Solution of the Schrödinger Equation by a Spectral Method

A new computational method for determining the eigenvalues and eigenfunctions of the Schrödinger equation is described. Conventional methods for solving this problem rely on diagonalization of a Hamiltonian matrix or iterative numerical solutions of a time independent wave equation. The new method, in contrast, is based on the spectral properties of solutions to the time-dependent Schrodinger equation. The method requires the computation of a correlation function <ψ( r, 0)| ψ( r, t)> from a numerical solution ψ( r, t). Fourier analysis of this correlation function reveals a set of resonant peaks that correspond to the stationary states of the system. Analysis of the location of these peaks reveals the eigenvalues with high accuracy. Additional Fourier transforms of ψ( r, t) with respect to time generate the eigenfunctions. The effectiveness of the method is demonstrated for a one-dimensional asymmetric double well potential and for the two-dimensional Hénon-Heiles potential.