Joint entropy of quantum damped harmonic oscillators

We use the dynamical invariant method and a unitary transformation to obtain the exact Schrödinger wave function, ψ n ( x , t ) , and calculate for n = 0 the time-dependent joint entropy (Leipnik’s entropy) 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.