Numerical simulation of the space dependent fractional Schrödinger equation for London dispersion potential type.

In this study, we apply the definition of one of the fractional derivatives definitions of increasing values of the variable, which is the fractional derivative of Riemann-Liouville, and the numerical-integral methods to find numerical solutions of the fractional Schrödinger equation with the time-independent form for Van Der Walls potential type. We use the dimensionless formalism of the fractional Schrödinger equation in the space-dependent form in case of London dispersion potential in the stationary state. The solutions are found for multiple values of the space-dependent fractional Schrödinger equation parameter with a certain value of the energy. We find that the numerical solutions are physically acceptable for some values of the space dependent fractional parameter of the fractional Schrödinger equation but are not physically acceptable for others for a specific case. The numerical solutions can be applied for the systems that obey London dispersion potential type, which is resulted from the polarization of the instantaneous multi-poles of two moieties, such as soft materials systems and fluids of the inert gases.