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Communications in Theoretical Physics

Relativistic Treatment of Spinless Particles Subject to aq-Deformed Morse Potential

Sami Ortakaya

Published Jun 1, 2013• 16 citations
Publisher

Abstract

The approximate analytical solutions of the Klein—Gordon equation with equal scalar and vector q-deformed Morse potential are presented for arbitrary ℓ-states by using Laplace integral transform. The energy eigenvalues and corresponding wave functions are obtained for n and ℓ values. In this study, in the non-relativistic limit c → ∞ , it has been also provided that the energy eigenfunctions for Klein—Gordon system turn into those for Schrödinger one.

DOI
10.1088/0253-6102/59/6/06

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The approximate analytical solutions of the Klein—Gordon equation with equal scalar and vector q-deformed Morse potential are presented for arbitrary ℓ-states by using Laplace integral transform. The energy eigenvalues and corresponding wave functions are obtained for n and ℓ values. In this study, in the non-relativistic limit c → ∞ , it has been also provided that the energy eigenfunctions for Klein—Gordon system turn into those for Schrödinger one.
Using Laplace integral transformation and the estimate transformation,the approximate analytical solutions of the 3-dimensional Schrdinger equation for q-deformed Morse potential is obtained.By means of expanding q-deformed Morse potential into the power series until the three order little term and smaller are ignored;sitting reasonably the parameter of wave function which is looked for makes the situation have essential change,so that the analytical solution without existence changes can turn into the approximate analytical solution existence;using reasonably terminal-value theorem and the convolution of Laplace integral transformation,and by applying the orthogonal condition of general Laguerre function,the explicit press of energy spectrum and the radial normalization eigenfunction of the quantum system are acquired.
Quantum relativistic solutions for a particle in a one-dimensional Morse potential are presented using methods previously proposed and applied to a particle in a harmonic oscillator. The methods lead to both numerical and analytical solutions, with the latter allowing smooth variation of the system parameters from non-relativistic to ultrarelativistic limits. Analytical expressions for the energy levels and wavefunctions are obtained, as solutions to a Schrödinger-type equation, including relativistic effects through a state dependent rescaled mass. The eigenstates of the Morse potential exhibit suitable and smooth behavior, and approach the corresponding harmonic oscillator solutions as the depth of the Morse potential well increases, as expected. A comparison is also presented between the relativistic harmonic oscillator obtained with this method, and the so-called Klein-Gordon oscillator.
We have obtained the approximate solutions of the Klein–Gordon equation with the Yukawa ring-shaped potential, by using the Nikiforov–Uvarov method for a special case of equal scalar and vector potentials. The energy eigenvalues for bound states and the corresponding wave functions are also obtained in a proper approximation. We have also shown that the results can be used to evaluate the energy eigenvalues of the Yukawa, angle-dependent, and Coulomb potentials. The numerical results are discussed and presented in the table and in the figure, which suggest their applicability to other systems. With the adjusted potential parameters given in the table, it is shown that the interaction of spinless (Klein–Gordon) particles with the Yukawa ring-shaped potential gives positive energy eigenvalues for the various quantum states.
In this research work, within the framework of relativistic and nonrelativistic noncommutative quantum mechanics, the deformed Klein–Gordon and Schrödinger equations were solved with the modified equal vector scalar Manning-Rosen potential that has been of significance interest in recent years using Bopp's shift method and standard perturbation theory in the first-order in the noncommutativity parameters in 3-dimensions noncommutative quantum mechanics. By employing the improved approximation of the centrifugal term, the relativistic and nonrelativistic bound state energies were obtained for some diatomic molecules such as (HCl, CH, LiH, CO, NO, O2, I2, N2, H2, and Ar2). The obtained energy eigenvalues appear as a function of the generalized Gamma function, the parameters of noncommutativity, and the parameters of studied potential, in addition to the atomic quantum numbers . In both relativistic and nonrelativistic problems, we show that the corrections on the spectrum energy are smaller than the main energy in the ordinary cases of RQM and NRQM. A straightforward limit of our results to ordinary quantum mechanics shows that the present result is consistent with what is obtained in the literature. We have seen that the improved approximation of the centrifugal term is better than the other approximations in finding the approximate analytical solutions of the Klein-Gordon and Schrödinger equations for the modified Manning–Rosen potential in RNCQM and NRNCQM.
Using Laplace integral transformation and the estimate transformation,the approximate analytical solutions of the 3-dimensional Schrdingere quation for q-deformed Morse potential are obtained.By means of expanding q-deformed Morse potential into the power series until the three order little terms and smaller are ignored;looking for reasonably the parameter of wave function, make its analytical solution not be existence change into the approximate analytical solution existence;using reasonably terminal-value theorem and the convolution of Laplace integral transformation,and by applying the orthogonal condition of general Laguerre function;the explicit press of energy spectrum and the radial normalization eigenfunction of the quantum system are acquired unl(ρ)= (n!(2β2)2κ+1/(2κ+n+1)) 1/2 e-β2ρρκL2κ n(2β2ρ)and meanwhile,proper discussion and some conclusions are presented.
The approximate solutions of the Klein-Gordon equation with the shifted Tietz-Wei potential function are thoroughly studied for Klein-Gordon equation with both 2 V and V potentials by using a suitable approximation scheme for the centrifugal term in the framework of the supersymmetric approach. The non-relativistic limits in both cases are obtained, and the numerical results are computed. We have equally investigated the Fisher information on the shifted Tietz-Wei potential function in the case of the non-relativistic limit of the Klein-Gordon equation with potential V because this is only possible in the non-relativistic regime. The Fisher information is observed to be inversely proportional to both C h and b h , where C h is the optimization parameter and b h = β (1 − C h ), with β being the Morse constant.