Supersymétrie en mécanique quantique relativiste et non relativiste et applications à des problèmes avec masse dépendante de la position par l’approche supersymétrique.

Abstract

This work concerns the application of the approach of supersymmetry in quantum mechanics (SUSYQM) to solve the one-dimensional Schrödinger, Klein-Gordon and Dirac equations for a particle of position-dependent mass subject to known potentials. Within the framework of the Schrödinger equation, we present the resolution of a model with a mass distribution that we can qualify as a Hulthén-type function and subject to a one-dimensional Morse-type potential. We solved it exactly by the generalized SUSYQM approach and obtained the spectrum of bound states and the corresponding eigenfunctions in an elegant algebraic way. In the second work, we solved an interesting model of Klein-Gordon equation with constant mass and mixing proportional scalar and vector potentials. The resolution was made using the approach of standard SUSYQM. But since the Klein-Gordon equation is not an eigenvalues equation and the effective potential explicitly depends on the energy, we used an interesting anzatz to successfully use standard SUSYQM unambiguously. Then, the energy levels are obtained in compact form with constraints on the problem parameters that must be satisfied for the physical solutions. Concerning the Dirac equation, we considered a particle whose mass depends on the position and placed in a vector potential. For a vector potential equal or opposite to the mass term, we reduced the upper component equation of the two-dimensional spinor to a position-dependent mass Schrödinger-type equation with an energy-dependent effective potential. By choosing the potential as a hyperbolic function and using generalized SUSYQM, combining with the previous ansatz, the spectrum and the eigenfunctions are explicitly obtained in a compact form.