A new kind of discretization scheme for solving a two-dimensional time-independent Schrödinger equation

In this paper we present a new kind of discretization scheme for solving a two-dimensional time-independent Schrödinger equation. The scheme uses a symmetrical multi-point difference formula to represent the partial differentials of the two-dimensional variables, which can improve the accuracy of the numerical solutions to the order of Δx2Nq+2Δx2Nq+2 when a (2Nq+1)(2Nq+1)-point formula is used for any positive integer NqNq with Δx=ΔyΔx=Δy, while Nq=1Nq=1 equivalent to the traditional scheme. On the other hand, the new scheme keeps the same form of the traditional matrix equation so that the standard algebraic eigenvalue algorithm with a real, symmetric, large sparse matrix is still applicable. Therefore, for the same dimension, only a little more CPU time than the traditional one should be used for diagonalizing the matrix. The numerical examples of the two-dimensional harmonic oscillator and the two-dimensional Henon–Heiles potential demonstrate that by using the new method, the error in the numerical solutions can be reduced steadily and extensively through the increase of NqNq, which is more efficient than the traditional methods through the decrease of the step size.