Recent progress in symplectic algorithms for use in quantum systems

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 H₂⁺ and elementary atmospheric reaction N(⁴S) + O₂ (X³∑_g ^-)→NO(X₂Π)+O(³P) calculated by means of Runge-Kutta methods and symplectic methods; the classical dissociation of the HF molecule and classical dynamics of H₂⁺ in an intense laser field; the symplectic form and symplectic-scheme shooting method for the time-independent Schrödinger equation; the computation of continuum eigenfunction of the Schrödinger equation; asymptotic boundary conditions for solving the time-dependent Schrödinger 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.