Eigensolutions of the Schrödinger equation with a class of Yukawa potentials via supersymmetric approach

Using the basic concept of the supersymmetric shape invariance approach and formalism, we obtained an approximate solution of the Schrödinger equation with an interaction of inversely quadratic Yukawa potential, Yukawa potential and Coulomb potential which we considered as a class of Yukawa potentials. By varying the potential strengths, we obtained a solution for Hellmann potential, Yukawa potential, Coulomb potential and inversely quadratic Yukawa potential. The numerical results we obtained show that the interaction of these potentials is equivalent to each of the potential.


Introduction
In the recent time, physicists have developed much interest in searching for the exponential-type potentials. The reason is that most of the exponential-type potentials play an important role in physics, e.g. Yukawa potential is used in plasma, solid-state and atomic physics [1]. As a result, many authors have solved both relativistic and non-relativistic wave equations with these potentials. For instance, Zhang et al. [2] obtained approximate solutions of the Schrödinger equation with the Generalized Morse potential model including a centrifugal term. Jia et al. [3] solved sixparameter exponential-type potentials. Onate [4] obtained relativistic and non-relativistic solutions of the inversely quadratic Yukawa potential. Falaye et al. [5] obtained bound state solutions of the Schrödinger equation with Manning-Rosen potential. Hassanabadi et al. [6,7] obtained Actual and general Manning-Rosen potential under spin and pseudospin symmetries of the Dirac equation; approximate solutions of the Schrödinger equation under Manning-Rosen potential in arbitrary dimensions via SUSY QM. Hamzavi et al. [8] obtained approximate spin and pseudospin solutions of the Dirac equation for inversely quadratic Yukawa potential and tensor interaction. Maghsoodi et al. [1] solved Dirac particles in the presence of Yukawa potential plus a tensor interaction in SUSY QM frame work. Ikhdair [9] obtained on the bound state solutions of the Manning-Rosen potential including an improved approximation to the orbital centrifugal term.
Motivated by the success in the exponential-type potentials, we attempt to investigate the solutions of the radial Schrödinger equation with a class of Yukawa potentials given as V r ð Þ ¼ Àbr þ rce Àdr À ae À2dr where a, b and c are potential strength and d is the screening parameter. The potential is obtained by the addition of Hellmann potential and inversely quadratic & C. A. Onate oaclems14@physicist.net; onateca12@gm.us Yukawa potential which is equivalent to the interaction of inversely quadratic Yukawa potential, Coulomb potential and Yukawa potential. This potential has its application where its components are useful. It is noted that the exact solution of the radial Schrödinger equation with potential (1) is not possible due to the presence of the inverse square term. Therefore, to obtain an approximate solutions, we employ a suitable approximation scheme. It is found that such approximation proposed by Greene and Aldrich [29] 1 is a good approximation to the centrifugal/inverse square term which is valid for d ( 1 for a short potential range. Our work is arranged as follows: in the following section, we obtain the bound state solutions. We present discussion and concluding remarks at the end of the article.

Bound state solutions
To study any quantum physical system, we solve the original Schrödinger equation given as [9,30,31] where E n;' is the energy, l is the particle mass and V(r) is the potential. Setting the wave functions w n;';m r ð Þ ¼ R n;' r ð ÞY ';m h;u ð Þ r , we obtained the following radial Schrödinger equation With Eqs. (1) and (2), (4) becomes where Since we are dealing with a Schrödinger-like equation that we solved by means of SUSY QM [6,[32][33][34][35], the first step in the SUSY approach is finding the solution of the Riccati equation [6]. Using the shape invariance formalism, it can be easily seen that whose solution is given as The ground state wave function R 0;' ðrÞ as where W(r) is called the superpotential function in supersymmetric quantum mechanics [36][37][38]. Substitute Eq. (7) into Eq. (6), we deduce the following relations With Eq. (7), we can construct the two partner potentials as follows: Using the shape invariance technique [6,[39][40][41][42], it can readily be shown that the two partner potentials are shape invariant. Therefore, their relationship is written as where a 0 is an old set of parameters in which the new set of parameters a 1 is obtained from and T(a 1 ) is a remainder that is independent of the variable r. Here, B = a 0 via mapping of the form a 1 ? a 0 -d, a 2 ? a 0 -2d, a 3 ? a 0 -3d. Thus, a generalization is drawn as a n ? a 0nd. In terms of the parameters of the problem, we obtain the following relations: The energy spectral can then be determine as follows: This gives energy equation as Wave function To obtain the un-normalized wave function, we define a variable of the form y ¼ e Àdr and substitute it into Eq. (5) to have Analyzing the asymptotic behavior of Eq. (22) at origin and at infinity, it can be tested that when r ? 0(y ? 1) and when r ? ?(y ? 0) Eq. (22) has a solution R n;' ðyÞ ¼ y where Now, taking a trial wave function of the form R n;' ðyÞ ¼ y g 1 À y ð Þ e f ðyÞ and substitute it into Eq. (22), we have f 00 ðyÞ þ f 0 ðyÞ 2g þ 1 À yð2g þ 2e þ 1Þ yð1 À yÞ þ f ðyÞ g þ e ð Þ 2 þN yð1 À yÞ ¼ 0: ð27Þ Equation (27) is a differential equation satisfied by the hypergeometric function. Thus, its solution is obtained as: Replacing the function f(y) with the hypergeometric function and write a complete radial wave function as: Results and discussion Some special cases of interest are studied here, when a = 0, our potential (1) reduces to Hellmann potential which reduces the energy Eq. (21) to When a = c = 0, the potential (1) reduces to Coulomb potential and the energy Eq. (21) turns to When a = b = 0, the potential (1) When b = c = 0, potential (1) reduces to inversely quadratic Yukawa potential V r ð Þ ¼ Àae À2dr r 2 ; ð36Þ and the energy Eq. (21) turns to In Table 1, we numerically reported the energy eigenvalues for a class of Yukawa potential 2p, 3p, 3d, 4p, 4d and 4f. In Table 2, we have reported the energy eigenvalues of these potentials for n ¼ 0; 2 ' ¼ 0; n ¼ 1; ' ¼ 0; 1; n ¼ 2; ' ¼ 0; 1; 2; n ¼ 3; 2 ' ¼ 0; 1; 2; 3 and n ¼ 4; ' ¼ 0; 1; 2; 3; 4: In Table 1, energy increases as the screening parameter increases. In Table 2, energy increases as n increases. In Table 3, we compared our result for the Hellmann potential with the result from two other methods.

Conclusions
We have obtained approximate solutions of the Schrödinger equation by combining inversely quadratic Yukawa potential, Yukawa potential and Coulomb potential. We deduced that from the energy equation of these combined potentials, the energy equation of Hellmann potential as well as these individual potential can be obtained. In Tables 1 and 2, we have numerically reported the equivalence of the energy of these potentials.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creative commons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.