Shannon entropy, Fisher information and uncertainty relations for log-periodic oscillators

We calculate the time-dependent Shannon (Sx and Sp) entropy and Fisher (Fx and Fp) information of three log-periodic oscillators. We obtain a general expression for Sx,p and Fx,p in the state n=0 in terms of ρ, a c-number quantity satisfying a nonlinear differential equation. For two out of three oscillators Sx,p and Fx,p depend on time, but Sx+Sp and FxFp do not. The other oscillator behaves as the time-independent harmonic oscillator where Sx,p and Fx,p are all constants. Relations among the Fisher information and the Stam and Cramer–Rao inequalities are also discussed.