Summary: Several finite difference schemes are discussed for solving the two-dimensional Schrödinger equation with Dirichlet’s boundary conditions. We use three fully implicit finite difference schemes, two fully explicit finite difference techniques, an alternating direction implicit procedure and the explicit formula of H. Z. Barakat and J. A. Clark [On the solution of the diffusion equation by numerical methods. ASME J. Heat Transf. 88, 421 (1966)]. Theoretical and numerical comparisons between four families of methods are described.
The main advantage of the alternating direction implicit finite difference technique is that the bandwidth of the sets of equations is a fixed small number that depends only on the nature of the computational molecule. This allows the use of very efficient and very fast techniques for solving the resulting tridiagonal systems of linear algebraic equations. The unique advantage of the Barakat and Clark technique is that it is unconditionally stable and is explicit in nature. Numerical results are presented followed by concluding remarks.