Application of formula method for bound state problems in Schrödinger equation

Abstract

By using Formula method for bound state problems, we have obtained the approximate analytical solution of the radial Schrӧdinger equation for Varshni potential model. The energy eigenvalues and the corresponding eigenfunctions are calculated in closed forms. In the process, eigen values for 1, 1, 2, a b a b = = = − = − and 1, 2 b a = − = − for 2 , 3 ,3 , 4 p p d p and 4 f states with three different values of the potential range were generated using the Varshni potential. Special cases of the potential are equally studied. Some numerical results have been presented which showed a good agreement with previous results obtained from another method.The study of exactly solvable problem has attracted much attention in theoretical physics since the development of quantum mechanics. For instance, the solutions of the non-relativistic Schrӧdinger equation for a hydrogen atom and a harmonic oscillator in three dimensions represent two typical examples in quantum mechanics (Schiff, 1955;Landau and Lifshitz, 1977). There are few traditional techniques or methods used to solve second-order differential equation in order to obtain the solutions of the quantum systems. These methods include Nikiforov-Uvarov method (Ebomwonyi et al., 2017; Ikot et al., 2015; Onate and Idiodi, 2015; Hamzavi and Ikhdair, 2012), asymptotic iteration method (Bayrak et al., 2007; Falaye et al., 2013; Bayrak and Boztosun, 2007; Ateser et al., 2007), supersymmetry shape invariance and solvable potential (Zarrinkamar et al., 2010; Jia et al., 2011; Hassanabadi, 2011; Onate et al., 2017), factorization method (Onyeaju et al., 2016), new exact quantization rule (Dong and Gonzalez-Cisneros, 2008; Gu et al., 2009) and others. Recently, Falaye et al. (2014) developed the Formula method for bound state problems from asymptotic iteration method. This method seems to be very effective, but the simplification in obtaining the energy equation is somehow technical; hence, resulting in little report on it in literature. This necessitates the present study; and in this paper, we intend to investigate the nonrelativistic Schrödinger equation with the Varshni potential function in the framework of the Formula method for bound state problems. To the best of our knowledge, this is the first time the Schrӧdinger equation is being studied with the Varshni potential function using the Formula method.The Varshni potential function is given as (Varshni, 1957;Oluwadare and Oyewumi, 2017) ( ) , r ab V a r e r δ − = −(1) Where, a and b are the potential strengths, δ is the screening parameter which controls the shape of the potential energy curve and r is the internuclear separation. The Varshni potential function is a short range repulsive potential energy function that plays an important role in both chemical and molecular physics (Varshni, 1957). The potential was used by Kaxiras and Pandey to describe the 2-body energy portion of multi-body condensed matter. This potential is used generally to describe bound states of the interaction of systems and has been applied in both...

Eigensolutions of the N-dimensional Schrödinger equation interacting with Varshni-Hulthén potential model

“…The numerical energy eigenvalues of Eqs. ( 34) and ( 35) for as presented in Table III, were computed for three different potential strengths; for 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d and 4f states and were found to be in conformity when compared with the work of Ebonwonyi et al [41] using formula method.…”

Nonrelativistic treatment of inversely quadratic Hellmann-Kratzer potential and thermodynamic properties

“…The method is gaining popularity because it is considered simple, accurate, and easy in applications. Recently, some authors have applied this method in various studies [34,41,42]. We consider the differential equation [43] of the form…”

Analyzing the effects of magnetic and Aharonov‐Bohm (AB) flux fields on the energy spectra and thermal properties of N2, NO, CO and H2 diatomic molecules

“…One of the potential that is significant in the history of molecular physics and quantum chemistry, and in describing molecular structure and molecules is Varshni potential [17]. It is a short‐range repulsive potential energy function that has been used successfully in describing the interaction of the bound state of systems in chemical, classical, modern, and molecular physics [18, 19]. The Varshni potential takes the form where are the potential strengths of Varshni potential, is the screening parameter and is the inter‐nuclear distance.…”