In the present paper we consider the semiclassical magnetic Schr\"odinger equation, which describes the dynamics of particles under the influence of a 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 derive ordinary differential equations of motion for the parameters of the variational solution. Moreover, we prove $L^2$-error bounds and observable error bounds for the approximating Gaussian wave packet.

翻译：本文考虑半经典磁薛定谔方程，该方程描述了粒子在磁场影响下的动力学行为。通过含时狄拉克-弗伦克尔变分原理，我们采用单个高斯波包对含时薛定谔方程的解进行近似。针对该近似，我们推导了变分解参数的常微分运动方程，并进一步证明了近似高斯波包的$L^2$误差界与可观测量误差界。