Two Novel Space-Time Polynomial Particular Solutions Methods for the Time-Dependent Schrödinger Equation

The conventional method of polynomial particular solutions is only applicable to partial differential equations on the real number field. Building on this method, this paper proposes two novel approaches for numerical simulation of the time-dependent Schrödinger equation. Under the assumption of treating the time variable as an ordinary spatial variable, the first approach approximates the real and imaginary parts of the equation using two different sets of linear combinations, which are then substituted into the corresponding original governing equations to form a coupled differential equation. The second approach is derived based on the basic form of complex coefficient partial differential equations and involves polynomial particular solutions with imaginary terms. Using the same numerical examples, compared to conventional finite difference method, Fourier spectral method and radial basis function collocation method, this algorithm's stability is not limited by the grid ratio of the time step and spatial step. It is not only simple and feasible but also suitable for solving high-dimensional problems.