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arXiv:1807.02109 · Advances in High Energy Physics

Dirac Equation in the Presence of Hartmann and Ring-Shaped Oscillator Potentials

Zahra Bakhshi

Published Jul 18, 2018• 3 citations
arXivHTMLPDF

Abstract

The importance of the energy spectrum of bound states and their restrictions in quantum mechanics due to the different methods have been used for calculating and determining the limit of them. Comparison of Schrödinger-like equation obtained by Dirac equation with the nonrelativistic solvable models is the most efficient method. By this technique, the exact relativistic solutions of Dirac equation for Hartmann and Ring-Shaped Oscillator Potentials are accessible, when the scalar potential is equal to the vector potential. Using solvable nonrelativistic quantum mechanics systems as a basic model and considering the physical conditions provide the changes in the restrictions of relativistic parameters based on the nonrelativistic definitions of parameters.

DOI
10.1155/2018/1940925

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The importance of the energy spectrum of bound states and their restrictions in quantum mechanics due to the different methods have been used for calculating and determining the limit of them. Comparison of Schrödinger-like equation obtained by Dirac equation with the nonrelativistic solvable models is the most efficient method. By this technique, the exact relativistic solutions of Dirac equation for Hartmann and Ring-Shaped Oscillator Potentials are accessible, when the scalar potential is equal to the vector potential. Using solvable nonrelativistic quantum mechanics systems as a basic model and considering the physical conditions provide the changes in the restrictions of relativistic parameters based on the nonrelativistic definitions of parameters.
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.
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.