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Physica Scripta

Path integral treatment of a Klein Gordon particle with generalized inverse Quadratic Yukawa potential

Salah Eddine Aid, Hocine Boukabcha, Mohamed Hemis

Published Sep 19, 2023• 3 citations
Publisher

Abstract

Abstract We solved the Klein–Gordon equation for a generalized inverse quadratic Yukawa (GIQY) potential via path integrals approach. We applied approximations to deal with the terms 1/r 2 and 1/r. A path integral representation of Green’s function relating to a particle moving in a mixture of equal vector and scalar potentials was established. To integrate Green’s function, a space-time transformation was successfully used, and the present problem was reduced to a previously known modified Pöschl-Teller potential problem. The integrated Green’s function contains important information about the considered system, from which we obtained energy eigenvalues and the corresponding normalized eigenfunctions for various values of n and ℓquantum numbers. Numerical results, Schrödinger solutions, and special cases such as the modified screened Coulomb plus inversely quadratic Yukawa potential, Kratzer potential, Yukawa potential, and Coulomb potential were also considered. These results are largely consistent with previous studies.

DOI
10.1088/1402-4896/acfb49

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Abstract We solved the Klein–Gordon equation for a generalized inverse quadratic Yukawa (GIQY) potential via path integrals approach. We applied approximations to deal with the terms 1/r 2 and 1/r. A path integral representation of Green’s function relating to a particle moving in a mixture of equal vector and scalar potentials was established. To integrate Green’s function, a space-time transformation was successfully used, and the present problem was reduced to a previously known modified Pöschl-Teller potential problem. The integrated Green’s function contains important information about the considered system, from which we obtained energy eigenvalues and the corresponding normalized eigenfunctions for various values of n and ℓquantum numbers. Numerical results, Schrödinger solutions, and special cases such as the modified screened Coulomb plus inversely quadratic Yukawa potential, Kratzer potential, Yukawa potential, and Coulomb potential were also considered. These results are largely consistent with previous studies.
The Green’s function for a Klein–Gordon particle under the action of vector plus scalar deformed Hulthén and Woods–Saxon potentials is evaluated by exact path integration. Explicit path integration leads to the Green’s function for different shapes of the potentials. From the singularities of the latter Green’s function, the bound states are extracted. For q≥1 and (1/α)ln q<r<∞, the analytic expression of the energy spectrum and the normalized wave functions for the l states are obtained within the framework of an approximation to the centrifugal term. When the deformation parameter q is 0<q<1 or q<0, it is found that the quantization conditions are transcendental equations involving the hypergeometric function that require a numerical solution for the s-state energy levels. Particular cases of these potentials are also discussed briefly.
Abstract: In this study, the Schrödinger equation was solved with a superposition of the Hulthen potential and generalized inverse quadratic Yukawa potential model using the Nikiforov-Uvarov (NU) method. For completeness, we also calculated the wave function. To validate our results, the numerical bound state energy eigenvalues was computed for various principal n and angular momentum l quantum numbers. With the aid of the Hellmann-Feynman theorem, the expressions for the expectation values of the square of inverse of position, r^(-2), inverse of position, r^(-1), kinetic energy, T ̂ and square of momentum, p ̂ are calculated. By adjusting the potential parameters, special cases of the potential were considered, resulting in Generalized Inverse Quadratic Yukawa potential, Hulthen potential, Coulomb potential, Kratzer potential, Inversely Quadratic Yukawa potential and Coulomb plus inverse square potential, respectively. Their energy eigenvalue expressions and numerical computations agreed with the literature. Keywords: Schrödinger equation, Hulthen potential (HP), Generalized inverse quadratic Yukawa potential (GIQYP), Nikiforov-Uvarov method. PACS: 03.65.−w, 03.65.Ca, 03.65.Ge.
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.
The coherent states for a particle in Kratzer type potentials are constructed by solving Feynman’s path integral. The action of the generalized Kratzer potential is transformed into two harmonic oscillators by Levi-Civita transformation to derive the parametric time coherent states. Green’s function, energy eigenvalues, and the corresponding wave functions for this potential are calculated. The evaluated results are reduced to the modified Kratzer potentials and Kratzer-Fues oscillators which are special cases of the generalized Kratzer potential.
In this paper, we investigate a central coordinate system within the framework of the Wigner–Dunkl deformation of quantum mechanics in d spatial dimensions, focusing on the dynamics of a relativistic spinless particle described by the Klein–Gordon equation. The analysis is carried out using Feynman’s path integral formalism in hyperspherical coordinates, where the Lagrangian includes an effective potential depending explicitly only on radial variables. This approach allows for the exact separation of variables in d-dimensional Dunkl space, enabling an analytical evaluation of the path integral. Particular emphasis is placed on the attractive Coulomb-type potential, for which the exact propagator is obtained in closed form. As a result, we derive the energy spectrum and corresponding wave functions of the Dunkl–Klein–Gordon system, expressed in terms of special functions that reflect the modified symmetries induced by the Dunkl operators.
We have obtained approximate bound state solutions of Schrödinger wave equation with modified quadratic Yukawa plus q-deformed Eckart potential Using Parametric Nikiforov-Uvarov (NU) method. However, we obtained numerical energy eigenvalues and un-normalized wave function using confluent hypergeometric function (Jacobi polynomial). With some modifications, our potential reduces to a well-known potential such as Poschl-Teller and exponential inversely quadratic potential. Numerical bound state energies were carried out using a well-designed Matlab algorithm while the plots were obtained using origin software. The result obtained is in agreement with that of the existing literature.