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Chinese Physics B

Approximate analytical solution of the Dirac equation with q -deformed hyperbolic Pöschl–Teller potential and trigonometric Scarf II non-central potential

Ade Kurniawan, Suparmi Suparmi, C. Cari

Published Feb 26, 2015• 22 citations
Publisher

Abstract

An approximate solution of the Dirac equation for a spin-1/2 particle under the influence of q -deformed hyperbolic Pöschl–Teller potential combined with trigonometric Scarf II non-central potential is studied analytically. It is assumed that the scalar potential equals the vector potential in order to obtain analytical solutions. Both radial and angular parts of the Dirac equation are solved using the Nikiforov–Uvarov method. A relativistic energy spectrum and the relation between quantum numbers can be obtained using this method. Several quantum wave functions corresponding to several states are also presented in terms of the Jacobi Polynomials.

DOI
10.1088/1674-1056/24/3/030302

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An approximate solution of the Dirac equation for a spin-1/2 particle under the influence of q -deformed hyperbolic Pöschl–Teller potential combined with trigonometric Scarf II non-central potential is studied analytically. It is assumed that the scalar potential equals the vector potential in order to obtain analytical solutions. Both radial and angular parts of the Dirac equation are solved using the Nikiforov–Uvarov method. A relativistic energy spectrum and the relation between quantum numbers can be obtained using this method. Several quantum wave functions corresponding to several states are also presented in terms of the Jacobi Polynomials.
Abstract Under the condition of the spin symmetry, we rigorously solve the Dirac equation with the q − deformed hyperbolic Pöschl-Teller potential plus a trigonometric Scarf II potential. According to the values of the deformation parameter q, four different cases are considered. For the two cases q ≥ 1 and q ≤ − 1 with 1 2 α ln q < r < + ∞ , the analytical energy spectra and the spinor wave functions associated with the l d − wave bound states are obtained using a suitable approximation to the centrifugal potential term. When 0 < q < 1 or −1 < q < 0, the spinor wave functions for the s − wave bound states are derived and we find that the quantization conditions are transcendental equations which can be solved numerically. The special case q = 0 is also discussed.
The Schrödinger equation for q-deformed Scarf II potential plus Pöschl-Teller potential and trigonometric Scarf potential was solved using Asymptotic Iteration Method (AIM). The combination of three potentials was substituted into the Schrödinger equation, then we separated the three-dimensional Schrodinger equation into three one-dimensional differential equation, consist of the radial part, polar part and azimuth part. The radial, polar and azimuth part differential equations were solved by reducing them into the hypergeometric intermediaries equation, to further we resolved it using AIM. By using AIM, the energy equation and the variable separation constant equations, λ 1 for the polar part and λ 2 for the azimuth part were obtained, where both are interrelated between the quantum numbers. The value of energies were calculated numerically using Matlab software, where the increasing in the radial quantum number n r causes increase and decrease in the energy. Radial, polar and azimuth wave functions were defined as hypergeometric functions and were visualized using Matlab software. The results show that the disturbance of Poschl-Teller potential and trigonometric Scarf potential give effect in probability
We study the problem of the relativistic motion of a 1/2-spin particle in an exactly solvable potential, which consists of the harmonic oscillator potential plus a novel angle-dependent potential, The analytic bound state solutions of the Dirac equation for this potential are obtained by using the Nikiforov–Uvarov method. The wave functions of the radial and angle-dependent parts of the Dirac equation are derived in the form of the Laguerre and Jacobi polynomials. The contribution of the angle-dependent potential to the relativistic energy spectra is discussed under the condition that the scalar potential is equal to or minus the vector potential.
The bound state solution of the Dirac equation for generalized Pöschl-Teller and trigonometric Pöschl-Teller non-central potentials was obtained using SUSY quantum mechanics and the idea of shape invariance potential.The approximate relativistic energy spectrum was expressed in the closed form.The radial and polar wave functions were obtained using raising and lowering of radial and polar operators.The orbital quantum numbers were found from the polar Dirac equation, which was solved using SUSY quantum mechanics and the idea of shape invariance.