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Journal of the Nigerian Society of Physical Sciences

Approximate Solutions of the Schrodinger Equation for a Momentum-Dependent potential

C. A. Onate, I. B. Okon, M. C. Onyeaju, Akaninyene Daniel Antia

Published May 29, 2022• 2 citations
Publisher

Abstract

The solution of one-dimensional Schrodinger equation for a newly proposed potential called modified shifted Deng-Fan momentum-dependent potential is obtained via supersymmetric approach. The expectation values of momentum and position were calculated using Hellmann Feynman Theorem. The effects of momentum-dependent parameter on the solutions of the system as well as the expectation values were studied. Finally, the special cases of the interacting potential were obtained.

DOI
10.46481/jnsps.2022.653

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The solution of one-dimensional Schrodinger equation for a newly proposed potential called modified shifted Deng-Fan momentum-dependent potential is obtained via supersymmetric approach. The expectation values of momentum and position were calculated using Hellmann Feynman Theorem. The effects of momentum-dependent parameter on the solutions of the system as well as the expectation values were studied. Finally, the special cases of the interacting potential were obtained.
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The approximate analytical solutions of the radial Schrӧdinger equation have been obtained with a newly proposed potential called Hellmann-generalized Morse potential. The potential is a superposition of Hellmann potential and generalized Morse or Deng-Fan potential. The Hellmann-generalized Morse potential actually comprises of three different potentials which includes Yukawa potential, Coulomb potential and Deng-Fan potential. The aim of combining these potentials is to have a wide application. The energy eigenvalue and the corresponding wave function are calculated in a closed and compact form using the parametric Nikiforov-Uvarov method. The energy equation for some potentials such as Deng-Fan, Rosen Morse, Morse, Hellmann, Yukawa and Coulomb potentials have also been obtained by varying some potential parameters. Some numerical results have been computed. We have plotted the behavior of the energy eigenvalues with different potential parameters and also reported on the numerical result. Finally, we computed the variance and information energy for the Hellmann-generalized Morse potential.
The solutions of the radial Schrödinger equation for a pseudoharmonic potential and Kratzer potentials has been studied separately in the past. The Kratzer potential despite different reports, the basic theoretic quantity such as Fisher information has not been reported. In this study, the solution of the radial Schrödinger equation for the combination of the pseudoharmonic and Kratzer potentials in the presence of a constant-dependent potential is obtained using the concepts and formalism of the supersymmetric and shape invariance approach. The position expectation value and momentum expectation value are calculated employing the Hellmann-Feynman Theory. The expectation values are used to calculate the Fisher information for both the position and momentum spaces in both the absence and presence of the constant-dependent potential. The obtained results revealed that the presence of the constant-dependent potential leads to an increase in the energy eigenvalue, as well as in the position and momentum expectation values. The presence of the constant-dependent potential also increases the Fisher information for both position and momentum spaces. Furthermore, the product of the position expectation value and the momentum expectation value along with the product of the Fisher information, satisfies both the Fisher’s inequality and the Cramer-Rao’s inequality.
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.