Entropia e informação de sistemas quânticos amortecidos

Abstract

In this work we analyze the solutions of the equations of motions for two Lane-Emden-Type Caldirola-Kanai oscillators. For these oscillators the mass varies as m(t) = t^a, where a > 0. We obtain the analytical expression of q·(t), q(t), and p(t) = m(t)q· for a = 2 and a = 4. These are damped-like harmonic oscillators with a time-dependent damping factor given by y(t) = a/t. We discuss the differences between the expressions for the hamiltonian and the mechanical energy for time-dependent systems. We also compared our results to those of the well-known Caldirola-Kanai oscillators. We use the quantum invariant method and a unitary transformation to obtain the exact Schrö wave function Ψn (q,t), and calculate for n = 0 the time-dependent joint entropy (Leipknik's entropy) and the position (Fq) and momentum (Fp) Fisher information for two classes of quantum damped harmonic oscillators. We observe that the joint entropy does not vary in time for the Caldirola-Kanai oscillator, while it decreases and tends to a constant value (ln (e/2)) for asymptotic times for the Lane-Emden ones. This is due to the fact that for the latter, the damping factor decreases as time increases. The results show that the time dependence of the joint entropy is quite complex and does not obey a general trend of monotonously increase with time and that Fq increases while Fp decreases with increasing time. Also, FqFp increases and tends to a constant value (4/h²) in the limit t ➜ ∞. We compare the results with those of the well-known Caldirola-Kanai oscillator.