Unitary transformation technique for the study of quantum dynamics of charged particle in time-dependent electromagnetic field

In this paper, we present a unitary transformation technique to solve the equation of quantum dynamics of a charged particle moving in a time-dependent electromagnetic field. The primary advantage of the technique lies in the connection of this technique with one of the essential features of the system which is the gauge. The gauge has been chosen in a way where the first three dimensional Hamiltonian transformed easily through this technique to a two dimensional coupled problem. This problem is central in several areas of physics, including solid-state and plasma physics, where the understanding of the behavior of charged particles in the presence of time-dependent electromagnetic fields is essential to explore the behavior of various phenomena. The complete solution of the Schrödinger equation is presented and the propagator is also determined using this technique. This technique is also used to shows how the problem can be reduced to a two-dimensional system of coupled harmonic oscillators, each subjected to a time-dependent linear potential. This reduction simplifies the analysis by transforming the problem into a form that can be mapped onto a well-established model, which has been previously studied using the path integral formalism.