Analytical and numerical solutions of the time fractional Schrödinger equation for generalized Morse potential

Abstract

This paper investigates the fractional Schrödinger equation (FSE) with the Caputo time fractional derivative for the generalized Morse potential, which has not yet been presented for this equation. This study depends on the analytical solution of the FSE by the method of integral transforms and the numerical solutions are presented by plotting the eigensolutions with the Python script. For this purpose, we apply a special ansatz solution together with the Fourier transform (for the space variable) and the Laplace transform (with respect to time) on the FSE and obtain the Gaussian hypergeometric differential equation. By applying the inverse Fourier transform on the solution of the hypergeometric function, the G-Meijer function in terms of the coordinate and the Laplace transformed variable are obtained. We then calculate the wave function of the time fractional Schrödinger using the inverse Laplace transform together considering the Schouten-Vanderpol theorem and some special circumstances of the problem. The obtained results show that for different values of the time fractional parameter, the probability of the particle presence is time-dependent, and in the limit case of [Formula: see text], the solutions obtained from the time FSE are consistent with the results of standard Schrödinger equation for the generalized Morse potential. The results also show that the amplitude of wave function of the particle presence decreases over time and the energy of the system decreases in small times for different values of the fractional parameter and for the large times, the amount of energy is almost constant.