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Acta Physica Sinica

Bound states of Klein-Gordon equation for ring-shaped non-spherical harmonic oscillator potentials

Fa‐Lin Lu, Chen Chang-Yuan, 盐城师范学院物理系，盐城 224002

Published Jan 1, 2004• 4 citations
Publisher

Abstract

By using the ordinary method of variable separation,the bound states of Klein-Gordon equation of the ring-shaped non-spherical harmonic oscillator with equal scalar and vector potentials are solved. The normalized angular wave function expressed in terms of the universal associatedLegendre polynomial and the normalized radial wave function expressed in terms of the confluent hyper-geometric function are presented. The exact energy spectrum equations are obtained.

DOI
10.7498/aps.53.1652

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By using the ordinary method of variable separation,the bound states of Klein-Gordon equation of the ring-shaped non-spherical harmonic oscillator with equal scalar and vector potentials are solved. The normalized angular wave function expressed in terms of the universal associatedLegendre polynomial and the normalized radial wave function expressed in terms of the confluent hyper-geometric function are presented. The exact energy spectrum equations 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.
In this paper, the bound state solution of the modified Klein–Fock–Gordon equation is obtained for the Hulthén plus ring-shaped-like potential by using the developed scheme to overcome the centrifugal part. The energy eigenvalues and corresponding radial and azimuthal wave functions are defined for any [Formula: see text] angular momentum case on the conditions that scalar potential is whether equal and nonequal to vector potential, the bound state solutions of the Klein–Fock–Gordon equation of the Hulthén plus ring-shaped-like potential are obtained by Nikiforov–Uvarov (NU) and supersymmetric quantum mechanics (SUSY QM) methods. The equivalent expressions are obtained for the energy eigenvalues, and the expression of radial wave functions transformations to each other is revealed owing to both methods. The energy levels and the corresponding normalized eigenfunctions are represented in terms of the Jacobi polynomials for arbitrary [Formula: see text] states. A closed form of the normalization constant of the wave functions is also found. It is shown that the energy eigenvalues and eigenfunctions are sensitive to [Formula: see text] radial and [Formula: see text] orbital quantum numbers.