Propagator of a non-relativistic quantum particle in an electric field

The aim of this thesis is to present a comprehensive investigation into the propagator calculation for a non-relativistic particle with charge q and time-dependent mass m(t) in a time-varying electric field ε(t). To achieve this goal, we have adopted four different methodologies, each supported by a theoretical framework that validates our approach. We begin by introducing the Path integral formulation, from its origins in the double-slit experiment to its mathematical expression for the transition amplitude, in Chapter 2. Next, in Chapter 3, we delve into the physics behind the Van Vleck-Pauli-Morette semi-classical formula for the propagator. In Chapter 4, we have introduced the time evolution operator and the Dyson series. Continuing through Chapter 5, we provide an overview of the Baker–Campbell–Hausdorff formula, starting from the Magnus expansion and explaining the main mathematical building blocks. Finally, in Chapter 6, we explain the method of characteristics for partial differential equations. In each chapter, we apply the theory to the specific case of a time-dependent-mass system in the presence of a linear time-dependent potential.