The Gaussian integration method of the Schrodinger equation and quantum 1-D theory of low gain free electron laser

We study the time-dependent solutions of Schrodinger equations ruled by different non-singular potentials. We employ a recendy 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 electronlaser dynamics. Finally, we comment on future developments of this line of research regarding the relevant comparison with other exact and approximate integration schemes. Published under license by AIP Publishing.