Commun. Comput. Phys., 2 (2007), pp. 1-53.


Recent Progress in Symplectic Algorithms for Use in Quantum Systems

Xue-Shen Liu 1*, Yue-Ying Qi 2, Jian-Feng He 3, Pei-Zhu Ding 1

1 Institute of Atomic and Molecular Physics, Jilin University, Changchun 130012, P. R. China.
2 Institute of Atomic and Molecular Physics, Jilin University, Changchun 130012, P. R. China/School of Electrical Engineering, Jiaxing University, Jiaxing 314001, P. R. China.
3 Institute of Atomic and Molecular Physics, Jilin University, Changchun 130012, P. R. China/Department of Physics, School of Science, Beijing Institute of Technology, Beijing 100081, P. R. China.

Received 20 December 2005; Accepted (in revised version) 5 April 2006
Available online 30 August 2006

Abstract

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 A$_{2}$B, Hydrogen ion H$_{2}^{ + }$ and elementary atmospheric reaction N($^{4}$S)+O$_{2}$(X$^{3}\Sigma _{g}^{ - }) \to $NO(X$^{2}\Pi )$+O($^{3}$P) calculated by means of Runge-Kutta methods and symplectic methods; the classical dissociation of the HF molecule and classical dynamics of H$_{2}^{ + }$ in an intense laser field; the symplectic form and symplectic-scheme shooting methods for the time-independent Schr\"{o}dinger equation; the computation of continuum eigenfunction of the Schr\"{o}dinger equation; asymptotic boundary conditions for solving the time-dependent Schr\"{o}dinger 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.


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PACS: 47.10.Df, 02.70.-c, 34, 95.30.-k
Key words: Quantum system, symplectic algorithm, classical trajectory; Schrodinger equation, intense laser field.

*Corresponding author.
Email: liuxs@jlu.edu.cn (X. S. Liu), jess1234yao@yahoo.com.cn (Y. Y. Qi), hjf@bit.edu.cn (J. F. He), dpz@mail.jlu.edu.cn (P. Z. Ding)
 

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