Solving Schr?dinger Equation Based On The Legendre-Fourier-Galerkin Spectral Method

Schr?dinger's equation is a basic hypothesis of quantum mechanics and a ba-sic equation of quantum mechanics.It reveals the basic laws of matter movement in the microphysics world.It is a powerful tool for atomic physics to deal with all non-relativistic issues.The problem is widely used,so it is particularly important to find the numerical solution of the Schr?dinger equation for the corresponding problem.However,given the initial boundary value conditions of the Schr?dinger equation and finding its corresponding numerical solution with high accuracy,there are still some problems.Based on the above It is of great significance to design an effective and high-precision numerical method to solve the Schr?dinger equationThis paper combines the high-precision Fourier-Galerkin spectral method and the Legendre-Galerkin spectral method to solve the Schr?dinger equation.The Schr?dinger equation is first transformed from the Cartesian coordinate system to the polar coordinate system for solving,and then the Fourier-Galerkin spectral method and Legendre-Galerkin The spectral method performs spectral expansion on the spatial direction of the two-dimensional Schr?dinger equation,and final-ly uses the Crank-Nicolson method to expand approximately in the time direc-tion.Among them,the base function selected by the Legendre-Galerkin spectral method has orthogonality and interpolation,which makes the algorithm highly accurate and greatly reduces the calculation amount.The Crank-Nicolson method also has the advantages of unconditional stability and high accuracyThe main tasks of this article are as followsFirstly,the Fourier-Galerkin spectral method is used to solve the elliptic equa-tion,and the Legendre-Galerkin spectral method is used to solve the Helmholtz equation.The two methods are described with high accuracy.The two spectral methods are combined to solve the two-dimensional Helmholtz equation.It shows that this method has high accuracy and stabilitySecondly,the combination method is used to solve the Schr?dinger equa-tion,and the Schr?dinger equation with zero boundary conditions is solved.The numerical solution given is the solution of the corresponding equationFinally,Matlab is used for numerical simulation,and the simulation image of the numerical solution is given by programming.The obtained results can show that this method has high accuracy and stability.