Solving the discretized time-independent Schrödinger 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.