Abstract
We study the time-dependent solutions of Schrödinger equations ruled by different non-singular potentials. We employ a recently proposed integration procedure, assuming a time-dependent Gaussian shape for the wave function. The method is independent of the specific form of the potential and allows a straightforward separation of the time and spatial variables. Here, we reconsider the integration method by the use of the formalism of two-variable Hermite polynomials providing a very simple derivation of the relevant physical quantities. This method is eventually exploited to study different problems including anharmonic oscillators and pendulum-like potentials. Regarding the case of periodic potentials, we touch on the application of the method to the quantum free-electron laser dynamics. Finally, we comment on future developments of this line of research regarding the relevant comparison with other exact and approximate integration schemes.
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