Numerical Methods For The Schr(o)dinger Equation

This dissertation is devoted to numerical methods for the Schrodinger equation. The following two parts are included:The first is about how to overcome the barrier for a finite difference method of the one-dimensional Schroinger equation defined on the infinite integration interval to obtain the nu-merical solutions accurate than the standard precision in most computing systems of 10-16. In chapter 2, the Obrechkoff one-step method implemented in the multi precision mode is employed to obtain the numerical solutions of the Woods-Saxon potential with errors less than 10-50 and 10-30 for the bound and resonant state, respectively, within a reasonable ef-ficiency. In this dissertation, I also investigate that how to use exponentially-fitting to im-prove the Obrechkoff one-step method in finding the numerical solutions of both the bound and resonant states of the Schroinger equation. The numerical experiments show that the exponentially-fitted method, when the number of fitted coefficients is not so much, can im-prove the precision of the eigenvalues in the bound and resonant state in the lower-order case, nevertheless the advantage in the precision is not so remarkable in the higher-order case. If the coefficients of the Obrechkoff one-step method are full fitted by the exponential function as shown in our work, these methods will surpass the non-fitted Obrechkoff one-step method in accuracy and efficiency considerably for finding out the numerical solutions of the high level resonant states. In chapter 3, a new P-stable Obrechkoff two-step method, which is based on the Obrechkoff one-step method, is presented. Compared to the scheme of M. Van Daele and G. Vanden Berghe, the present one, which contains the high-order derivatives of both even and odd order, can greatly reduce the error. Five numerical examples, which includes the Woods-Saxon potential, the Morse potential, the modified Poschl-Teller potential, the Stiefel-Betis problem, and the Duffing equation, are given to illustrate the performance of this method. Chapter 4 concerns the trigonometrically-fitted two-step method with multi-derivative for the numerical solution to the one-dimensional Shrodinger equation. In this chapter, a general formula of the Phase-lag for the Obrechkoff two-step method is presented.The second is for accurate numerical solutions of the time-dependent Schrodinger equa-tion (TDSE). We present an improved space discretization scheme for the numerical solutions of the TDSE. Compared to the scheme of van Dijk and Toyama,the present one, which con- tains more terms of second-order partial derivatives, can greatly reduce the error resulting from the integration over the space. For a (2l+1)-point formula with (2l+1) terms of second-order partial derivatives, the local truncation error can decrease from the order of (â–³x)2l to (â–³x)4l, while the previous one contains only one term of second-order partial derivative. In addition, we employ the high-order Pade approximant for the time evolution operator. Two well-known numerical examples and the corresponding error analysis demonstrate that the present scheme has the advantage in the precision and efficiency over the previous one.