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Journal of Southwest China Normal University

Exact Solutions of Bound States of the Schrdinger Equation with the Ring-Shaped Hartmann Potential

Zidong Chen

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

The Schrdinger equation with the ring-shaped Hartmann potential is solved by using the usual method of variable separation in spherical polar coordinates. The exact energy equation is obtained by then solved angle equation and radial equation. The normalized angle wavefunction and radial wavefunction are presented. The general calculation formulas for radial matrix elements and radial average values of the Hartmann potential are obtained.

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The Schrdinger equation with the ring-shaped Hartmann potential is solved by using the usual method of variable separation in spherical polar coordinates. The exact energy equation is obtained by then solved angle equation and radial equation. The normalized angle wavefunction and radial wavefunction are presented. The general calculation formulas for radial matrix elements and radial average values of the Hartmann potential are obtained.
By using the usual method of variable separation, bound states of Klein-Gordon equation with equal Hartmann scalar and vector potentials are solved. Properties of the system relate to three quantum numbers and parameters of Hartmann potential. The normalized angle wave function expressed in terms of the universal associated-Legendre polynomial and the normalized radial wave function expressed in terms of the confluent hypergeometric function are presented. The exacl energy spectrum equations are obtained. Bound states of Klein-Gordon equation for Hydrogen potential are special cases for the Hartmann potential in the Letter.
Under strong coupling conditions,the non-relativistic motion of particles and spin zero or half-integer the nature of relativistic particles in the system with the Schrdinger equation and Dirac equation can respectively describe.A new ring-shaped harmonic oscillator potential is proposed in this paper.The exact bound solution of Dirac equation and Schrdinger equation for the above potential is obtained under the condition of equal scalar potentials and vector potentials.They can be separated into an corresponding to the wave function equations of angular and radial wave function equation with the method of separation variables.The normalized angle wave function and radial wave function were expressed respectively in terms of the generalized associated-Legendre function and the confluent hyper-geometric function are presented,that the normalized radial wave function,and then by the radial bound state wave function are satisfied with the boundary conditions,obtains the exact bound state energy spectrum equations.The exact energy spectrum equations are obtained and meanwhile,proper discussion and some important conclusions are presented.
Abstract Using the proper quantization rule, we investigate the exact solution of Dirac equation for Hartmann and the ring-shaped non-spherical harmonic oscillator potentials under the condition of equal scalar and vector potentials. By considering the proper quantization condition within angular and radial variables, the exact relativistic energy spectra are obtained for each system. Then by the mean of suitable changes of variables, the corresponding spinor wave-functions are constructed where the normalization constants are exactly calculated. We also derived the non-relativistic limit of energy spectra.
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.