LowOrder Polynomial Approximation of Propagators for the TimeDependent Schrödinger Equation
Abstract
Polynomial approximations for propagating the timedependent Schrödinger equation are studied. These methods are motivated by the numerical demands of systems with timedependent Hamiltonian operators. Firstorder and secondorder Magnus expansions are tested for approximating the time ordering operator. The polynomials considered are based on a reduced basis space which is obtained by iterating the Hamiltonian operator on an initial wavefunction. The approximate polynomials are obtained by minimizing the error in the propagation. One such approach which minimizes the residuum outside the reduced space is proved to be equivalent to the short time iterative Lanczos procedure. A second approach which uses a different optimization scheme (the residuum method) is found to be somewhat superior to the Lanczos procedure. Numerical examples are used to illustrate the convergence of these methods. The secondorder Magnus approximation is found to be a significant improvement over the firstorder approximation regardless of method.
 Publication:

Journal of Computational Physics
 Pub Date:
 May 1992
 DOI:
 10.1016/00219991(92)90318S
 Bibcode:
 1992JCoPh.100..179T
 Keywords:

 Approximation;
 Polynomials;
 Schroedinger Equation;
 Time Dependence;
 Hamiltonian Functions;
 Optimization;
 Propagation;
 Atomic and Molecular Physics