Numerical solution of one-dimensional time-independent Schr�dinger equation by using symplectic schemes

Abstract

The symplectic schemes are extended to the solution of one-dimensional time-independent Schrödinger equation. The Schrödinger equation is first transformed into a Hamiltonian canonical equation by means of the Legendre transformation, and then two methods are developed to solve the numerical solution of the one-dimensional time-independent Schrödinger equation: the symplectic scheme-matrix eigenvalue method and the symplectic scheme-shooting method. Both methods are applied to the calculations of a one-dimensional harmonic oscillator, the hydrogen atom, and a double-well anharmonic oscillator. It is shown that the numerical results of the two methods are nearly the same and are in good agreement with the exact ones when the step length is taken to be properly small. The computation with the symplectic scheme-shooting method spends less computer time than that with the symplectic scheme-matrix eigenvalue method. And thus the symplectic scheme-shooting method is a better numerical method for the calculation of the eigenvalue problem. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 343–349, 2000