Abstract
In this paper, the general state of quantum mechanics equations that can be typically expressed by nonlinear fractional Schrödinger models will be solved based on an attractive efficient analytical technique, namely the conformable residual power series (CRPS). The fractional derivative is considered in a conformable sense. The desired analytical solution is obtained using conformable Taylor series expansion through substituting a truncated conformable fractional series and minimizing its residual errors to extract a supportive approximate solution in a rapidly convergent fractional series. This adaptation can be implemented as a novel alternative technique to deal with many nonlinear issues occurring in quantum physics. The effectiveness and feasibility of the CRPS procedures are illustrated by verifying three realistic applications. The obtained numerical results and graphical consequences indicate that the suggested method is a convenient and remarkably powerful tool in solving different types of fractional partial differential models.
Highlights
Modern physics was born after classical mechanics failed in explanation of various physical phenomena, including microscopic scales, like photoelectric effect, black body radiation, and the stability of the atoms depending on the fact that all physical quantities of a bound system are restricted to discrete values quantization
The path integral approach and the perturbation technique based on it, Feynman diagrams, became powerful tools in quantum mechanics and quantum field theory, solid-state and quantum liquid theory, equilibrium and non-equilibrium statistical physics, theory of turbulence and chaotic phenomena, theory of random processes and polymer physics, mathematics, chemistry, and economic studies
The application of conformable residual power series (CRPS) technique is extended to construct the approximate solution of the following nonlinear time-fractional Schrödinger equation (FSE): (1)
Summary
Modern physics was born after classical mechanics failed in explanation of various physical phenomena, including microscopic scales, like photoelectric effect, black body radiation, and the stability of the atoms depending on the fact that all physical quantities of a bound system are restricted to discrete values quantization. This gives us the motivation to look for numerical solutions for these systems In this analysis, the application of conformable residual power series (CRPS) technique is extended to construct the approximate solution of the following nonlinear time-fractional Schrödinger equation (FSE):. The CRPS is applied for both linear and nonlinear cases with zero and nonzero trapping potential In this light, it is assumed that fractional initial value problems (IVPs), Equations (1) and (2), have a unique and sufficiently smooth solution in the domain of interest. There is no general theory for finding a closed-form solution of these nonlinear differential equations To avoid such difficulty associated with the nonlinear terms, computer transform techniques, linearization process, or simplifying assumptions can be used to obtain solutions and consider linear approximations, and a empt to reach geometric and arithmetic interpretations that enable us to understand these phenomena further.
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