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Reports on Mathematical Physics

Exact Analytical Solutions of the Wave Function for Some q-Deformed Potentials

M. Sebawe Abdalla, Hichem Eleuch, T. Barakat

Published Apr 1, 2013• 22 citations
Publisher
DOI
10.1016/s0034-4877(13)60031-2

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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.
The Dirac equation of q-deformed hyperbolic Manning Rosen potential in D dimension was solved by using Supersymmetric Quantum Mechanics (SUSY QM). The D dimensional relativistic energy spectra were obtained by using SUSY QM and shape invariant properties and D dimensional wave functions of q-deformed hyperbolic Manning Rosen potential were obtained by using the SUSY raising and lowering operators. In the nonrelativistic limit, the relativistic energy spectra for exact spin symmetry case reduced into nonrelativistic energy spectra and so for the wave functions. In the classical regime, the partition function, the vibrational specific heat, and the vibrational mean energy of some diatomic molecules were calculated from the non-relativistic energy spectra with the help of error function and imaginary error function.
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.