Résumé:
This thesis uses Feynman’s path integrals formalism to exp lore quantum systems from
b oth relativistic and non-relativistic p ersp ectives. We have provided a brief overview of
the p ath integral approach an d the Duru-Kleinert space-time transformation. Three problems have b een examined for the non-relativistic regime: the trigonometric Pöschl-Teller,
the mo dified Pöschl-Teller and the Generalized Inverse Quadratic Yu kawa (GIQY) Potentials. Our approach to the fi rst two problems involves expanding the available s-states
solutions using the Greene-Aldrich approximation scheme to comp ensate for the centrifu gal term. However, in addition to approximating the 1/r and 1/r2 terms, the Generalized
Inverse Quadratic Yukawa Potential treatment relied on an appropriate space-time transformation that allowed the propagator to be reduced to that of a modified Pöschl-Teller
problem. We have compared the results to some previous approaches and it was satisfactory. For the relativistic regime, two problems have been considered: we have investigated
the problem of a spinless particle subjected to Generalized Inverse Quadratic Yukawa potential. Path integral representation and its corresponding Green’s function has been
derived with the help of the previous space-time transformation. Particular cases were
also considered, which made it possible to make comparison with other results obtained
differently. As a second relativistic problem we have dealt with a spin-1/2 particle in vector and scalar potentials of GIQY type. Both spin and pseudospin symmetries were taken
into account. From the four coupled partial differential equations included in Dirac equation, two Schrödinger’s like equations has been derived . For every equation, a Green’s
function has been evaluated. Thanks to a space-time transformation, we have been able
to integrate the Green’s function and deduce the discrete spectrum energy levels and
the upper and lower components of corresponding wave functions. numerical results and
special cases were also presented in the last of this wo
