Quantum mechanics came into existence many decades ago to salvage the failure of classical mechanics, not being able to explain some of the physical phenomena such as Compton effects, specific heat capacity, blackbody radiation. From this point of view, many theoretical physicists have been investigating the exact and approximate solutions of the Schrödinger equation for some potentials of physical interests ^1,2 . The solutions of the Schrödinger equation play a vital role in many branches of modern physics and chemistry ³ . This is because it contains all the necessary information needed for the full description of a quantum state such as the probability density and entropy of the system ⁴ .

The Schrödinger equation with many physical potentials model have been investigated in recent times with different analytical methods such as Nikiforov-Uvarov (NU) method ^5-8 , asymptotic iteration method (AIM) ^9-14 , supersymmetric quantum mechanics(SUSYQM) ^15-18 among others ^19-22 . One of such potential models is the screened Coulomb potential, which is given by Eq. 1 ²³ .

(1)

The screened Coulomb potential, also known as the Yukawa potential is greatly important, with applications cutting across nuclear Physics and Condensed-matter Physics ²³ . Here, its usage is involved in short-ranged interactions ^24-26 . The screened-Coulomb potential is known to be the potential of a charged particle in a weakly non-ideal plasma. It also describes the charged particle effects in a sea of conduction electrons in solid-state physics ²⁷ .

An approximate solution of the Schrödinger equation interacting with an inversely quadratic Yukawa potential has been obtained using SUSYQM ¹⁵ , where the screened Coulomb potential was obtained as a special case by varying the potential strength. Also, an approximate analytical solution of the radial Schrödinger equation for the screened Coulomb potential has been obtained, with energy eigenvalues and its corresponding eigenfunctions computed in closed forms ²⁸ .

Several researchers have also devoted great attention to investigate the quantum systems of the energy dependence of different potentials ^29-31 . Hassanabadi et al. ^32,33 studied the exact solutions of D-dimensional Schrödinger and Klein-Gordon equations using the Nikiforov-Uvarov method. Also, Lombard et al. ³⁴ investigated the wave equation energy-dependent potential for confined systems. Numerous applications of the energy-dependent potential of wave equations have been seen in the spectrum of confined systems and heavy quark confinement in nuclear and molecular physics ^35,36 . Recently, Budaca ³⁷ studied an energy-dependent Coulomb-like potential within the framework of Bohr Hamiltonian. The author further reported that the energy dependence on the coupling constant of the potential drastically changes the analytical properties of wave function and the corresponding eigenvalues of the system. Also, Boumali and Labidi ³⁸ investigated the Shannon and Fisher information in the Klein-Gordon equation with energy-dependent potential.

In this research, we seek to investigate the influence of the energy-dependent screened

Coulomb potential defined as in Eq. 2,

(2)

where is the energy slope parameter, is the depth of the potential, and is the range of the potential. The effects of the energy dependence on the screened Coulomb potential have not been considered before in any literature, to the best of our knowledge. It can be deduced that when , the potential of Eq. 2 reduces to the Screened Coulomb potential. When g = 0 as a → 0 the potential of Eq. 2 reduces to the Coulomb potential. Using the conventional NU method, we will derive the ℓ-wave bound state solutions and their eigenfunctions of the Schrödinger equation for the energy-dependent screened Coulomb potential, both analytically and numerically. Special cases are also considered and our results are compared with existing literature for confirmation sake.

The organization of this work is as follows: In section 2, we determine the eigensolutions of the energy-dependent screened Coulomb potential by employing a new form of Greene-Aldrich approximation scheme and Nikiforov-Uvarov method. Section 3 is devoted to discuss the results obtained and compare to results in relevant literature. The conclusion of the work is presented in section 4.

The radial part of the Schrödinger equation in D-dimension ³⁹ is given by Eq. 3.

(3)

where μ is the reduced mass, E_nℓ is the non-relativistic energy eigenvalues to be determined. Substituting Eq. 2 into Eq. 3 gives Eq. 4.

(4)

In solving Eq. 4, we invoke a new form of Greene-Aldrich approximation scheme ²³ to deal with the centrifugal term since ℓ ≠ 0. The approximations schemes are given by Eq. 5 and 6.

(5)

(6)

The plots explaining the rationality and validity of the above Eq. 5 and 6 and their expansions are given in Fig. 1(a–d). Figure 1a shows the plots of f ₁(r), f ₂(r) andf ₃(r) as they vary with , when the screening parameter α is taken to be 0.5. In addition, we employ 1/r from f ₁(r) as given in Eq. 7.

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Figure 1. Figure 1. The plots of expressions (a) f1(r), f2(r) and f3(r) as functions of r with α = 0.5; (b) f4(r), f5(r) as functions of r with α = 0.5; (c) f1(α), f2(α), f3(α) as functions of α with r = 5; (d) f4(α), f5(α) as functions of α with r = 5nm.

Here, we have ignored the second expansion term α(1-e ^-2ar )/(12e ^-ar ) when expanding Eq. 6. Furthermore, the rationality and validity of this expansion is given in Fig. 1b, with α = 0.5. in Figs. 1c and 1d, we observe that the approximation f ₃(α) corresponds to that of f ₁(α) when α ≤ 0.05. As such, we see that the approximation f ₃(α) is better than that of f ₂(α).

With the above approximation schemes, Eq. 4 becomes Eq. 8.

(8)

where γ is given by Eq. 9.

(9)

By using the coordinate transformation of Eq. 10.

(10)

Eq. 8 becomes the differential equation of the form given in Eq. 11.

(11)

where ε ² is given by Eq. 12.

(12)

By comparing Eq. 11 and Eq. A1 (see Appendix), we have the following parameters, Eq. 13:

(13)

Substituting Eq. 13 into Eq. A8 (see Appendix), we get ℼ(z), Eq. 14:

(14)

where a, b and c are given by Eq. 15.

(15)

We can obtain the constant k by expressing the discriminant under the square root of Eq. 14 to be equal to zero. As such, we have Eq. 16

(16)

Substituting Eq. 16 into Eq. 14 yields Eq. 17:

(17)

According to NU method ⁴⁰ , we choose the expression ℼ(z)_ which the function τ(z) has a negative derivative. This is given by Eq. 18

(18)

with 𝜏(𝑧) being obtained with Eq. 19.

(19)

By recalling Eq. A9 (see Appendix), we define the constant λ (Eq. 20) as

(20)

Substituting Eq. 20 into Eq. A10 (see Appendix) and carrying out algebraic simplifications, where τ’(z) is given by Eq. 21

(21)

and σ”(z), Eq. 22

(22)

we obtain Eq. 23

(23)

where χ is given by Eq. 24

(24)

Substituting Eq. 12 into Eq. 23 yields a complicated transcendental energy eigenvalue equation of the energy dependent screened Coulomb potential in D-dimensions as (Eq. 25):

(25)

Eq. 25 can also be expressed by Eq. 26.

(26)

where γ and χ are given in Eqs. 9 and 24, respectively.

To obtain the special case, we first rewrite Eq. 26 to the form of Eq. 27:

(27)

As α → 0andg → 0, Eq. 2 reduces to the standard Coulomb potential of the form of Eq. 28.

(28)

Setting the parameters D = 3 as α → 0andg → 0 we obtain the energy eigenvalue equation (Eq. 29):

(29)

This result is very consistent with the result obtained in Eq. 101 of Birkdemir et al. ⁴¹ . Also, taking the natural units (ћ ² = μ = 1) and setting D = 3andg = 0, the energy eigenvalue expression of Eq. 26 can be reduced to Eq. 30.

(30)

The result of Eq. 30 is consistent with the result obtained in Eq. 18 of Dong et al. ²³ .

To obtain the corresponding wave functions, we substitute ℼ(z)_ and σ(z) from Eqs. 18 and 13, respectively into Eq. A4 (see Appendix) and solve the first-order differential equation. This gives Eq. 31.

(31)

The weight function ρ(z) from Eq. A6 (see Appendix) can be obtained Eq. 32

(32)

From the Rodrigues relation of Eq. A5 (see Appendix), we obtain Eq. 33 and Eq. 34.

(33)

(34)

where P_n ^(θ,ϑ) is the Jacobi Polynomial.

Substituting Φ(z)andy_n(z) from Eqs. 31 and 34, respectively into Eq. A2 (see Appendix), we obtain Eq. 35:

(35)

where G is given by Eq. 36.

(36)

From the definition of the Jacobi Polynomials ⁴² results Eq. 37.

(37)

In terms of hypergeometric Polynomials, Eq. 35 can be written as Eq. 38.

(38)

In this study, the energy eigenvalues for the energy-dependent screened Coulomb potential as they vary with the screening parameter were computed for different quantum states as shown in Tab. 1. The existence of the energy slope parameter in Eq. 26 results in two different energy spectra for a particular quantum state and a specific screening parameter. The screening parameter varies in an inverse version with the duo energy spectra. The energy eigenvalues for the energy-dependent screened Coulomb potential as a function of screening parameters in higher dimensions were also computed, as shown in Tabs. 2 and 3, respectively. By employing Eq. 30, we have also computed the energy eigenvalues in three dimensions and in the absence of the energy slope parameter, as shown in Tab. 4. For the two different potential depths considered, we obtained the bound state energy eigenvalues for different quantum states of 2p-4f , as they vary with different screening parameters. It can be seen that the energy eigenvalues for the different potential depths decrease as the screening parameter increases, at each quantum state. Our analytical result of Eq. 30 and its corresponding numerical results of Tab. 4 are very consistent with the results obtained by Dong et al. ²³ . We have also computed the energy eigenvalues of Eq. 26 for higher dimensions, as shown in Tabs. 5 and 6, respectively. The level of effects imposed by the potential parameters of Eq. 2 on the energy eigenvalues of Eq. 26 are shown in Figs. 2-7. For different dimensions, the energy slope parameter causes an interwoven interaction of the curves, as compared to the situation when the energy slope parameter is zero. Figs. 8 and 9 show the variation of energy eigenvalues for the energy-dependent screened Coulomb potential with the different quantum numbers, for different values energy slope parameter considered. The trend of the relationship shows that the energy eigenvalues decrease as the quantum numbers increase. In Fig. 10, we also plotted the variation of the energy eigenvalues with the energy slope parameter for different dimensions.

Table 1. Table 1. Eigenvalues (Enℓ) for the energy-dependent screened Coulomb potential as a function of the parameter α for 2p-4f states, with D = 3.

Table 2. Table 2. Eigenvalues (Enℓ) for the energy-dependent screened Coulomb potential as a function of the parameter α for 2p-4f states, with D = 4.

Table 3. Table 3. Eigenvalues (Enℓ) for the energy-dependent screened Coulomb potential as a function of the parameter α for 2p-4f states, with D = 5

Table 4. Table 4. Eigenvalues (Enℓ) of Eq. 29 as a function of the parameter for 2p-4f states, with D = 3andg = 0.

Table 5. Table 5. Eigenvalues (Enℓ) of Eq. 25 as a function of the parameter for 2p-4f states, with D = 4andg = 0.

Table 6. Table 6. Eigenvalues (Enℓ) of Eq. 26 as a function of the parameter for 2p-4f states, with D = 5andg = 0.

Figure 2. Figure 2. Energy Eigenvalue Variation with α for different dimensions with g = 0,A = 5,n = 2,ℓ = 1.

Figure 3. Figure 3. Energy Eigenvalue variation with α for different dimensions with g = -1,A = 5,n = 2,ℓ = 1 .

Figure 4. Figure 4. Energy Eigenvalue Variation with α for different dimensions with g = 1,A = 5,n = 2,ℓ = 1 .

Figure 5. Figure 5. Energy Eigenvalue Variation with A for different dimensions with g = 0,α = 100,n = 2,ℓ = 1 .

Figure 6. Figure 6. Energy Eigenvalue Variation with A for different dimensions with g = -1,α = 100,n = 2,ℓ = 1.

Figure 7. Figure 7. Energy Eigenvalue Variation with A for different dimensions with g = 1,α = 100,n = 2,ℓ = 1 .

Figure 8. Figure 8. Energy Eigenvalue Variation with ℓ for different values of g with D = 3,α = 1,A = 5,n = 0.

Figure 9. Figure 9. Energy Eigenvalue Variation with n for different values of g with D = 3,α = 1,A = 5,ℓ = 1 .

Figure 10. Figure 10. Energy Eigenvalue Variation with g for different dimensions, with α = 10,A = 100,n = 2,ℓ = 1.

We have solved the Schrödinger equation with the energy-dependent screened Coulomb potential in higher dimensions. With the use of the conventional Nikiforov-Uvarov method and a new form of Greene-Aldrich approximation, the energy eigenvalues and its corresponding eigenfunctions were obtained approximately. Numerical values of the energy eigenvalues were also obtained with natural units, in three dimensions. We have also elucidated the variation of these energy spectra with different potential parameters and the energy slope parameter. The effect of the energy slope parameter is clearly seen in the existence of duo energy spectra, as compared to when the energy slope parameter is diminished. Special cases have been deduced, and these results agree perfectly with available literature.