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Journal of Physics Conference Series

The construction of partner potential from the general potential anharmonic inD-dimensional Schrodinger system

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

Published Mar 1, 2018• 0 citations
Publisher

Abstract

The Schrodinger equation is the fundamental equation in quantum physics. The characteristic of the particle in physics potential field can be explained by using the Schrodinger equation. In this study, the solution of 4 dimensional Schrodinger equation for the anharmonic potential and the anharmonic partner potential have done. The method that used to solve the Schrodinger equation was the ansatz wave method, while to construction the partner potential was the supersymmetric method. The construction of partner potential used to explain the experiment result that cannot be explained by the original potential. The eigenvalue for anharmonic potential and the anharmonic partner potential have the same characteristic. Every increase of quantum orbital number the eigenvalue getting smaller. This result corresponds to Bohrn's atomic theory that the eigenvalue is inversely proportional to the atomic shell. But the eigenvalue for the anharmonic partner potential higher than the eigenvalue for the anharmonic original potential.

DOI
10.1088/1742-6596/997/1/012002

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The Schrodinger equation is the fundamental equation in quantum physics. The characteristic of the particle in physics potential field can be explained by using the Schrodinger equation. In this study, the solution of 4 dimensional Schrodinger equation for the anharmonic potential and the anharmonic partner potential have done. The method that used to solve the Schrodinger equation was the ansatz wave method, while to construction the partner potential was the supersymmetric method. The construction of partner potential used to explain the experiment result that cannot be explained by the original potential. The eigenvalue for anharmonic potential and the anharmonic partner potential have the same characteristic. Every increase of quantum orbital number the eigenvalue getting smaller. This result corresponds to Bohrn's atomic theory that the eigenvalue is inversely proportional to the atomic shell. But the eigenvalue for the anharmonic partner potential higher than the eigenvalue for the anharmonic original potential.
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The solution of the Schrodinger equation with the physical potential is important part in quantum mechanics. The anharmonic potential is one of the physical potential that is interesting to be explored. Many methods have been developed to solve the Schrodinger equation. The Ansatz wave function method is one of the many methods that used to solve the Schrodinger equation. The Ansatz wave function method was used to solve the radial part of Schrodinger Equation. In this study we present the solution of the radial part Schrodinger equation in D-dimensional system with potential V(r)=ar2+br+cr-1+dr-2+er-3+fr-4. The important part in Ansatz wave function method is to determine the wave function Ansatz corresponding to the shape of the anharmonic potential that used. The Ansatz wave function determined by using supersymmetric approximation. By substituting the wave function Ansatz into Schrodinger equation and using algebraic we obtain the radial eigenfunction and the energy, and then the eigenfunction visualized.
The harmonic oscillator is a quantum mechanical system that represents one of the most basic potentials. In order to understand the behavior of a particle within this system, the time-independent Schrödinger equation was solved; in other words, its eigenfunctions and eigenvalues were found. The first goal of this study was to construct a family of single parameter potentials and corresponding eigenfunctions with a spectrum similar to that of the harmonic oscillator. This task was achieved by means of supersymmetric quantum mechanics, which utilizes an intertwining operator that relates a known Hamiltonian with another whose potential is to be built. Secondly, a generalization of the technique was used to work with the time-dependent Schrödinger equation to construct new potentials and corresponding solutions.
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.
A method for calculating the analytical solutions of the one-dimensional Schr\"odinger equation is suggested. A general discussion of the possible forms of the potentials and wave functions that are necessary to get the analytical solution is presented. In general, the analytical solutions appear in multiplets corresponding to the quantum number n of the harmonic oscillator. As an application, known solutions for the anharmonic oscillators are critically recalculated and a few additional results are found. Analytical solutions are also found for the generalized Morse oscillators. \textcopyright{} 1996 The American Physical Society.