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Taş Ahmet, Havare Ali. Bound states resulting from interaction of the non-relativistic particles with the multiparameter potential. Chinese Physics B, 2017, 26(10): 100301  
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Bound states resulting from interaction of the non-relativistic particles with the multiparameter potential
Taş Ahmet †, Havare Ali
Mersin University, Department of Physics, Mersin 33343, Turkey

 

† Corresponding author. E-mail: aahmet.tas@gmail.com

Abstract

In this study, we present the analytical solutions of bound states for the Schrödinger equation with the multiparameter potential containing the different types of physical potentials via the asymptotic iteration method by applying the Pekeris-type approximation to the centrifugal potential. For any n and l (states) quantum numbers, we derive the relation that gives the energy eigenvalues for the bound states numerically and the corresponding normalized eigenfunctions. We also plot some graphics in order to investigate effects of the multiparameter potential parameters on the energy eigenvalues. Furthermore, we compare our results with the ones obtained in previous works and it is seen that our numerical results are in good agreement with the literature.

PACS: 03.65.Nk;03.65.Ge;02.30.Gp;21.60.-n
Keyword:multiparameter potential;Schrödinger equation;bound states;asymptotic iteration method
1. Introduction

One of the most important tasks discussed in physics is to obtain the exact analytical solutions, which describe the bound and scattering states, of radial Schrödinger equation with arbitrary angular quantum numbers (l = 0 and ) for various potentials.[1–16] For the s-wave states (the case of l = 0), the analytical solutions for the bound states can be achieved directly. On the other hand, for the l-wave states (the case of ), it is necessary to use a convenient approximation to the centrifugal term () such as the Pekeris approximation[17] and the approximation scheme proposed by Greene and Aldrich.[18] In order to obtain the relation for the energy eigenvalues which is used to get numerical results of the bound states, one needs to solve the second order differential equations. There are many solution methods of the differential equations. Some of these are the standard method,[19–21] the Nikiforov–Uvarov method (NU),[22–24] the supersymmetric shape invariance method (SUSY),[25–29] and the asymptotic iteration method (AIM).[30,31]

Some of the potentials can be generalized to explain the interactions consisting of more than one process.[32–36] Therefore, in our study, we introduce the multiparameter potential which can be used to describe the interactions in the nuclei and structures of the diatomic and polyatomic molecules, in the following form:

All of the parameters in the potential are real. The shape of the potential changes according to values of the parameters. In special cases, the multiparameter potential is reduced to potentials that have many applications in the relativistic and non-relativistic quantum mechanics such as the Manning–Rosen,[37] Hulthen,[38] Eckart-type,[39] Rosen–Morse,[40] Woods–Saxon,[41] Morse,[42] Kratzer–Fues,[43] q-parameter hyperbolic Pöschl–Teller,[44] and Yukawa[45] potentials.

The content of this study is regulated as follows: in the next section, the asymptotic iteration method is presented. In Section 3, analytical solutions for the bound states of the radial Schrödinger equation with the multiparameter potential is obtained within the framework AIM. Also in same section, for any n and l quantum numbers, the relation for energy eigenvalues is derived in a closed form and normalized eigenfunctions are obtained in terms of the hypergeometric functions by using the normalization condition of the wave functions. The special potentials which are obtained from the multiparameter potential are acquainted in Section 4 and calculated the energy eigenvalues of bound state energy eigenvalues for these potentials numerically. Finally, we discuss the results in Section 5.

2. Asymptotic iteration method

The asymptotic iteration method has been suggested to solve the homogenous linear second-order differential equation which must be in the following form[46]

where . The and are sufficiently differentiable with respect to x. To obtain a general solution for this equation, the following procedure is supposed.

We differentiate Eq. (2) with respect to x, we obtain

where
The second derivative of Eq. (2) becomes
where
Finally, by taking (k + 1)-th and (k + 2)-th derivative of Eq. (2), , we find the following equations respectively
where

Therefore, from Eq. (7) we can obtain the following relation

For sufficiently large value of k, if
which is the “asymptotic” aspect of the method, in that case, equation (9) reduces to
which leads to the general solution of Eq. (2) as
where C1 and C2 are integration constants. For any given potential, the radial part of the Schrödinger equation is transformed into the form of Eq. (2). Then, the and coefficients are found and and parameters are calculated by using recurrence relations defined by Eq. (8). Then, taking into account the condition given by Eq. (10), the energy eigenvalues are obtained if the problem is the exactly solvable. Equation (10) can be expressed in the following manner:

We want to obtain the exact solutions of the radial Schrödinger equation for which the relevant second order homogenous linear differential equation takes the following general form[47]

The exact solutions for this equation are given by
where
The energy eigenfunctions can be obtained by using Eq. (14) directly.

3. Bound state solutions of the Schrödinger equation: Calculation of the energy eigenvalues and eigenfunctions

The Schrödinger equation in spherical coordinate for a particle with energy E moving in a external potential is written as the following form (in natural units ):

If the wave function is selected as
and the necessary calculations are done, the radial part of the Schrödinger equaiton is obtained as the following form
Substituting Eq. (1) into Eq. (18) gives,

To find the analytical solutions of this equation we must use the following approximation which is called the Pekeris approximation.[48] This approximation is applied to the centrifugal term

where C0, C1 and C2 are constants and their values are calculated by expanding the terms in Eq. (20) as follows:
The terms in Eq. (20) are expanded in a series about because the nuclear distance r does not fluctuate very far from the equilibrium position at L.[49] Therefore the approximation in Eq. (20) gives good result only for . Substituting Eq. (20) into Eq. (19), we obtain
where the parameters ϵ1, ϵ2, ϵ3, ϵ4, and ϵ5 are defined as follows:
If we use a new variable in Eq. (22) we obtain the following equation:
where

Note that equation (24) has second degree singular pole points at y = 0 and y = 1. In order to remove these singularities and get a solution for Eq. (24) with AIM, we write the reasonable physical wave functions as follows:

with boundary conditions
We use the approximation here in order to ensure the boundary conditions of wave functions. If , and then . By substituting Eq. (26) into Eq. (24), we obtain the second-order homogeneous equation which is compatible with Eq. (2)
Now we can use the AIM to solve this equation, comparing Eq. (27) with Eq. (2) and using the recurrence relations which are defined in Eq. (8), one obtains the values of λn and Sn as follows:
If we write the above equations into the condition defined in Eq. (13), we acquire
with the generalization of these relations

Then, by using notations (25), the energy eigenvalues of the Schrödinger equation with the multiparameter potential is obtained as follows:

By using this relation, the energy eigenvalues can be calculated for any n and l quantum numbers. The energy values obtained for some values given in the parameters are presented in Table 1.

Table 1.

The bound state energy eigenvalues for the multiparameter potential for , , , , p = 1.5, q = 3.5, , and in atomic units .

.

Let us now obtain the wave functions. By comparing Eq. (27) with Eq. (14), we obtain

where and . Then, the wave function () yields
where
By using Eq. (26), Eq. (33) and Eq. (34), the total radial wave functions of the Schrödinger equation with the multiparameter potential can be written as the following form
where is the Gauss hypergeometric function and Cnl is the normalization constant and it can be calculated by using the normalization condition of the wave function. Then, we obtain
If we replace Eq. (35) with the above equation, we obtain
The integral in Eq. (37) is calculated for different n values using the Mathematica software program for and as follows:
Therefore, we achieve

4. Numerical results for the special cases

The potentials that have many applications in the quantum mechanics can be obtained from the multiparameter potential under the certain conditions. For some of these potentials, the energy eigenvalues are calculated and compared with data obtained in earlier studies. On the other hand, there are no numerical results in the literature for some potentials, so the numerical values given to the parameters are chosen in this study and the energy eigenvalues obtained cannot been compared.

5. Conclusion

In this paper, we have obtained the analytical solutions of the bound state for the Schrödinger equation with the multiparameter potential via the AIM. In order to solve the radial Schrödinger equation for , the Pekeris-type approximation has been applied to the centrifugal potential. Taking into account the boundary conditions of the wave function, the relation obtained for the energy eigenvalues has been derived and by using the normalization condition for the wave functions, we have calculated the normalization constant in terms of the gamma functions. By giving appropriate values to the parameters, the energy eigenvalues for the bound states have been calculated for both l = 0 and states numerically and are given in Table 1.

Furthermore, the effects of potential parameters on energy have been explored. The variation Enl with respect to b, L, p, q, A, B, C, and D parameters are presented in Figs. 1–4 for 1s, 2p, and 3d states. It is seen from these figures that the energy values Enl decrease as L, B, and C increase and increase as b, p, q, A, and D increase.

Figure 1. The effect of parameters b and L on the energy eigenvalues. Figure 1(a) shows Enl varying with b for , , , , , , and fm. Figure 1(b) displays Enl varying with L for , , , , , , and in atomic units . The 1s, 2p, and 3d cases are described with black line, dashed line, and dot-dashed line, respectively.
Figure 2. The effect of parameters p and q on the energy eigenvalues. Figure 2(a) shows Enl varying with p for , , , , , , and fm. Figure 2(a) displays Enl varying with q for , , , , , , and fm in atomic units . The 1s, 2p, and 3d states are described with black line, dashed line, and the dot-dashed line, respectively.
Figure 3. The effect of parameters A and B on the energy eigenvalues. Figure 3(a) shows Enl varying with A for , , , , , , and fm. Figure 3(b) displays Enl varying with B for , , , , , , and fm in atomic units . The 1s, 2p, and 3d states are described with black line, dashed line, and the dot-dashed line, respectively.
Figure 4. The effect of parameters C and D on the energy eigenvalues. Figure 4(a) shows Enl varying with C for , , , , , , and fm. Figure 4(b) displays Enl varying with D for , , , , , and fm. in atomic units . The 1s, 2p and 3d states are described with black line, dashed line, and dot-dashed line, respectively.

The graphics given in Fig. 5 shows that how the change the radial bound state wave functions and the radial probability density for different orbitals according to distance from nucleus r. First, we have investigated the relationship between the radial quantum numbers nr and the radial node points which are defined as points where the electron density is zero. As seen in the figures, the radial wave functions of states 1s, 2p, and 3d have 1, 2, and 3 nodes, respectively and the distances from the nucleus of these node points are different from each other. Another consequence of observed is that the radial wave functions of the bound states are the relationship between amplitudes of the wave function and the corresponding energy. When the energy values of the states in Fig. 5 are compared, it is seen that and also one can see in Fig. 5 that there is the similar relationship between the amplitudes of the states as (where the A denotes amplitude). On the other hand, we have explored to the radial probability density functions of states. When looking at the graphs on the right side of Fig. 5, it is seen that different peaks are occurred. The largest peak is called the probability density at the maximum away from the nucleus, while the low peaks are defined as the probability densities for the penetration distance. When we compare the graphics in Fig. 5, we see that the maximum distances from the nucleus are fm ( m), fm, and fm for 1s, 2p, and 3d states, respectively.

Figure 5. Variation of the bound states radial wave function and radial probability density with respect to distance from nucleus r for , , , , , , , and in atomic units . The orbitals energy are , , and for 1s, 2p, and 3d, respectively.

Finally, we have studied the special potentials, which are defined in the literature, have been obtained from the multiparameter potential. For these potentials, which are the Manning–Rosen, Hulthen, Eckart-type, Rosen–Morse, Woods–Saxon, Morse, Kratzer–Fues, q-parameter hyperbolic Pöschl–Teller, and Yukawa potentials, the energy eigenvalues for the bound state have been obtained. Also the numerical results have been computed and given in Tables 2–5. In addition to these, to show the accuracy of our results, the numerical energy eigenvalues for these potentials have been compared with the results obtained in previous studies. One can clearly see from Tables 2–4 that the results are in good agreement with other results in the literature.

Surely this paper will be a good reference in the literature due to the fact that the multiparameter potential model describes both core and molecular structures unlike the other potential models. So, the results obtained in this work can be applied to different research areas. For instance, by using the energy eigenvalue formula defined in Eq. (31), one can explore thermodynamic and optical properties of the atoms and molecules.[66–75]

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