求解含时薛定谔方程的四阶梯度辛积分方法
Fourth Order Gradient Symplectic Integrator Methods for Solving the Time-Dependent Schrödinger Equation
摘要：
我们表明，该方法分裂的运营商${\rm e}^{ε(T+V)}$四阶纯正系数产生优秀的算法求解时间相关的薛定谔方程。这些算法需要知道势和势的梯度。一个四阶算法每次迭代只需要四个快速傅立叶变换。在一维散射问题中，这些新算法的四阶误差系数比基于传统Ruth-Forest辛积分器的负系数四阶算法小约500倍.这些算法可以使用5到10倍大的时间步长产生常规二阶或四阶算法的收敛结果。将这些正系数算法迭代到6阶也比将Ruth-Forest算法迭代到6阶或直接使用Yoshida的6阶算法A产生更好的收敛算法。
英文摘要：
We show that the method of splitting the operator ${\rm e}^{ε(T+V)}$ to fourth order with purely positive coefficients produces excellent algorithms for solving the time-dependent Schrödinger equation. These algorithms require knowing the potential and the gradient of the potential. One 4th order algorithm only requires four Fast Fourier Transformations per iteration. In a one dimensional scattering problem, the 4th order error coefficients of these new algorithms are roughly 500 times smaller than fourth order algorithms with negative coefficient, such as those based on the traditional Ruth-Forest symplectic integrator. These algorithms can produce converged results of conventional second or fourth order algorithms using time steps 5 to 10 times as large. Iterating these positive coefficient algorithms to 6th order also produced better converged algorithms than iterating the Ruth-Forest algorithm to 6th order or using Yoshida's 6th order algorithm A directly.
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