Propagator of a Charged Particle with a Spin in Uniform Magnetic and   Perpendicular Electric Fields
  We construct an explicit solution of the Cauchy initial value problem for the time-dependent Schroedinger equation for a charged particle with a spin moving in a uniform magnetic field and a perpendicular electric field varying with time. The corresponding Green function (propagator) is given in terms of elementary functions and certain integrals of the fields with a characteristic function, which should be found as an analytic or numerical solution of the equation of motion for the classical oscillator with a time-dependent frequency. We discuss a particular solution of a related nonlinear Schroedinger equation and some special and limiting cases are outlined. 
Ricardo Cordero-Soto, Raquel M. Lopez, Erwin Suazo, and Sergei K.   Suslov