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Journal of Physics Conference Series

Energy analysis of four dimensional extended hyperbolic Scarf I plus three dimensional separable trigonometric noncentral potentials using SUSY QM approach

Suparmi Suparmi, C. Cari, Utama Alan Deta, Jeffry Handhika

Published Nov 1, 2016• 1 citation
Publisher

Abstract

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.

DOI
10.1088/1742-6596/776/1/012077

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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.
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.
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.
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Abstrak Tujuan dari penelitian ini adalah menyelesaikan persamaan Schrodinger potensial non-sentral Scarf hiperbolik plus Rosen-Morse trigonometrik menggunakan metode Supersimetri Mekanika Kuantum (SUSI MK).Spektrum energi dan fingsi radial diperoleh dari penyelesaian persamaan Schrodinger bagian radial, sedangkan fungsi gelombang bagian sudut dan bilangan kuntum orbital diperoleh dari persamaan Schrodinger bagian sudut polar.Spektrum energi dan bilangan kuantum orbital ditentukan dengan sifat shape invariance .Penentuan fungsi gelombang tingkat dasar bagian radial ditentukan dengan sifat lowering operator dan fungsi gelombang tereksitasi ditentukan dengan sifat raising operator .Baik untuk bagian radial maupun bagian polar ditentukan dengan menggunakan sifat lowering operator dan raising operator . Abstract The purpose of this research is to solve the Schrodinger equation of non-central potential Scarf hyperbolic plus Rosen-Morse trigonometric method Supersymmetry Quantum Mechanics (SUSY QM). Energy spectrum and radial functions derived from the completion of the radial part of the Schrodinger equation, while the wave functions of the corners and orbital quantum numbers obtained from the Schrodinger equation part of the polar angle. Energy spectrum and orbital quantum number is determined by the nature of the shape invariance. Determination of the wave functions of the radial part of the base rate is determined by the nature of lowering operators and the excited wave function is determined by the nature of the raising operator. Both for the radial and polar parts determined by using the properties of lowering and raising operators. Keywords: Schrodinger equation, the non-central potential Scarf hiperbolic plus Rosen-Morse trigonometric, Supersymmetry Quantum Mechanics