Polynomial Solution of Non-Central Potentials

Abstract

We show that the exact energy eigenvalues and eigenfunctions of the Schrödinger equation for charged particles moving in certain class of noncentral potentials can be easily calculated analytically in a simple and elegant manner by using Nikiforov and Uvarov (NU) method. We discuss the generalized Coulomb and harmonic oscillator systems. We study the Hartmann Coulomb and the ring-shaped and compound Coulomb plus Aharanov-Bohm potentials as special cases. The results are in exact agreement with other methods.

Any ℓ-state solutions of the Eckart potential via asymptotic iteration method

“…On the other hand it is also known that the Schrödinger equa- * E-mail: fbjames11@physicist.net tion does not admit an exact solution for an arbitrarystate ( = 0). In such cases the centrifugal barrier must be dealt with either by solving numerically or by using approximation schemes [11][12][13][14][15][16][17][18]. The analytical solution of the Schrödinger equation with the Eckart-type potential has been studied by many researchers [1,10].…”

Exact solution of D-dimensional Schrödinger equation generated from certain central power-law potentials

“…Considerable effort has been made in the past several decades to solve the time-independent Schrödinger equation for central multi-term potentials containing negative powers of the radial coordinate in two as well as three dimensions [6,[10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. However, analysis of such problems in arbitrary D dimensions is limited to the best of our knowledge [22,23,26,27].…”

Exact solutions of the D-dimensional Schrödinger equation for a ring-shaped pseudoharmonic potential

“…Recently, the solution was also carried out by using an orthogonal polynomial solution method and also by performing a proper transformation procedure [32]. In addition, the analytical solutions of the D-dimensional radial SE with some diatomic molecular potentials like pseudoharmonic [32] and modified Morse or Kratzer-Fues [33] potential have been solved by selecting a suitable ansatz to the wave function [34][35][36].…”

“…Recently, Chen and Dong [64] found a new ring-shaped (non-central) potential and obtained the exact solution of the SE for the Coulomb potential plus this new ringshaped potential which has possible applications to ringshaped organic molecules like cyclic polyenes and benzene. The complete exact energy bound-state solution and the corresponding wave functions of a class of non-central potentials [65] have been solved recently by means of the Nikiforov-Uvarov (NU) method [65][66][67][68][69][70][71][72]. Very recently, Cheng and Dai [73], proposed a new potential consisting of a modified Kratzer's potential [75] plus the new proposed ring-shaped potential in [64].…”

“…[64,73] are closely relevant to each other as they deal with a Coulombic field interaction except for an additional centrifugal potential barrier acting as a repulsive core which for any arbitrary angular momentum prevents collapse of the system in any dimensional space due to this additional perturbation to the original angular momentum barrier. In very recent works [76][77][78], we have obtained analytically the exact energy eigenvalues and the corresponding normalized wave functions of the SE in D-dimensions with the proposed ring-shaped modified Kratzer potential [76] and the ring-shaped modified Coulomb potential [77,78] using the NU method [65][66][67][68][69][70][71][72][73][75][76][77][78]. The purpose of this work is to solve the SE for the pseudoharmonic plus ring-shaped potential offered in [64].…”