Osciladores log-periódicos e tipo Caldirola-Kanai

• Autor(a) principal: Bessa, Vagner Henrique Loiola
• Data de Publicação: 2012
• Tipo de documento: Dissertação
• Idioma: por
• Título da fonte: Repositório Institucional da Universidade Federal do Ceará (UFC)
• dARK ID: ark:/83112/0013000024hhr
• Texto Completo: http://www.repositorio.ufc.br/handle/riufc/13625
• topic: Mecânica quântica; Oscilador harmônico dependente do tempo; Operador invariante; Quantum mechanics; Time-dependent harmonic oscillators; Invariant operator

In this work we present the classical and quantum solutions of two classes of time-dependent harmonic oscillators, namely: (a) the log-periodic and (b) the Caldirola-Kanai-type oscillators. For class (a) we study the following oscillators: (I) $m(t)=m_0frac{t}{t_0}$, (II) $m(t)=m_0$ and (III) $m(t)=m_0ajust{frac{t}{t_0}}^2$. In all three cases $omega(t)=omega_0frac{t_0}{t}$. For class (b) we study the Caldirola-Kanai oscillator (IV)where $omega(t)=omega_0$ and $m(t)=m_0 ext{exp}ajust{gamma t}$ and the oscillator with $omega(t)=omega_0$ and $m(t)=m_0ajust{1+frac{t}{t_0}}^alpha$, for $alpha=2$ (V) and $alpha=4$ (VI). To obtain the classical solution for each oscillator we solve the respective equation of motion and analyze the behavior of $q(t)$, $p(t)$ as well as the phase diagram $q(t)$ vs $p(t)$. To obtain the quantum solutions we use a unitary transformation and the Lewis and Riesenfeld quantum invariant method. The wave functions obtained are written in terms of a function ($ ho$) which is solution of the Milne-Pinney equation. Futhermore, for each system we solve the respective Milne-Pinney equation and discuss how the uncertainty product evolves with time.