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Journal of Physics Conference Series

Schrödinger equation solution for q-deformed Scarf II potential plus Pöschl-Teller potential and trigonometric Scarf potential

Fery Widiyanto, Suparmi Suparmi, C. Cari, Fuad Anwar, Mohtar Yunianto

Published Nov 1, 2017• 2 citations
Publisher

Abstract

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

DOI
10.1088/1742-6596/909/1/012036

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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
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.
The non-relativistic energies and wave functions of extended hyperbolic Scarf I plus separable non-central shape invariant potential in four dimensions are investigated using Supersymmetric Quantum Mechanics (SUSY QM) Approach. The three dimensional separable non-central shape invariant angular potential consists of trigonometric Scarf II, Manning Rosen and Poschl-Teller potentials. The four dimensional Schrodinger equation with separable shape invariant non-central potential is reduced into four one dimensional Schrodinger equations through variable separation method. By using SUSY QM, the non-relativistic energies and radial wave functions are obtained from radial Schrodinger equation, the orbital quantum numbers and angular wave functions are obtained from angular Schrodinger equations. The extended potential means there is perturbation terms in potential and cause the decrease in energy spectra of Scarf I potential.
This research’s aim are to determine energy spectra and wave function of Schrodinger Equation in D-Dimensions for non central shape invariant potentials, that is, Scarf hyperbolic plus non central Poschl-Teller potential and Scarf trigonometric plus non central Rosen-Morse potential with Nikivarof-Uvarov method. Nikivarof-Uvarov method are the method for solving a second order of differential equation which transform that equation into hypergeometric equation using a correct subtitution of variable. Research was done through a literature studies to solve the Schrodinger equation D-Dimensions analitically for Scarf hyperbolic plus non central Poschl- Teller potential and Scarf trigonometric plus non central Rosen-Morse potential. Energy spectra and wave function obtained by Nikivarof-Uvarov method. After obtaining energy levels and compared with the other methods in the special conditions, the wave function can be obtained by Nikivarof-Uvarov method. Energy spectra of the Scarf hyperbolic plus non-central Poschl Teller potential and Scarf trigonometric plus non-central Rosen Morse potential on DDimensions which obtained by Nikiforov-Uvarof methods have a results consistently with previous research on special condition. Energy spectra and wave functions of both combination of potentials are determined approximately for the value of orbital quantum number l ≠ 0. Keywords: Schrodinger Equation in D-Dimensions, Non Central Shape Invariant Potentials, and Nikivarof-Uvarov Method.
The Hypergeometric method was used to describe radial equations by probeneus approaximation at a singular regular point. Schrodinger equations radial part was reduced into differential equations with variable substitution. By using q-deformed momentum parameter and Coulomb potential as well as variable substitution, the second order of Schrodinger equation with centrifugal approach for q-deformed momentum in Coulomb potential can be solved with Hypergeometric method. The analytical result of energy spectrum and wave function with quantum number and q-deformed variations, while the numerical result for energy spectrum can be calculated by using Matlab 2013 software.
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.
Spatially quadratic deformation radial momentum of D-dimensional Klein-Gordon equation with three-dimensional Harmonic Oscillator and Inverse Quadratic potential was investigated using Hypergeometric Method. By using appropriate variable transformation, D-Dimensional Klein-Gordon equation for q deformed momentum was reduced into one order dimensional Schrodinger like equation with Pöschl-Teller like potential. The relativistic energy and wave function equations were obtained from the solution of Schrodinger like equation. Then, the relativistic energy and wave function were calculated and visualized using Matlab R2013 software, respectively. The q deformed parameter had the effect to the relativistic energy and wave function. The increase of q deformed parameter causes the increase of energy spectra and the decrease of amplitude.