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Journal of Physics Conference Series

The solution of 4-dimensional Schrodinger equation for Scarf potential and its partner potential constructed By SUSY QM

Wahyulianti, Suparmi Suparmi, C. Cari, Kristiana Nathalia Wea

Published Nov 1, 2017• 1 citation
Publisher

Abstract

The angular part of 4-dimensional Schrodinger equation for Scarf potential was solved by using the Nikiforov-Uvarov method and Supersymmetric Quantum Mechanic method. The determination of the ground state wave function has been used Nikiforov-Uvarov method and by applying the parametric generalization of the hypergeometric type equation. By using manipulation of the properties and operators of the Supersymmetric Quantum Mechanic method the partner potential was constructed. The ground state wave functions of original Scarf potential is different than the ground state wave functions of the construction result potential.

DOI
10.1088/1742-6596/909/1/012034

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The angular part of 4-dimensional Schrodinger equation for Scarf potential was solved by using the Nikiforov-Uvarov method and Supersymmetric Quantum Mechanic method. The determination of the ground state wave function has been used Nikiforov-Uvarov method and by applying the parametric generalization of the hypergeometric type equation. By using manipulation of the properties and operators of the Supersymmetric Quantum Mechanic method the partner potential was constructed. The ground state wave functions of original Scarf potential is different than the ground state wave functions of the construction result potential.
The 4-Dimensional Schrodinger's Equation is solved using the Supersymmetric Quantum Mechanics (SUSY QM) to obtain the first three energy spectrum and the ground state wave functions both in radial and angular parts. The non-central potential consisted of radial Scarf potential, and mix trigonometry potential such us Poschl-Teller, Rosen Morse, and Scarf for the angular part were used in this study. The energy equation of the 4-dimensional Schrodinger equation system was used to analyze some thermodynamic properties (vibrational mean energy and specific heat) of the system.
The radial part of 4-dimensional Schrodinger equation for Polynomial Inverse potential and its partner potential was solved by using the Ansatz Wavefunction method. By applying of the properties and operators of the Supersymmetric Quantum Mechanic method the partner potential was constructed. The eigenvalues and eigenfunctions of the Polynomial Inverse potential and the eigenvalues and eigenfunctions of its partner potential were different. The partner potential of Polynomial Inverse potential for n > 1 has eigenvalue greater than Polynomial Inverse potential as its original potential.
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.
In this paper, we show that the exact energy eigenvalues and eigenfunctions of the Schrödinger equation for charged particles moving in a certain class of noncentral potentials can be easily calculated analytically in a simple and elegant manner by using Supersymmetric method (SUSYQM). We discuss the Poschl-Teller plus Scarf non-central potential systems. Then, by operating the lowering operator we get the ground state wave function, and the excited state wave functions are obtained by operating raising operator repeatedly. The energy eigenvalue is expressed in the closed form obtained using the shape invariant properties. The results are in exact agreement with other methods.Keyword: supersymmetry, non-central potentials, poschl teller plus scarf.
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.
In this paper, we show that the exact energy eigenvalues and eigenfunctions of the Schrödinger equation for charged particles moving in a certain class of noncentral potentials can be easily calculated analytically in a simple and elegant manner by using Supersymmetric method (SUSYQM). We discuss the Poschl-Teller plus Scarf non-central potential systems. Then, by operating the lowering operator we get the ground state wave function, and the excited state wave functions are obtained by operating raising operator repeatedly. The energy eigenvalue is expressed in the closed form obtained using the shape invariant properties. The results are in exact agreement with other methods.Keyword: supersymmetry, non-central potentials, poschl teller plus scarf.