The focus of our interest will be the coupling of atomic and molecular systems to laser fields, whose maximal strength is of the order of the field exerted on an electron in the ground state of the hydrogen atom. This restriction allows us to describe the field matter interaction non-relativistically by using the time-dependent Schrödinger equation (TDSE) [1]. Analytical solutions of this linear partial differential equation are scarce, however, even in the case without external driving.
In this chapter, we continue laying the foundations for the later chapters by reviewing some basic properties of the time-dependent Schrödinger equation and by discussing two analytically solvable cases. In the following, we will then consider some general ways to rewrite, respectively, solve the time-dependent Schrödinger equation. Formulating the solution with the help of the Feynman path integral will allow us to consider an intriguing approximate, so-called semiclassical approach to the solution of the time-dependent Schrödinger equation by using classical trajectories. The last part of this chapter deals with numerical solution techniques, that will be referred to in later chapters.
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(2008). Time-Dependent Quantum Theory. In: Theoretical Femtosecond Physics. Springer Series on Atomic, Optical, and Plasma Physics, vol 48. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77897-4_2
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