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The European Physical Journal Plus

Generalized tanh-shaped hyperbolic potential: bound state solution of Schrödinger equation

H. I. Ahmadov, E. A. Dadashov, N. Sh. Huseynova, V. H. Badalov

Published Feb 1, 2021• 11 citations
Publisher
DOI
10.1140/epjp/s13360-021-01202-8

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In this study, the bound state energy eigenvalues and the corresponding wave functions of the deformed Poe schlTeller and Hyperbolic Kratzer-like potentials have been obtained by the NikiforovUvarov method using the deformed hyperbolic functions (sinh q (x) 4 1 (ex ( qe~x), cosh q (x) 4 and that 1 (ex ) qe~x), sech q (x) 4 1/cosh q (x) tanh q (x) 4 sinh q (x)/cosh q (x)) were introduced for the —rst time by Arai (J. Math. Anal. Appl. 158, 63 (1991)). It is also observed that, the energy eigenvalues of these iinewˇˇ type deformed hyperbolic potentials bring up the criteria for the shape invari- ance of the potentials according to the deformation parameter q.
We solve exactly one-dimensional Schrödinger equation for the generalized asymmetric Manning-Rosen (GAMAR) type potential containing the different types of physical potential that have many application fields in the nonrelativistic quantum mechanics and obtain the solutions in terms of the Gauss hypergeometric functions. Then we determine the solutions for scattering and bound states. By using these states we calculate the reflection and transmission coefficients for scattering states and achieve a correlation that gives the energy eigenvalues for the bound states. In addition to these, we show how the transmission and reflection coefficients depend on the parameters which describe shape of the GAMAR type potential and compare our results with the results obtained in earlier studies.
In this paper, Schrodinger equation has been solved analytically with position dependent mass by the Rosen-Morse potential. The position dependent mass defined as the function 1/(1 — tanh(η x )). Then corresponding position dependent mass substituted into Schrodinger equation. After that the obtained equation compared with associated Jacobi differential equation. Therefore, the eigenvalue and eigenfunctionhave been calculated, and from there the bound state has been found in terms of quantum numbers and the coefficients of potential. Also, the wave function has been obtained in terms of the associated Jacobi equation by μ = —3 and v = 2.