Influência do sistema Aharonov-Bohm-Coulomb na dinâmica do oscilador de Dirac e de uma partícula com massa efetiva dependente da posição

Abstract

In this dissertation, we study the influence of the Aharonov-Bohm-Coulomb (ABC) system on the relativistic and nonrelativistic quantum dynamics of the Dirac oscillator (DO) and of a Dirac particle with position-dependent effective mass (PDEM). To obtain the bound-state solutions of these two problems, we use the projection operators left-handed and right-handed. With respect to the results of the first problem, we verified that Dirac spinor is written in terms of the biconfluent Heun functions and the relativistic energy spectrum depends on the parameters Z and Φ which characterize the ABC system and of the cyclotron frequency ωc characterizes a uniform magnetic field. We observed that Z has the function of increase the values of this spectrum, while ωc has the function of decrease. In particular, we note a peculiar quantum effect: the angular frequencies of the OD are quantized in terms of quantum numbers. In addition to these frequencies being real and positive quantities, are linearly proportional to ωc and increase quadratically with Z and the energies of the system. By consequence of this unusual result, the energies of the ground state are greater of the that the of the first excited state. When we analyze the nonrelativistic limit of this first problem, we obtain the equation of the quantum harmonic oscillator (QHO) with a strong spin-orbit coupling in the presence of a uniform magnetic field and under the influence of the ABC system. Unlike of the DO, the frequencies of the QHO increase linearly with the energies. Now, with respect to the results of the second problem, we verify that the Dirac spinor is written in terms of the generalized Laguerre polynomials and the relativistic energy spectrum depends of Z, ω and of the parameter k that characterizes the PDEM. We observed that ω and k have the function of increase the values of this spectrum, while Z has the function of decrease. In special, when we take the constant mass limit (κ → 0), the results of the relativistic ABC system are recovered. However, we verified that even in the absence of the ABC system (Φ = Z = 0), the energy spectrum of the free particle with MEDP still remains quantized. On the other hand, when we analyze the nonrelativistic limit of our last problem, we obtain the Schrödinger equation (SE) for a particle with PDEM under the influence of the ABC system, where we verify that the energy spectrum of such an equation is reduced to the spectrum of the nonrelativistic ABC system in the limit (Φ = Z = 0), while for Φ = Z = 0, reduces to the spectrum of the free particle with MEDP.