium systems provides a solid understating for the quantitative test of quantum chromodynamic (QCD) theory and the standard model [11]. As a subatomic system, a quarkonium that is composed of a heavy quark-antiquark (qq) pair has attracted attention of particle physicists since the first half 1970, and are just a few studies of them. In the most of these studies, the system is examined via Schrödinger equation in nonrelativistic quantum chromodynamics (NQCD), assuming that quarks are spinless [45]. However, in the present study we will assume that the quarks are spinless.

In the recent developments, researchers have focus in investigating the spectra of the above mentioned quarks. For instance, Ibekwe et al. [1] analytically solved the radial Schrödinger equation with an exponential, generalized, anharmonic Cornell and generated the expression for the mass spectra of heavy quarkonium systems. Omugbe et. al. [3] obtained mass spectrum of mesons via the WKB approximation method. Abu-Shady et al. [5] obtained the exact solution of the N-dimensional radial SE with the generalized Cornell potential and applied the result to obtain the mass spectra for the system. Ciftci and Kisoglu [7] generated energy eigenvalues for an exact SE and derived the mass of a heavy quark–antiquark system (quarkonium) using the asymptotic iteration method (AIM). Yazarloo and Mehraban [11] studied the B and Bs mesons spectra and their decay properties within the framework of a non-relativistic potential model using a new potential model for the interaction of mesonic systems. Abu-Shady and Ezz-Alarab [15] studied the N-radial Schrödinger equation analytically using an exact-analytical iteration method and used the results to calculate the thermodynamic properties and mass of mesons. Kumar and Fakir [30] analytically obtained the energy eigenvalues and normalized eigen-functions of the radial SE in N-dimensional space for the quark–antiquark interaction potential. Maksimenko and Kuchin [46] generated the mass spectrum of the SE for a potential comprised of the sum of a harmonic oscillator potential, a linear potential and a Coulomb potential. Abu-Shady et al. [47] studied the masses and thermodynamic properties of heavy mesons in the non-relativistic quark model using the Nikiforov–Uvarov method.

2 Screened Kratzer Potential (SKP)

Ikot et al. [48] proposed screened Kratzer potential of the form

$$ V(r) = - 2D_{e} \left( {\frac{A}{r} - \frac{B}{{2r^{2} }}} \right)e^{ - \alpha r} . $$
(1)

where \(A = r_{e} \,{\text{and}}\,B = r_{e}^{2}\), De is the dissociation energy, re is the equilibrium bond length, r is the interatomic distance and α is the screening parameter. The authors reported that the screened Kratzer potential is a generalized potential composed of Kratzer, screened Coulomb and Coulomb potential. The screened Kratzer potential reduces to the standard Kratzer potential when α → 0, screened Coulomb potential when B = 0, and the Coulomb potential when B = 0 and α → 0. The SKP is advantageous over Coulomb potential as its solution is more generalized and has wider application in molecular and chemical physics.

The authors obtained the energy eigenvalues and the corresponding normalized eigenfunctions of a newly proposed screened Kratzer potential and applied the energy eigenvalues of the potential and investigated the vibrational partition function and other thermodynamic functions are also obtained in closed form for the diatomic molecules.

This presentation is aimed at extending the study of the screened Kratzer potential to investigate mass spectroscopy of some heavy quarkonium particles. The work is organized as: Sect. 2 presents a brief introduction to the screened Kratzer potential (SKP), Sect. 3 studies the potential with the radial SE and presents the bound state energy eigenvalue for the potential, Sect. 4 presents the bound state energies of the special cases (Coulomb and Kratzer potentials). In Sect. 5, we derive the mass spectra expression of the heavy quarkonium systems. The results of the work are discussed in Sect. 6. A brief conclusion is presented in Sect. 7. We made possible recommendation in Sect. 8.

3 Bound state solution with radial Schrödinger equation

Taylor expanding the exponential term of the potential and ignore terms greater than r3 our Eq. (1) becomes

$$ V(r) = \frac{{ADe\alpha^{3} }}{3}r^{2} + \left( { - A\alpha^{2} - \frac{{BD_{e} \alpha^{3} }}{6}} \right)r - \left( {2D_{e} A + D_{e} B\alpha } \right)\frac{1}{r} + \frac{{D_{e} B}}{{r^{2} }} + 2D_{e} A\alpha + \frac{{D_{e} B\alpha^{2} }}{2}, $$
(2)

the inverse square term, \({{D_{e} B} \mathord{\left/ {\vphantom {{D_{e} B} {r^{2} }}} \right. \kern-\nulldelimiterspace} {r^{2} }}\) makes the potentials more singular and produces better confinement which is a necessary condition for the investigation of quark system [6].

We consider the radial Schrödinger equation of the form [1, 18]

$$ \frac{{^{{\mathop {\text{d}}\nolimits^{2} }} \Psi \left( r \right)}}{{{\text{d}}r^{2} }} + \frac{2}{r}\frac{{{\text{d}}\Psi \left( r \right)}}{{{\text{d}}r}} + \left[ {\frac{2\mu }{{\hbar^{2} }}\left( {E - V} \right) - \frac{{l\left( {l + 1} \right)}}{{r^{2} }}} \right]\Psi \left( r \right) = 0 $$
(3)

where l is the magnetic quantum number taking the values 0,1,2,3,4…, μ is the reduced mass of a diatomic molecule, r is the internuclear separation, and E denotes the energy eigenvalues of the system.

Now we substitute Eqs. (2),  (3) and obtain

$$\begin{aligned} & \frac{{^{{\mathop {\text{d}}\nolimits^{{2}} }} \Psi \left( r \right)}}{{{\text{d}}r^{2} }} + \frac{2}{r}\frac{{{\text{d}}\Psi \left( r \right)}}{{{\text{d}}r}}\\ & \quad + \left[ {\frac{2\mu }{{\hbar^{2} }}\left( {E - \left( \begin{gathered} V(r) = \frac{{AD_{e} \alpha^{3} }}{3}r^{2} \left( { - A\alpha^{2} - \frac{{BD_{e} \alpha^{3} }}{6}} \right)r \hfill \\ - \frac{{\left( {2D_{e} A + D_{e} B\alpha } \right)}}{r} + \frac{{D_{e} B}}{{r^{2} }} + 2D_{e} A\alpha + \frac{{D_{e} B\alpha^{2} }}{2} \hfill \\ \end{gathered} \right) - \frac{{l\left( {l + 1} \right)}}{{r^{2} }}} \right)} \right]\Psi \left( r \right) = 0 \end{aligned} $$
(4)

Equation (4) can be written as

$$ \left[ {\frac{{^{{\mathop {\text{d}}\nolimits^{2} }} }}{{{\text{d}}r^{2} }} + \frac{2}{r}\frac{{\text{d}}}{{{\text{d}}r}} - \frac{{L\left( {L + 1} \right)}}{{r^{2} }} + \left( {\varepsilon - Ur^{2} - Vr + W\frac{1}{r} + X\frac{1}{{r^{2} }}} \right)} \right]\Psi \left( r \right) = 0. $$
(5)

with

$$ \varepsilon = \frac{2\mu }{{\hbar^{2} }}\left( {E - \left( {2D_{e} A\alpha + \frac{{D_{e} B\alpha^{2} }}{2}} \right)} \right), $$
(6)
$$ U = \frac{{2\mu AD_{e} \alpha^{3} }}{{3\hbar^{2} }}, $$
(7)
$$ V = \frac{2\mu }{{\hbar^{2} }}\left( { - A\alpha^{2} - \frac{{BD_{e} \alpha^{3} }}{6}} \right), $$
(8)
$$ W = \frac{2\mu }{{\hbar^{2} }}\left( {2D_{e} A + D_{e} B\alpha } \right), $$
(9)
$$ L\left( {L + 1} \right) = l\left( {l + 1} \right) + \frac{2\mu }{{\hbar^{2} }}X $$
(10)

where

$$L = - \frac{1}{2} + \frac{1}{2}\sqrt {\left( {2l + 1} \right)^{2} + {{8\mu X} \mathord{\left/ {\vphantom {{8\mu X} {\hbar^{2} }}} \right. \kern-\nulldelimiterspace} {\hbar^{2} }}}. $$
(11)

Now make an anzats wave function [1, 6]

$$ \Psi \left( r \right) = \mathop e\nolimits^{{ - \alpha r^{2} - \beta r}} F\left( r \right) $$
(12)

where α and β are positive constants. Using this wave function on Eq. (5), it becomes

$$ \begin{aligned} & F^{{\prime \prime }} \left( r \right) + \left[ { - 4\alpha r - 2\beta + \frac{2}{r}} \right]F^{{\prime }} \left( r \right) \\ & \quad + \left[ {\left( {4\alpha^{2} - U} \right)r^{2} + \left( {4\alpha \beta - V} \right)r + \left( {W - 2\beta } \right)\frac{1}{r} - \frac{{L\left( {L + 1} \right)}}{{r^{2} }} + \left( {\varepsilon - 6\alpha + \beta^{2} } \right)} \right]F\left( r \right) = 0. \\ \end{aligned} $$
(13)

Due to the singularities in Eq. (13), the fractional wave function of the form [1, 6, 49]

$$ F\left( r \right) = \sum\limits_{n = 0}^{\infty } {a_{n} } r^{2n + L} . $$
(14)

is considered suitable to solve Eq. (13). Taking the first and the second derivatives of Eq. (14) and substitute alongside with Eq. (14) into our Eq. (13), we obtain

$$ \begin{aligned} & \sum\limits_{n = 0}^{\infty } {a_{n} } \left\{ {\left[ {\left( {2n + L} \right)\left( {2n + L - 1} \right) + 2\left( {2n + L} \right) - L\left( {L + 1} \right)} \right]} \right.r^{2n + L - 2} \\ & \quad + \left[ { - 2\beta \left( {2n + L + 1} \right) + W} \right]r^{2n + L - 1} + \left[ { - 4\alpha \left( {2n + L} \right) + \varepsilon + \beta^{2} - 6\alpha } \right]r^{2n + L} \\ & \quad + \left[ {4\alpha \beta - V} \right]r^{2n + L + 1} + \left[ {4\alpha^{2} - U} \right]\left. {r^{2n + L + 2} } \right\} = 0. \\ \end{aligned} $$
(15)

Equation (15) is linearly independent, thus each of the terms is separately equal to zero, having in mind that r is a non-zero function; therefore, it is the coefficient of r that is zero. With this in mind, we obtain the relation for each of the terms as given below

$$ \left( {2n + L} \right)\left( {2n + L + 1} \right) - L - L^{2} = 0, $$
(16)
$$ \varepsilon = 2\alpha \left( {4n + 2L + 3} \right) - \beta^{2} , $$
(17)
$$ W = 2\beta \left( {2n + L + 1} \right), $$
(18)
$$ \alpha = \frac{\sqrt U }{2}, $$
(19)
$$ \beta = \frac{V}{4\alpha }. $$
(20)

We proceed to obtaining the energy eigenvalues expression using Eqs. (8) and (17). By comparing the two equations, we have

$$ \frac{2\mu }{{\hbar^{2} }}\left[ {E - \left( {2D_{e} A\alpha + \frac{{D_{e} B\alpha^{2} }}{2}} \right)} \right] = 2\alpha \left( {4n + L + 3} \right) - \beta^{2} . $$
(21)

Substituting the expression for \(L\),\(\alpha\), and \(\beta\) into Eq. (21) yields the energy eigenvalue expression as

$$ \begin{aligned} E_{nl} = & \sqrt {\frac{{\hbar^{2} AD_{e} \alpha^{3} }}{6\mu }} \left( {4n + 2 + \sqrt {\left( {2l + 1} \right)^{2} + \frac{{8\mu D_{e} B\,}}{{\hbar^{2} }}} } \right) \\ & \quad - \frac{{2\mu \left( {2D_{e} A + D_{e} B\alpha } \right)^{2} }}{{\hbar^{2} }}\left( {4n + 1 + \sqrt {\left( {2l + 1} \right)^{2} + \frac{{8\mu D_{e} B\,}}{{\hbar^{2} }}} } \right)^{ - 2} + 2D_{e} A\alpha + \frac{{D_{e} B\alpha^{2} }}{2}. \\ \end{aligned} $$
(22)

To make Eq. (22) more manageable, we set

$$ \begin{aligned} & \sigma = \frac{{AD_{e} \alpha^{3} }}{3},\,\kappa = - A\alpha^{2} - \frac{{BD_{e} \alpha^{3} }}{6},\,\gamma = 2D_{e} A + D_{e} B\alpha , \\ & \chi = D_{e} B\,,\xi = 2D_{e} A\alpha + \frac{{D_{e} B\alpha^{2} }}{2}, \\ \end{aligned} $$
(23)

resulting in

$$ 2m + \sqrt {\frac{{\hbar^{2} \sigma }}{2\mu }} \left( {4n + 2 + \sqrt {\left( {2l + 1} \right)^{2} + \frac{8\mu \chi }{{\hbar^{2} }}} } \right) - \frac{{2\mu \gamma^{2} }}{{\hbar^{2} }}\left( {4n + 1 + \sqrt {\left( {2l + 1} \right)^{2} + \frac{8\mu \chi }{{\hbar^{2} }}} } \right)^{ - 2} + \xi . $$
(24)

4 Special cases of the potentials

To check the correctness of the result obtained from this method, we carry out suitable adjustments to the screened Kratzer potential parameters resulting in potential models, such as the Kratzer and Coulomb potentials as special cases. From the results of the SKP potential we can deduce solutions to a special type of interacting potentials which has direct applications in practical problems solving in physics and chemistry.

4.1 The Kratzer potential

The Kratzer potential has extensively been used to describe molecular structures and interactions [1, 6]. Now setting the values of screening parameter \(\alpha = 0,\) results in

$$ \sigma = \kappa = 0,\;\gamma = 2D_{e} A,\,\chi = D_{e} B,\;\xi = 0, $$
(25)

And the SKP reduces to the Kratzer potential as

$$ V\left( r \right) = \frac{{D_{e} r_{e}^{2} }}{{r^{2} }} - \frac{{2D_{e} r_{e} }}{r}, $$
(26)

also, the energy expression of the SKP reduces to the energy eigenvalues of the Kratzer potential as

$$ E_{nl} = - \frac{{2\mu D_{e}^{2} r_{e}^{2} }}{{\hbar^{2} }}\left( {n + \frac{1}{4} + \sqrt {\left( {l + \frac{1}{2}} \right)^{2} + \frac{{2\mu D_{e} r_{e}^{2} }}{{\hbar^{2} }}} } \right)^{ - 2} . $$
(27)

The numerical of the eigenspectra is presented in Tables 2 and 3 which is in good agreement with the ones existing in the literature [48, 49].

4.2 The Coulomb potential

Now setting the values of \(B = 0,\) screening parameter \(\alpha = 0,\) results in

$$ \sigma = \kappa = 0,\;\gamma = 2D_{e} A,\,\chi = 0,\;\xi = 0, $$
(28)

The SKP reduces to Coulomb potential as

$$ V\left( r \right) = - \frac{{2D_{e} r_{e} }}{r}. $$
(29)

Using Eq. (28), on the SKP energy expression, we obtain the energy eigenvalues for the Coulomb potential as

$$ E_{nl} = - \frac{{2\mu \left( {D_{e} r_{e} } \right)^{2} }}{{\hbar^{2} \left( {2n + l + 1} \right)^{2} }}. $$
(30)

Equation (30) is in good agreement with the ones existing in literature [48, 49].

5 Mass spectra

In this section, we derive the mass spectra of the heavy quarkonium systems such as charmonium and bottomonium having quark and antiquark of the same flavor. To determine the mass spectra we use the following relation [1, 3, 7].

$$ M = m_{1} + m_{2} + E_{nl} , $$
(31)

but,

$$ m_{1} = m_{2} = m_{b} , $$
(32)

Resulting in the expression

$$ M = 2m_{b} + E_{nl} . $$
(33)

where \(m_{b}\) is the mass of the particle under investigation and \(E_{nl}\) is the derived energy eigenvalues.

Substituting Eq. (24) into Eq. (33) we obtain

$$ \begin{aligned} M & = 2m + \sqrt {\frac{{\hbar^{2} \sigma }}{2\mu }} \left( {4n + 2 + \sqrt {\left( {2l + 1} \right)^{2} + \frac{8\mu \chi }{{\hbar^{2} }}} } \right) \\ & \quad - \frac{{2\mu \gamma^{2} }}{{\hbar^{2} }}\left( {4n + 1 + \sqrt {\left( {2l + 1} \right)^{2} + \frac{8\mu \chi }{{\hbar^{2} }}} } \right)^{ - 2} + \xi . \\ \end{aligned} $$
(34)

Tables 3 and 4 present the numerical values of Eq. (34) for some quarkonium systems charmonium and bottomonium, respectively. The results of the mass spectra are in consonance with the experimental values and other theoretical works of similar investigation.

6 Discussion of Results

This work has analytically solved the radial SE with the screened Kratzer potential, using the series expansion method, and the bound state energy spectra of the SE are obtained. To validate the method employ in this study, we deduce Kratzer and Coulomb potentials as special cases and compute the numerical values of the special cases for HCl and LiH diatomic molecules for different quantum numbers, n and l and the results agree with the table in Ref. [48]. and [49] and other theoretical studies of the same potentials. Tables 1 and 2 show the comparison of the numericals of Kratzer potential for HCl and LiH, respectively, with other theoretical works of the same investigation. It is observed from Tables 1 and 2 that the numerical values of the Kratzer potential for the diatomic molecules increase as quantum numbers increase. Tables 3 and 4 present the numerical values of the mass spectra for some quarkonium systems charmonium and bottomonium, respectively. The results of the mass spectra are in consonance with the experimental values and other theoretical works of similar investigation. Figure 1 is shape of mass spectra of bottomonium for different quantum numbers n and l. Figure 2 is shape of mass spectra of charmonium for different quantum numbers n and l.

Table 1 Comparison of the Energy eigenvalues of the Special case (Kratzer potential) for HCl (eV)
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Table 2 Comparison of the Energy eigenvalues of the Special case (Kratzer potential) for LiH
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Table 3 mass spectra of charmonium with the mass \(m_{c} = 1.209\;{\text{GeV}},\chi = 0.023,\xi = 0.7264,\gamma = 3.9422\,{\text{and}}\,\sigma = 0.00524\)
Full size table
Table 4 mass spectra of bottomonium with the mass \(m_{b} = 4.822\;{\text{GeV}},\;\chi = 0.52,\;\xi = 0.00836,\;\gamma = 3.0884\,{\text{and}}\,\sigma = 0.0123\)
Full size table
Fig. 1
figure 1

Plot of mass spectra of charmonium for different quantum number l

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Fig. 2
figure 2

Plot of mass spectra of bottomonium for different quantum number l

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7 Conclusion

This work breaks down the radial Schrödinger equation (SE) with screened Kratzer potential using the series expansion method and obtains the bound state energy spectra for the potential. Kratzer and Coulomb potentials are obtained as special cases and the numerical values of the special cases for HCl and LiH diatomic molecules for different quantum numbers n and l are computed and the results agree with similar theoretical investigations of the same potential ref. [48] and other theoretical studies Ref. [49]. In addition, with the application of the energy spectra of the screened Kratzer potential, an expression for the mass spectra of heavy quarkonium systems (charmonium and bottomonium) is obtained and the numerical results are computed. The results agree with the experimental values and theoretical studies in previous works. It is also observed that the values numerically improved in comparison with recent works Ref. [50, 51]. and reference there in, and as well. Finally, the mass spectra are plotted showing the variation of the mass spectra with quantum number (n) for different quantum mechanical states l. In recent investigations, some authors have accounted for the effect of spin dependence in quarkonium studied. Worthy of note are the works of Charturvedi and Rai [50] and Kher and Rai [51] and some of the references therein. We therefore recommend that spin should be included in subsequent investigation of quarkonium system.