Résumé:
"This thesis provides a comprehensive discussion of the problem of a hydrogenic system in a
curve, approaching the Feynman path integral space.
Two types of curved spaces are considered. The first is a spherical space curve , is a curve
characterized by a space of positive curvature constant and the second space is a hyperbolic
curve which is a continuous space with negative curvature .
The propagator for a particle in a symmetrical potential in three-dimensional space having a
general metric tensor defined in polar coordinates on the basis of developed spherical
harmonics using the formulation based on the definition of the product form '' ' . The
calculation depends on the radial propagator specification of the function f (r) which depends
on the metric.
In both cases (spherical and hyperbolic curve space) , the radial Green function is built in a
compact form . The spectrum energy and the wave functions properly standardized are
respectively extracted by the poles and residues of the Green's function radial. In the case of
movement in spherical curve space , the particle has only bound states and that of hyperbolic
space curve, there are also scattering states with continuous energy spectrum extends from ( (
ℏ ² ) / ( 2MR ²) ) + ( ( Ze ² ) / R ) to infinity . In the limit of flat space, is when R tends to
infinity, the spectrum energy and wave functions of the hydrogen atom (Z = 1) are exactly
recovered. In particular, the problem of the missing factor in the expression of the wave
function of continuous states in flat space is clarified."
