SOLVING THE DISCRETIZED TIME-INDEPENDENT SCHRODINGER-EQUATION WITH THE LANCZOS PROCEDURE

A new method is presented to find bound state solutions of the one‐, two‐, or three‐dimensional Schrödinger equation. The equation is turned into a sparse matrix eigenvalue problem by representing the potential energy surface and the wave function on a grid. The Laplacian is represented by a high (10th) order finite difference formula. Eigenvalues are found by the Lanczos procedure [J. Cullum and R. A. Willoughby, J. Comp. Phys. 44, 329 (1981)] and transition probabilities (Franck–Condon factors) are found by the recursive residue generation method [A. Nauts and R. E. Wyatt, Phys. Rev. Lett. 51, 2238 (1983)]. Examples are given for the 1D Morse oscillator and the 2D Hénon‐Heiles potential. Numerical convergence can be checked easily and highly accurate results can be obtained. The algorithm is fast, easy to implement, and vectorizable. © 1990, American Institute of Physics. All rights reserved.