Time dependence of joint entropy of oscillating quantum systems

The time dependent entropy (or Leipnik's entropy) of harmonic and damped harmonic oscillators is extensively investigated by using time dependent wave function obtained by the Feynman path integral method. Our results for simple harmonic oscillator are in agrement with the literature. However, the joint entropy of damped harmonic oscillator shows remarkable discontinuity with time for certain values of damping factor. According to the results, the envelop of the joint entropy curve increases with time monotonically. This results is the general properties of the envelop of the joint entropy curve for quantum systems.