Symbolic-Numerical Algorithm for Solving the Time-Dependent Schrödinger Equation by Split-Operator Method
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A new computational approach is proposed for the solution of the time-dependent Schrödinger equation (TDSE), in which a symbolic algorithm named GATEO and a numerical scheme based on the finite-element method (FEM) are effectively composed. The GATEO generates the multi-layer operator-difference scheme for TDSE and evaluates the effective Hamiltonian from the original time-dependent Hamiltonian by means of the Magnus expansion and the Pade-approximation. In order to solve the TDSE with the effective Hamiltonian thus obtained, the FEM is applied to a discretization of spatial domain which brings the difference scheme in operator form to the one in algebraic form. The efficiency and accuracy of GATEO and the numerical scheme associated with FEM is confirmed in the second-, fourth-, and sixth-order time-step computations for certain integrable atomic models with external fields.
KeywordsGauge Transformation Algebraic Problem Gaussian Wave Packet Potential Function Versus Partial Splitting
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