As we know, the Schrödinger equation has two parts: the first part contains the first-order time derivative, and the second part has the second-order space derivative. This prominent formalism of quantum mechanics is a basis for the fractional quantum mechanics. It is an extension of the Feynman path integral formalism. The fractional derivative Schrödinger equation was generalized by Laskin two decades ago. Feynman’s path integral first introduced the concept of fractal into quantum mechanics. This concept for the first time was introduced by Mandelbrot in many research items.
Recently, the word fractional has been widely used in various sciences, especially, physics and mathematics. The mathematical appearance of this equation is similar to a diffusion equation that can be derived by taking into account probability distributions. Also, Feynman applied a path integral approach for a Gaussian probability distribution function to achieve the Schrödinger equation. This equation is obtained in a different form in quantum physics, for example, in canonical quantization of the quantum mechanics, time evolution of the wave function leads to the Schrödinger equation. The most famous equation in quantum mechanics that can explain the behavior of particles in Hilbert spaces is the Schrödinger equation. Finally, having solved some examples by using the cubic B-splines for the spatial variable, we show the plots of approximate and exact solutions with the noisy data in figures. In addition, we prove the convergence of the method and compute the order of the mentioned equations by getting an upper bound and using some theorems. Moreover, by applying a finite-difference formula to time discretization and cubic B-splines for the spatial variable, we approximate the inhomogeneous nonlinear time-fractional Schrödinger equation the simplicity of implementation and less computational cost can be mentioned as the main advantages of this method. First, we begin from the original Schrödinger equation, and then by the Caputo fractional derivative method in natural units, we introduce the fractional time-derivative Schrödinger equation. We study the inhomogeneous nonlinear time-fractional Schrödinger equation for linear potential, where the order of fractional time derivative parameter α varies between \(0 < \alpha < 1\).
0 kommentar(er)
