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Molecular Physics

Eigen solutions of the Schrodinger equation with variable mass under the influence of the linear combination of modified Woods–Saxon and Eckart potentials in toroidal coordinate

Suparmi Suparmi, C. Cari, Suci Faniandari

Published Jun 22, 2020• 14 citations
Publisher

Abstract

The solutions of the Schrodinger equation was investigated in space-times with nonspherical topology in the presence of the linear combination of modified Woods–Saxon and Eckart potentials. The variable mass and Supersymmetric Quantum Mechanics method were used to solve the Schrodinger equation in toroidal coordinate. This solution was used to obtain the nonrelativistic energy eigenvalue and the wave function of the system. Using the proper super potential to deal with the hyperbolic form of the effective potential, the Schrodinger equation in the radial direction part was solved analytically for modified Woods–Saxon plus Eckart potentials. The energy eigenvalue and the un-normalised ground state wave function equations were obtained. The results showed that the energy value depends on the quantum number, the parameters of potentials, and variable separation constants.

DOI
10.1080/00268976.2020.1781946

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The solutions of the Schrodinger equation was investigated in space-times with nonspherical topology in the presence of the linear combination of modified Woods–Saxon and Eckart potentials. The variable mass and Supersymmetric Quantum Mechanics method were used to solve the Schrodinger equation in toroidal coordinate. This solution was used to obtain the nonrelativistic energy eigenvalue and the wave function of the system. Using the proper super potential to deal with the hyperbolic form of the effective potential, the Schrodinger equation in the radial direction part was solved analytically for modified Woods–Saxon plus Eckart potentials. The energy eigenvalue and the un-normalised ground state wave function equations were obtained. The results showed that the energy value depends on the quantum number, the parameters of potentials, and variable separation constants.
Schrodinger equation on a toroidal coordinate was proposed in theoretical physics to get the information and the behavior of the system of particle. It was solved just recently in case of a charged scalar particle interacting with a uniform magnetic field, a uniform electric field, and a neutral charge constrained to the surface. The methodology used in the referred work was to solve the Schrodinger equation using an approach outlined in the Whittaker-Watson treatise, which deals with an infinite-dimensional eigenvalue problem and specific particular values of the applied field for eigenfunction problem. In contrast, in the quantum mechanical problem, one had an infinite-dimensional generalized eigenvalue problem. This study aimed to obtain the non-relativistic energy eigenvalue and the radial wave function of the Schrodinger equation under the influence of Yukawa potential. The Supersymmetric Quantum Mechanics (SUSY QM) method was used as a basis to tackle the primary objective of this paper to study the problem of a particle with variable mass in toroidal coordinate. The proper super potential was used to deal with the hyperbolic form of effective potential, and the energy spectra were calculated for different quantum numbers, potential depth, and potential parameters. The radial wave function equation for ground and excited state were obtained. The results showed that the increasing value of the quantum numbers caused the energy spectra of the system to increase to the highest value when the quantum number was equal to the potential parameter, which means the most effective energy value was produced, then it was decreased afterward. While the energy value did not depend on the change of the potential parameter. This property could be used to produce this equation as an application of the previous results, the Schrödinger eigenfunction was used as the starting points to solve the other equation in the same geometrical setting and potential.
In this work, the analytical solutions of the [Formula: see text]-dimensional radial Schrödinger equation are studied in great detail for the Wood–Saxon potential by taking advantage of the Pekeris approximation. Within a novel improved scheme to surmount centrifugal term, the energy eigenvalues and corresponding radial wave functions are found for any angular momentum case within the context of the Nikiforov–Uvarov (NU) and Supersymmetric quantum mechanics (SUSYQM) methods. In this way, based on these methods, the same expressions are obtained for the energy eigenvalues, and the expression of radial wave functions transformed each other is demonstrated. In addition, a finite number energy spectrum depending on the depth of the potential [Formula: see text], the radial [Formula: see text] and orbital [Formula: see text] quantum numbers and parameters [Formula: see text] are defined as well.
In this work, we obtained an exact solution to Schrodinger equation using q-deformed Woods-Saxon plus modified Coulomb potential Using conventional Nikiforov-Uvarov method.We also obtained the energy eigen value and its associated total wave function .This potential with some suitable conditions reduces to two well known potentials namely: the Yukawa and coulomb potential.Finally, we obtained the numerical results for energy eigen value with different values of q as dimensionless parameter.The result shows that the values of the energies for different quantum number(n) is negative(bound state condition) and increases with an increase in the value of the dimensionless parameter(arbitrary constant).The graph in figure (1) shows the different energy levels for a particular quantum number.
A general solution of the Schrödinger equation in the potential representation has been obtained in the form of integral equations. In this representation, the wave function for positive and negative energies or bound states can be expressed as a product of the unperturbed solution for model potential and the function which depends on the additional potential or potential perturbation. Here we have proved that this method is equivalent to the method of variation of constants for negative energies. The linearly independent solutions of Schrödinger equation for harmonic oscillator potential have been obtained for derivation of integral equations, which are used for finding eigenfunctions and eigenvalues for Woods–Saxon potential. Eigenvalues obtained by numerical iterations of these integral equations are in good agreement with results obtained by the discretization method. The kernels of the obtained integral equations are proportional to the perturbation or difference of Woods–Saxon and harmonic oscillator potentials.