Constrained parameters and exact solution of Schrödinger equation of charged particle in a time-dependent electric field: A unitary transformation approach

We present an exact analytical solution for the quantum dynamics of a charged particle subjected to both a time-dependent electric field and a static magnetic field aligned along the $z$-direction. Using a systematic approach based on successive unitary transformations, we reduce the original three-dimensional problem to a two-dimensional system of decoupled, time-dependent harmonic oscillators. This technique produces free parameters that allow us to impose constrains to derive the exact solution of Schrödinger equation with the time-dependent Hamiltonian through the explicit derivation of the quantum propagator and shows the equivalence of this approach to established path integral methods for such systems. The developed framework provides new insights into quantum systems with time-dependent electromagnetic fields and offers analytical solutions.