1. Introduction

One of the interesting problems in quantum mechanics is to get exact solutions of the Schrӧdinger equation. To do this, a real potential is often selected to serve as the driving force of the energy eigenvalues and the eigenfunctions of the Schrӧdinger equation^1-3. These state solutions reveal the particle dynamics in non-relativistic quantum mechanics². Numerous researchers have investigated the bound states of the Schrӧdinger equation using variety of potentials and quantum formalism. Some of these potentials play critical roles in many fields of Physics such as Molecular Physics, Solid State and Chemical Physics⁴. The Manning-Rosen potential has been studied in-depth and have also been utilized in quantum systems and Yukawa potential, and its classes have been studied in Schrӧdinger formalism^5,6.

In this work, using the Wentzel, Kramers and Brillouin (WKB) quantum approximation, we shall investigate the bound state solutions of the Schrӧdinger equation using a combination of potentials known as the Inversely Quadratic Yukawa/Attractive Coulomb potential plus a Modified Kratzer potential (IQYCKFP).

2. The WKB Theoretical Approximation

In this section, we consider the quasiclassical solution of the Schrӧdinger’s equation for the spherically symmetric potentials. Given the Schrӧdinger equation for a spherically symmetric potentials of Equation 3 as

(1)

The total wave function in Equation 3 can be defined as

(2)

And by decomposing the spherical wave function in Equation 1 using Equation 2 we obtain the following equations:

(3)

(4)

(5)

where M², 𝑀_𝑧² are the constants of separation and, at the same time, integrals of motion. The squared angular momentum M² = (𝑙 + 1/2 )² ђ² .

Considering Equation 6, the leading order WKB quantization condition appropriate to Equation 3 is

(6)

where r₂ & r₁ are the classical turning point known as the roots of the equation

(7)

Equation 9 is the WKB quantization condition which is subject for discussion in the preceding section. Consider Equations 5-7 in the framework of the quasi-classical method, the solution of each of these equations in the leading ђ approximation can be written in the form

(8)

3. Solutions of the Schrödinger Equation

The Wentzel, Kramers and Brillouin surmise has been of tremendous importance to physicist, chemist, mathematician as regards quantum mechanics in view of the fact that it gives approximate solutions to linear differential equations. The inversely quadratic Yukawa/attractive Coulomb plus Kratzer Fues potential can be expressed thus

(9)

The sum of these potentials can be written as

(10)

(11)

(12a)

Equation 12a stands for the classical formula for momentum.

(12b)

Upon, substituting Equations 11 and 12a into 12b i.e. the (WKB) we have

(13)

(14)

(15)

(16)

(16b)

Upon substituting the representations made into Equation 16 we have

(17)

Factoring out root of A, we have

(18)

x represent M/Ãand y N/Ã

(19)

(20)

Where we obtain the classical turning points r_a and r_b from the terms inside the square roots as;

(21)

Recall

(22)

(23)

(24)

Upon substituting the coefficients of M, N, Ã intoEquation 24 to obtain the energy eigenvalue.

(25a)

(25b)

The above equation results in the bound state energy spectrum with respect to quantum numbers of a vibrating-rotating diatomic molecule subject to the (IQYCKFP) potential. Thus, its corresponding wave function is given as

(26)

4. Discussion

Having obtained the Energy Eigen Value and its corresponding (ψ) using the WKB approach for the Schrödinger equation with the (IQYCKFP), we understood that if we set up parameters.

(27)

5. Conclusions

It is much easy to show that Equation 19 has resulted to a bound state energy spectrum of a vibrating rotating diatomic molecule subject to the inversely quadratic Yukawa plus attractive coulomb potential.

Similarly, if D_e different 0, V₀ = 0 and A different Z_e² = 0

(28)

Equation 28 results to a bound state energy spectrum subject to Kratzer Fues potential.

Acknowledgments

6. Acknowledgements

The authors are thankful to Professor Benedict I. Ita for the scientific encouragement leading to the success of this manuscript.

7. References

[1] Antia, A. D, Essien, I. E., Umoren, E. B., Eze, C. C., Approximate solution of the non-relativistic Schrödinger equation with inversely quadratic Yukawa plus Mobius square potential via parametric Nikiforov-Uvarov method, advances in Physics theories and application, 44 (2015) 1-13. https://iiste.org/Journals/index.php/APTA/article/view/23029/23549.

[2]Ita, B. I., Louis, H., Nzeata-Ibe, N., Ikeuba, A., Ozioma, A. U., Thomas, M. O., Pigweh, A. I., Michael, M. O., Approximate 𝒍-states solutions to the Schrödinger equation with Manning-Rosen plus Hellmann potential via WKB approximation scheme, Sri Lankan Journal of Physics 19 (1) (2018) 37-45. https://doi.org/10.4038/sljp.v19i1.8050.

[3] Onate, C. A., Adebimpe, O., Lukman, A. F., Adama, I. J., Okoro, J. O., Davids, E. O., Approximate eigensolutions of the attractive potential via parametric Nikiforov-Uvarov method, Heliyon 4 (11) (2018). https://doi.org/10.1016/j.heliyon.2018.e00977.

[4] Hitler, L., Iserom, I. B., Tchoua, P., Ettah, A. A., Bound state solutions of the Klein-Gordon equation for the more general exponential screened Coulomb potential plus Yukawa (MGESCY) potential using Nikiforov-Uvarov method, J. Phys. Math. 9 (1) (2018). https://doi.org/10.4172/2090-0902.1000261.

[5] Ikot, A. N., Hassanabadi, H., Maghsoodi, E., Zarrinkamar, S., Relativistic symmetries of Hulthén potential incorporated with generalized tensor interactions, advances in high energy physics 2013 (2013) 1-10. https://doi.org/10.1155/2013/910419.

[6] Ita, B. I., Louis, H., Akakuru, O. U., Magu, T. O., Joseph, I., Tchoua, P., Amos, P. I., Effiong, I., Nzeata, N. A., Bound state solutions of the Schrödinger equation for the more general exponential screened Coulomb potential plus Yukawa (MGESCY) potential using Nikiforov-Uvarov method, Journal of Quantum Information Science 8 (1) (2018) 24-45. https://doi.org/10.4236/jqis.2018.81003.

Notas de autor

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