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Journal of Applied Physics

Exact analytic solutions of the Schrödinger equations for some modified q-deformed potentials

M. Sebawe Abdalla, Hichem Eleuch

Published Jun 18, 2014• 23 citations

Abstract

In this paper, we introduce the exact solution for the wave function in the presence of potential energy, consisting of combination between q-deformed hyperbolic and exponential function with different argument. The functions we have used in the present communication can be regarded as a generalization of the Arai q-deformed function (modified q-deformed Morse potential). In this context, we have restricted our discussion for some particular cases of the q-deformed hyperbolic functions. This is due to the difficulty for dealing with most of the arguments included in the potential functions. For the most particular cases, the energy eigenfunctions are obtained, and the behavior is also discussed. It has been shown that the wave functions are sensitive to the variation in the value of q-deformed parameter as well as the strength of the potential parameter λ. Furthermore, the energy eigenvalues are also considered for some particular cases where the argument of the exponential function plays a strong role effecting its value.

DOI: 10.1063/1.4883296

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Using Laplace integral transformation and the estimate transformation,the approximate analytical solutions of the 3-dimensional Schrdinger equation for q-deformed Morse potential is obtained.By means of expanding q-deformed Morse potential into the power series until the three order little term and smaller are ignored;sitting reasonably the parameter of wave function which is looked for makes the situation have essential change,so that the analytical solution without existence changes can turn into the approximate analytical solution existence;using reasonably terminal-value theorem and the convolution of Laplace integral transformation,and by applying the orthogonal condition of general Laguerre function,the explicit press of energy spectrum and the radial normalization eigenfunction of the quantum system are acquired.

Abstract In this work, the exactly solvable Schrödinger equation for s‐states of a class of multiparameter exponential‐type potential (E‐tP) is used to obtain the approximate ℓ ≠ 0 bound state solutions for the corresponding q‐deformed radial potentials. To deal with the effective potential, the Pekeris approximation for the centrifugal term is applied. The proposal has the advantage that, depending on the choice of the parameters involved in the E‐tP, several q‐deformed exponential potentials are obtained here as particular cases. That is, our proposition indicates that it is not necessary to use specialized methods to solve the Schrödinger equation for specific q‐deformed exponential potentials. At this regard, in order to show the usefulness of the proposed method, we consider the ℓ ≠ 0 bound state solutions of the q‐deformed Tietz, Hulthén, Manning‐Rosen, Shioberg, quadratic exponential, Wei and Hua among other potential models. Furthermore, with the example of the Hua potential we are contributing to solve the recent controversy on the energy spectra of this q‐deformed model. Moreover, in some applications considered in this work the results can be used to correct similar findings already published. Besides, the proposal is useful when considering new q‐deformed potentials with hypergeometric wavefunctions to be used in quantum chemical applications of diatomic molecules.

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