Sobre aplicações da teoria quântica de invariantes a sistemas hamiltonianos dependentes do tempo - Lima, Alberes Lopes de (2008)
In this thesis, we use linear invariants and the method of dynamical invariants to obtain exact solutions of the Schr¨odinger equation for the generalized time-dependent forced harmonic oscillator in terms of the solutions of a second order ordinary differential equation that describes the amplitude of the classical unforced damped oscillator. In addition, we construct gaussian wave packet solutions and calculate the fluctuations in coordinate and momentum as well as the quantum correlations between them. It is shown that the width of the Gaussian packet, fluctuations and correlations do not depend on the external force. As a particular case, we consider the forced Caldirola-Kanai oscillator. In addition, we use the Coulomb gauge, linear invariants and the method of dynamical invariants in the framework of the Schr¨odinger equation to obtain a quantum description of the light propagation through a homogeneous conducting linear media with no charge density. We obtain exact wave functions for this problem in terms of solutions of a second order ordinary differential equation which describes the amplitude of the classical damped harmonic oscillator. Furthermore, we construct gaussian wave packet solutions and calculate the fluctuations in coordinate and momentum as well as the quantum correlations for every mode of the quantized electromagnetic field. We also use quadratic invariants together method of dynamical invariants to study the light propagation in a conducting media. We obtain exact solutions of the time-dependent Schr¨odinger equation for this case and construct coherent and squeezed states for the quantized electromagnetic waves. Yet, we evaluate the quantum fluctuations in coordinate and momentum as well as the uncertainty product for every mode of the electromagnetic field. Finally, we use quadratic invariants and the dynamical invariant method for obtain exact wavefunctions of the Schr¨odinger equation for a particle trapped by oscillating fields.