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Pseudospectral method with symplectic algorithm for the solution of time-dependent Schrodinger equations

CHINESE PHYSICS 2007年第7期16卷 1822-1826页

作者： Bian Xue-Bin Qiao Hao-Xue Shi Ting-Yun Chinese Acad Sci Wuhan Inst Phys & Math State Key Lab Magnet Resonance & Atom & Mol Phys Wuhan 430071 Peoples R China Wuhan Univ Dept Phys Wuhan 430072 Peoples R China Chinese Acad Sci Grad Sch Beijing 100049 Peoples R China

A pseudospectral method with symplectic algorithm for the solution of time-dependent Schrodinger equations (TDSE) is introduced. The spatial part of the wavefunction is discretized into sparse grid by pseudospectral method and the time evolution is given in symplectic scheme. This method allows us to obtain a highly accurate and stable solution of TDSE. The effectiveness and efficiency of this method is demonstrated by the high-order harmonic spectra of one-dimensional atom in strong laser field as compared with previously published work. The influence of the additional static electric field is also investigated.

关键词： pseudospectral method symplectic algorithm high-order harmonic generation

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symplectic multi-level method for solving nonlinear optimal control problem

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Applied Mathematics and Mechanics(English Edition) 2010年第10期31卷 1251-1260页

作者：彭海军高强吴志刚钟万勰 Department of Engineering Mechanics State Key Laboratory of Structural Analysis of Industrial EquipmentDalian University of Technology School of Aeronautics and Astronautics State Key Laboratory of Structural Analysis of Industrial EquipmentDalian University of Technology

By converting an optimal control problem for nonlinear systems to a Hamiltonian system,a symplecitc-preserving method is *** state and costate variables are approximated by the Lagrange *** state variables at two ends of the time interval are taken as independent *** on the dual variable principle,nonlinear optimal control problems are replaced with nonlinear ***,in the implementation of the symplectic algorithm,based on the 2N algorithm,a multilevel method is *** the time grid is refined from low level to high level,the initial state and costate variables of the nonlinear equations can be obtained from the Lagrange interpolation at the low level grid to improve *** simulations show the precision and the efficiency of the proposed algorithm in this paper.

关键词： nonlinear optimal control dual variable variational principle multi-level iteration symplectic algorithm

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Ionization of 1-D model atom in an intense laser field based on asymptotic boundary conditions and symplectic algorithm

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JOURNAL OF MATHEMATICAL CHEMISTRY 2006年第1期39卷 133-149页

作者： Qi, YY Liu, XS Liu, XY Ding, PZ Jiaxing Univ Sch Elect Engn Jiaxing 314001 Peoples R China Jilin Univ Inst Atom & Mol Phys Changchun 130012 Peoples R China NE Normal Univ Dept Math Changchun 130024 Peoples R China

In this paper the asymptotic boundary condition (ABC) of 1-D model atom in the intense laser field at the spatial sufficiently far distance is presented using Fourier transformation on the condition that the initial state is local and the atomic potential in the model falls off rapidly. On the basis of this ABC, the symplectic algorithm is developed for computing the model atom in the intense laser field. The ABC and symplectic algorithm are applied to compute the ionization behaviors for 1-D Poschl-Teller short-range potential. The numerical results illustrate that the ABC and the symplectic algorithm presented are reasonable and effective for 1-D model atom in the intense laser field.

关键词： asymptotic boundary condition symplectic algorithm 1-D Poschl-Teller short-range potential ionization probability

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The enhancement of efficiency of high-order harmonic in intense laser field based on asymptotic boundary conditions and symplectic algorithm

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JOURNAL OF MATHEMATICAL CHEMISTRY 2006年第3-4期39卷 451-463页

作者： Zhang, Chunli Liu, Xueshen Ding, Pezhu Qi, Yueying Jilin Univ Inst Atom & Mol Phys Changchun 130012 Peoples R China Jiaxing Univ Sch Elect Engn Jiaxing 314001 Peoples R China

Asymptotic boundary condition (ABC) of laser-atom interaction presented recently is applied to transform the initial value problem of the time-dependent Schradinger equation (TDSE) in infinite space into the initial and boundary value problem in the finite space, and then the TDSE is discretized into linear canonical equations by substituting the symmetry difference quotient for the 2-order partial derivative. The canonical equation is solved by symplectic algorithm. The ground state and the equal weight coherent superposition of the ground state and the first excited state have been taken as the initial conditions, respectively, while we calculate the population of bound states, the evolution of average distance and the high-order harmonic generation (HHG). The conversion efficiency of HHG can be enhanced by initial coherent superposition state and moderate laser intensities.

关键词： asymptotic boundary condition symplectic algorithm high-order harmonic generation superposition state

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Continuum eigen-functions of 1-D time-independent Schrodinger equation solved by symplectic algorithm

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INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 2005年第1期101卷 21-26页

作者： Qi, YY Liu, XS Ding, PZ Jilin Univ Inst Atom & Mol Phys Changchun 130012 Peoples R China

We present the symplectic algorithm for solving the continuum eigen-functions of 1-dimensional (1D) time-independent Schrodinger equation, which keeps Wronskian conservation of linearly independent continuum eigen-functions. This symplectic algorithm has been applied to the computation of the linearly independent continuum eigen-functions of both 1D soft-coulomb potential and Poshl-Teller short-range potential as well as the radial continuum eigen-functions of hydrogen atom. The numerical results using the symplectic algorithm are in agreement with the existing theories. (C) 2004 Wiley Periodicals, Inc.

关键词： continuum eigen-functions symplectic algorithm time-independent Schrodinger equation Wronskian

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Computation of quasiclassical trajectories by symplectic algorithm for the N(⁴S) + O₂(X³Σ_g^-) → NO(X²Π) + O(³P) reaction system

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JOURNAL OF MATHEMATICAL CHEMISTRY 2005年第2期37卷 127-138页

作者： He, JF Hua, W Liu, XS Ding, PZ Jilin Univ Inst Atom & Mol Phys Changchun 130012 Jilin Peoples R China

Computation of quasiclassical trajectories for the N(S-4) + O-2(X(3)Sigma(g) over bar). NO(X(2)Pi) +O(P-3) atmospheric reaction system, based on a new ground potential energy surface reported by R. Sayos et al., has been performed in this work by means of both the fourth-order explicit symplectic algorithm (S4) and the fourth-order Runge-Kutta scheme (RK4), and then computed results of two schemes are compared. It is shown that RK4 cannot preserve energy conservation and symplectic structure of the reaction system, which results in the bad veracity of the trajectory calculation. RK4 cannot rightly reflect both the colliding mode and the reaction mode of the trajectories. Moreover, the amplitudes of vibration of the reactant molecule and the product molecule become gradually small with the time increasing, and their rotation-vibrational levels in fact vary during the integration. For these reasons, RK4 cannot assure the accuracy of the quasiclassical trajectory (QCT) study of the atmospheric reaction. However, S4 maintains these characteristics and can actually describe the circumstance of the reaction system. S4 is better than RK4 is prospective in the QCT study of the chemical reaction.

关键词： symplectic algorithm quasiclassical trajectory chemical reaction

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Modeling and computation for the high-speed rotating flexible structure

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JOURNAL OF VIBRATION AND ACOUSTICS-TRANSACTIONS OF THE ASME 2008年第4期130卷 041005-1-041005-1页

作者： Huang, Yong-an Yin, Zhou-ping Xiong, You-lun Huazhong Univ Sci & Technol State Key Lab Digital Mfg Equipment & Technol Wuhan 430074 Peoples R China Huazhong Univ Sci & Technol Sch Mech Sci & Engn Wuhan 430074 Peoples R China

This paper is presented to improve the modeling accuracy and the computational stability for a high-speed rotating flexible structure. The differential governing equations are derived based on the first-order approximation coupling (FOAC) model theory in the framework of the generalized Hamiltonian principle. The semi-discrete model is obtained by the finite element method, and a new shape function based on FOAC is established for the piezoelectric layers. To increase the efficiency, accuracy, and stability of computation, first, the second-order half-implicit symplectic Runge-Kutta method is presented to keep the computational stability of the numerical simulation in a long period of time. Then, the idea of a precise integration method is introduced into the symplectic geometric algorithm. An improved symplectic precise integration method is developed to increase accuracy and efficiency. Several numerical examples are adopted to show the promise of the modeling and the computational method.

关键词： rotating flexible structure multibody dynamics piezoelectric material Hamiltonian principle symplectic algorithm precise integration method

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The quantitative analysis of enhancement of high-order harmonics in two-color intense laser fields

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JOURNAL OF MATHEMATICAL CHEMISTRY 2008年第4期43卷 1429-1436页

作者： Zhang, Chunli Liu, Xueshen Ding, Peizhu Jilin Univ Inst Atom & Mol Phys Changchun 130012 Peoples R China

The time-dependent Schrodinger equation of the interaction of laser pulse with He(+) is solved by using the asymptotic boundary condition and symplectic algorithm in fundamental laser-field and two-color laser fields. We find that the conversion efficiency of high-order harmonic generation (HHG) is higher in the two-color laser fields than in the fundamental laser field, especially for the combination of omega(0) - 19 omega(0). To explain these phenomena, the ionization, the average distance, the probability of first excited sate, and the transition probability are calculated. We give the qualitative and quantitative analysis for the enhancement of conversion efficiency of HHG.

关键词： high-order harmonic generation two-color laser field symplectic algorithm transition probability quantitative analysis

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Static/dynamic Analysis of Functionally Graded and Layered Magneto-electro-elastic Plate/pipe under Hamiltonian System

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Chinese Journal of Aeronautics 2008年第1期21卷 35-42页

作者：代海涛北京航空航天大学航空科学与工程学院

The 3-dimensional couple equations of magneto-electro-elastic structures are derived under Hamiltonian system based on the Hamilton principle. The problem of single sort of variables is converted into the problem of double sorts of variables, and the Hamilton canonical equations are established. The 3-dimensional problem of magneto-electro-elastic structure which is investigated in Euclidean space commonly is converted into symplectic system. At the same time the Lagrange system is converted into Hamiltonian system. As an example, the dynamic characteristics of the simply supported functionally graded magneto-electro-elastic material （FGMM） plate and pipe are investigated. Finally, the problem is solved by symplectic algorithm. The results show that the physical quantities of displacement, electric potential and magnetic potential etc. change continuously at the interfaces between layers under the transverse pressure while some other physical quantities such as the stress, electric and magnetic displacement are not continuous. The dynamic stiffness is increased by the piezoelectric effect while decreased by the piezomagnetic effect.

关键词： functionally graded magneto-electro-elastic material Hamiltonian system symplectic algorithm

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Recent Progress in symplectic algorithms for Use in Quantum Systems

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Communications in Computational Physics 2007年第1期2卷 1-53页

作者： Xue-Shen Liu Yue-Ying Qi Jian-Feng He Pei-Zhu Ding Institute of Atomic and Molecular Physics Jilin UniversityChangchun 130012P.R.China. School of Electrical Engineering Jiaxing UniversityJiaxing 314001P.R.China. Department of Physics School of ScienceBeijing Institute of TechnologyBeijing 100081P.R.China.

In this paper we survey recent progress in symplectic algorithms for use in quantum systems in the following topics:symplectic schemes for solving Hamiltonian systems;Classical trajectories of diatomic systems,model molecule A2B,Hydrogen ion H+2 and elementary atmospheric reaction N(4S)+O2(X 3Σ−g)→NO(X 2Π)+O(3P)calculated by means of Runge-Kutta methods and symplectic methods;the classical dissociation of the HF molecule and classical dynamics of H+2 in an intense laser field;the symplectic form and symplectic-scheme shooting method for the time-independent Schr¨odinger equation;the computation of continuum eigenfunction of the Schr¨odinger equation;asymptotic boundary conditions for solving the time-dependent Schr¨odinger equation of an atom in an intense laser field;symplectic discretization based on asymptotic boundary condition and the numerical eigenfunction expansion;and applications in computing multi-photon ionization,above-threshold ionization,Rabbi oscillation and high-order harmonic generation of laser-atom interaction.

关键词： Quantum system symplectic algorithm classical trajectory Schr¨odinger equation intense laser field.

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