Analytical approximations to thel-wave solutions of the Schrödinger equation with the Eckart potential

Abstract

The bound-state solutions of the Schrödinger equation with the Eckart potential with the centrifugal term are obtained approximately. It is shown that the solutions can be expressed in terms of the generalized hypergeometric functions 2 F 1 (a, b; c; z). The intractable normalized wavefunctions are also derived. To show the accuracy of our results, we calculate the eigenvalues numerically for arbitrary quantum numbers n and l. It is found that the results are in good agreement with those obtained by other methods for short-range potential (large a). Two special cases for l = 0 and β = 0 are also studied briefly.

Bound States of the Klein-Gordon for Exponential-Type Potentials in D-Dimensions

“…Recently, the study of exponential-type potentials has attracted much attention from many authors (for example, cf, ). These physical potentials include the Woods-Saxon [7,8], Hulthén [9][10][11][12][13][14][15][16][17][18][19][20][21][22], modified hyperbolic-type [23], ManningRosen [24][25][26][27][28][29][30][31], the Eckart [32][33][34][35][36][37], the Pöschl-Teller [38] and the Rosen-Morse [39,40] potentials.…”

“…The bound state solutions of the s-wave Klein-Gordon (KG) equation with equally mixed Rosen-Morse-type (Eckart and Rosen--Morse well) potentials have been studied [52]. The bound state solutions of the s-wave Dirac equation with equal vector and scalar Eckart-type potentials in terms of the basic concepts of the shape-invariance approach in the SUSYQM have also been studied [34][35][36][37]. The spin symmetry and pseudospin symmetry in the relativistic Eckart potential have been investigated by solving the Dirac equation for mixed potentials [38].…”

“…Unfortunately, the wave equations for the Eckart-type potential can only be solved analytically for zero angular momentum states because of the centrifugal potential term. Some authors [32][33][34][35][36][37][38] studied the analytical approximations to the bound state solutions of the Schrödinger equation with Eckart potential by using the usual existing approximation scheme proposed by Greene and Aldrich [53] for the centrifugal potential term. This approximation has also been used to study analytically the arbitrary l -wave scattering state solutions of the Schrödinger equation for the Eckart potential [54,55].…”

Energy States of Some Diatomaic Molecules: The Exact Quantisation Rule Approach

“…Few of these methods include the Feynman integral formalism [5,16], asymptotic iteration method (AIM) [1][2][3][4][5][17][18][19][20][21], functional analysis approach [22][23][24], exact quantization rule method [25][26][27][28][29][30][31][32][33], proper quantization rule [27,34], Nikiforov-Uvarov (NU) method [35][36][37][38], supersymetric quantum mechanics [40,[40][41][42][43][44][45], etc.…”

Closed Analytical Solutions of the D-Dimensional Schrödinger Equation with Deformed Woods–Saxon Potential Plus Double Ring-Shaped Potential

“…Recently, the asymptotic iteration method (AIM) [15][16][17] an elegant, efficient technique to solve second-order homogeneous differential equations, has been the subject of extensive investigation in recent years, particularly when dealing withe non central potential. The Schrödinger equation has been investigated for several potentials as the Woods-Saxon potential [18][19][20], harmonic oscillator potential [21], Hulthén potential [22][23][24][25], Kratzer potential [26], generalized q-deformed Morse potential [27], modifed Woods-Saxon potential [28], Makarov potential [29], deformed Woods-Saxon Potential [30], Pseudoharmonic potential [31,32], Yukawa potential [33,34] and Eckart potential [35,36]. Very recently, the Schrödinger equation in generalized D dimensions for different potentials is getting more attention with the aim of generalizing the solutions to multidimensional space for many potentials [37][38][39][40][41][42][43][44][45][46].…”