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Communications in Theoretical Physics

Exact Solutions of Schrödinger Equation with Improved Ring-Shaped Non-Spherical Harmonic Oscillator and Coulomb Potential

A. N. Ikot, Ita O. Akpan, Tamunoimi Michael Abbey, HASSAN HASSANABADI

Published May 1, 2016• 25 citations
Publisher

Abstract

Abstract We propose improved ring shaped like potential of the form, V(r, θ) = V(r) + (ħ 2 /2Mr 2 )[(β sin 2 θ + γ cos 2 θ + λ) / sin θ cos θ] 2 and its exact solutions are presented via the Nikiforov–Uvarov method. The angle dependent part V(θ) = (ħ 2 / 2 Mr 2 )[(β sin 2 θ + γ cos 2 θ + λ) / sin θ cos θ] 2 , which is reported for the first time embodied the novel angle dependent (NAD) potential and harmonic novel angle dependent potential (HNAD) as special cases. We discuss in detail the effects of the improved ring shaped like potential on the radial parts of the spherical harmonic and Coulomb potentials.

DOI
10.1088/0253-6102/65/5/569

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Abstract We propose improved ring shaped like potential of the form, V(r, θ) = V(r) + (ħ 2 /2Mr 2 )[(β sin 2 θ + γ cos 2 θ + λ) / sin θ cos θ] 2 and its exact solutions are presented via the Nikiforov–Uvarov method. The angle dependent part V(θ) = (ħ 2 / 2 Mr 2 )[(β sin 2 θ + γ cos 2 θ + λ) / sin θ cos θ] 2 , which is reported for the first time embodied the novel angle dependent (NAD) potential and harmonic novel angle dependent potential (HNAD) as special cases. We discuss in detail the effects of the improved ring shaped like potential on the radial parts of the spherical harmonic and Coulomb potentials.
Most of Schrdinger equations in quantum mechanics haven't analytic solutions except the problems of well with infinite depth,one-dimensional linear harmonic oscillator and Coulomb's potential as well as three-dimensional isotropic harmonic oscillator.These Schrdinger equations include the various equations in the non-spherical ring shaped oscillator potential that haven't analytic solutions. We can only solve these problems approximately in our research activities, but the solving process of these problems is often exceptional complex.The limitation handicaps us to lucubrate the idiographic problems.This paper adopts NU Method(A.F.Nikiforov-V.B.Uvarov Method) to solve the angular equation of the Schrdinger equation in the non-spherical ring shaped oscillator potential V(r,θ)=μω2r2/2+2α/(2μr2)+2βcosθ/(2μr2sin2θ).We decompose the Schrdinger equation in the non-spherical ring shaped oscillator potential step by step until we obtains the analytic solutions of the equation,thus we obtain the analytic solution of the angular eigenfunction,and simplify the process of solutions.Moreover,this paper adopts special functions in the radial equation at the same time to broaden our study in solving the Schrdinger equations.
The general solutions of Schrodinger equation for non central potential are obtained by using Nikiforov Uvarov method. The Schrodinger equation with general non central potential is separated into radial and angular parts and energy eigenvalues and eigenfunctions for these potentials are derived analytically. Non central potential is reduced to Coulomb and Hartmann potential by making special selections, and the obtained solutions are compared with the solutions of Coulomb and Hartmann ring shaped potentials given in literature.
The general solutions of the Schrödinger equation for a non-central potential are obtained by using the Nikiforov–Uvarov method. The Schrödinger equation with general non-central potential is separated into radial and angular parts, and energy eigenvalues and eigenfunctions are derived analytically. By making special selections, the non-central potential is reduced to Coulomb and Hartmann ring-shaped potentials, and the obtained results are compared with the solutions of Coulomb and Hartmann potentials given in the literature.