Integrale de chemin en mecanique quantique

Abstract

The purpose of this work is the study of some non-relativistic quantum mechanics systems in the context of Feynman path integral approach by using the coordinate time transformations technique and mathematical tools necessary to solve them as simple as possible. Whenever possible, the wave functions and the corresponding spectra are compared with those obtained in the framework of classical and quantum mechanics. The second chapter is devoted to the study of the motion of a free particle but constrained to move on the conical surface by means of a constraint which represent the equation of the cone. We have adopted the mid-point principle and with suitable coordinate and time transformations, variables were separated, the spectrum and wave functions of bound states have been accurately deduced. In the third chapter we have reconsidered the problem discussed above. This same particle is subjected to the action of an inverse quadratic oscillator. As the potential has a singularity at the origin, it seemed necessary to reject it at infinity by using a spatial transformation followed by a temporal one. Green's function of this problem is reduced to that associated to the Morse potential whose solution has long been known. The discrete spectrum and wave functions of bound states were obtained. The fourth chapter deals with the study, in the phase space path integral approach, of two problems on a circle which are the singular oscillator and the singular Coulomb problems. The technique used is based mainly on the delta functional and on the Hamiltonian formalism. The expression of the propagator of the singular oscillator has been developed with maximum detail and clarity. The discrete spectrum of the energy is exact and fully consistent with the literature. Through a duality transformation, we have established a relation-ship between the singular oscillator and the singular Coulomb systems. This link is at the origin of the similarity of the two propagators expressions form. For both problems, the propagator has been reduced to that of the well-known Pöschl-Teller problem discussed earlier in the context of the Schrödinger formulation and in the configuration space path integral. In other words, the duality transformation has allowed us and without making calculations to deduce the solutions of the singular Coulomb problem from those of singular oscillator one.