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Eclética Química

Solving Schrödinger equation for two dimensional potentials using supersymmetry

Elso Drigo Filho

Published Dec 8, 1997• 0 citations
Publisher

Abstract

The formalism of supersymmetric Quantum Mechanics can be extended to arbitrary dimensions. We introduce this formalism and explore its utility to solve the Schrödinger equation for a bidimensinal potential. This potential can be applied in several systems in physical and chemistry context , for instance, it can be used to study benzene molecule.

DOI
10.26850/1678-4618eqj.v22.1.1997.p67-73

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The formalism of supersymmetric Quantum Mechanics can be extended to arbitrary dimensions. We introduce this formalism and explore its utility to solve the Schrödinger equation for a bidimensinal potential. This potential can be applied in several systems in physical and chemistry context , for instance, it can be used to study benzene molecule.
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The Schrödinger equation under the ManningRosen potential is solved in arbitrary dimension via the quantum mechanical idea of supersymmetry.The Pekeris approximation is used to overcome the inconsistency of the potential with the centrifugal term.Comments on the energy eigenvalue behavior versus dimension are included.The inter-dimensional degeneracy for various orbital quantum number l and dimensions D are studied.The expectation values of some physical parameters are reported via the FeynmanHellmann theorem.
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.