Joint entropy of quantum damped harmonic oscillators

• We obtain the exact wave functions for quantum damped harmonic oscillators.
• For the Caldirola–Kanai and Lane–Endem oscillators we calculate the joint entropy.
• The joint entropy (SjSj) does not vary in time for the Caldirola–Kanai oscillator.
• SjSj decreases and goes to ln(e2) with increasing time for the Lane–Endem oscillators.

We use the dynamical invariant method and a unitary transformation to obtain the exact Schrödinger wave function, ψn(x,t)ψn(x,t), and calculate for n=0n=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(e2)) 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.