Advances in High Energy Physics

Volume 2017 (2017), Article ID 9671816, 24 pages

https://doi.org/10.1155/2017/9671816

## Approximate Solutions of Schrodinger Equation with Some Diatomic Molecular Interactions Using Nikiforov-Uvarov Method

^{1}Theoretical Physics Group, Department of Physics, University of Uyo, Uyo, Nigeria^{2}Theoretical Physics Group, Department of Physics, University of Ibadan, Ibadan, Nigeria

Correspondence should be addressed to Ituen B. Okon

Received 7 February 2017; Revised 24 February 2017; Accepted 19 March 2017; Published 30 May 2017

Academic Editor: Saber Zarrinkamar

Copyright © 2017 Ituen B. Okon et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We used a tool of conventional Nikiforov-Uvarov method to determine bound state solutions of Schrodinger equation with quantum interaction potential called Hulthen-Yukawa inversely quadratic potential (HYIQP). We obtained the energy eigenvalues and the total normalized wave function. We employed Hellmann-Feynman Theorem (HFT) to compute expectation values , , , and for four different diatomic molecules: hydrogen molecule (H_{2}), lithium hydride molecule (LiH), hydrogen chloride molecule (HCl), and carbon (II) oxide molecule. The resulting energy equation reduces to three well-known potentials which are as follows: Hulthen potential, Yukawa potential, and inversely quadratic potential. The bound state energies for Hulthen and Yukawa potentials agree with the result reported in existing literature. We obtained the numerical bound state energies of the expectation values by implementing MATLAB algorithm using experimentally determined spectroscopic constant for the different diatomic molecules. We developed mathematica programming to obtain wave function and probability density plots for different orbital angular quantum number.

#### 1. Introduction

The study of diatomic molecules is very significant and applicable in many areas of chemical and physical sciences. The excitation of atoms of some diatomic molecules especially the homonuclear diatomic molecules is the principle used in spectrophotometric technique. Diatomic molecules contains two atoms per molecule and can either be homonuclear if it contains two atoms of the same kind per molecule or be heteronuclear if its contains two atoms of different kind per molecule [1–5]. Bound state solutions of relativistic and nonrelativistic wave equation arouse a lot of interest for decades. Schrodinger wave equations constitute nonrelativistic wave equation while Klein-Gordon and Dirac equations constitute the relativistic wave equations [6–10]. Bound state solutions predominantly have negative energies because the energy of the particle is less than the maximum potential energy [11]. The quantum interaction potential (HYIQP) can be used to compute the bound state energies for both homonuclear and heteronuclear diatomic molecules. Other potentials have been used in studying bound state solutions like the following: Hulthen, Poschl-Teller, Eckart, Coulomb, Hylleraas, pseudoharmonic, and scarf II potentials and many others [6, 12–19]. These potentials are studied with some specific methods and techniques like the following: asymptotic iteration method, Nikiforov-Uvarov method, supersymmetric quantum mechanics approach, formular method, exact quantisation, and many more [19–29]. This article is divided into seven sections. Section 1 is the introduction; Section 2 is the brief introduction of conventional Nikiforov-Uvarov method. In Section 3, we presented the radial solution to Schrodinger wave equation using the proposed potential and obtained both the energy eigenvalue and their corresponding normalized wave function. In Section 4, we have deductions of three well-known potentials from the proposed potential. The numerical results of two of the potentials are compared to that of existing literature. In Section 5, we applied HFT to compute the expectation values for different diatomic molecules. In Section 6, we present numerical results of the expectation values by implementing MATLAB algorithm. Section 7 gives the analytical solutions of finding the normalization constant using confluent hypergeometric function while Section 8 gives the general conclusion of the article.

The proposed quantum interaction potential is given bywhere is the potential depth, is the adjustable parameter known as the screening parameter. , , and are all spectroscopic constants which are molecular bond length, molecular constant, and potential range, respectively.

Figure 1 shows the graph of quantum interaction potential for various values of the screening parameter which decays exponentially. Meanwhile, Figure 2 shows the graph of individual potentials plotted in the same scale with the quantum interaction potential (HYIQP). From this MATLAB plot, it can be seen that HYIQP is best suitable in describing bound state energies of diatomic molecules. In elementary quantum mechanics, the wave function implicitly described the behaviour of quantum mechanical systems. The developed potential model is used to study the behaviour of four diatomic molecules, namely, hydrogen, carbon (II) oxide, lithium hydride, and hydrogen chloride molecules. The wave function and probability density plots, gotten from mathematica programming as seen in Figures 3–8 for wave function plots and Figure 9 for probability density plots for orbital angular quantum numbers , and 5, respectively, is very significant in investigating the behaviour of diatomic molecules. This plot shows that carbon (II) oxide with maximum peak is very flammable and a highly toxic gas though industrially it is a good reducing agent. Lithium hydride possesses similar characteristic being a toxic and poisonous gas. However, the various plot shows that hydrogen chloride and hydrogen molecules are less reactive as compared to lithium hydride and carbon (II) oxide molecules.