标题： 高斯变分逼近的磁性薛定谔方程时间积分 显示英文标题

标题： Time-integration of Gaussian variational approximation for the magnetic Schrödinger equation

摘要： 本文我们研究了半经典的磁性薛定谔方程，该方程描述了带电粒子在电磁场影响下的动力学。 通过时变的狄拉克-弗伦克尔变分原理，时变薛定谔方程的解由单一高斯波包近似。 对于这一近似，我们使用变分解参数的常微分运动方程，并将经典力学中的二阶波里斯算法推广到量子力学情形。 此外，我们提出了经典四阶龙格-库塔方法的一种修改版本。 数值实验探索了参数收敛性和几何性质。 此外，我们以潘宁阱的解析解作为基准进行对比。 显示英文摘要

摘要： In the present paper we consider the semiclassical magnetic Schr\"odinger equation, which describes the dynamics of charged particles under the influence of a electro-magnetic field. The solution of the time-dependent Schr\"odinger equation is approximated by a single Gaussian wave packet via the time-dependent Dirac--Frenkel variational principle. For the approximation we use ordinary differential equations of motion for the parameters of the variational solution and extend the second-order Boris algorithm for classical mechanics to the quantum mechanical case. In addition, we propose a modified version of the classical fourth order Runge--Kutta method. Numerical experiments explore parameter convergence and geometric properties. Moreover, we benchmark against the analytical solution of the Penning trap.