Treatment of quantum motions by path integral approach.

Abstract

This thesis is devoted to the study of non-relativistic quantum systems with explicitly time and position-time dependent coefficients in the framework of the Feynman's path integrals formalism. We have presented a systematic method for constructing the propagator of time-dependent systems in both configuration and phase spaces. As application, we have considered the problem of harmonic oscillator with both mass and frequency being arbitrary functions of time. The treatment has been based on the use of explicitly time-dependent coordinate transformations as well as of time transformations, which permit to reduce the propagator to that with constant mass and frequency. We have illustrated the general result by choosing some models of varying mass and frequency. On the other hand, we have extended the space-time transformations technique to bring the problem of a particle with time-dependent mass moving in two- dimensional space and subjected to Coulomb plus inverse quadratic potential to a stationary problem. Then, polar coordinates were adequate for evaluating the Green's function and exactly deducing the discrete spectrum energy levels and the relating wave functions. We have been also interested in developing a systematic procedure to study one-dimensional path integral in phase space for a class of position-time dependent masses and time dependent potentials. Thanks to an explicitly time dependent canonical transformation, we have been able to absorb the time dependence of the Hamiltonian. As application, we have considered two different mass distributions each associated with a chosen potential so that the corresponding path integral have been exactly solved. We have also obtained exact propagators for a particle confined in infinite square well and further subjected to some potentials. The Green's function have been constructed for each situation thanks to an appropiate point canonical transformation. Finally, we have found the path integral solution for an electrically charged particle in orbit around a dyon. Judicious regulating functions have permitted to express the promotor as a product of two partial kernels that are the problems of Morse and Pöschl-Teller potentials.