  Exploring the analogy between quantum mechanics and statistical mechanics we
formulate an integrated version of the Quantropy functional [1]. With this
prescription we compute the propagator associated to Boltzmann-Gibbs statistics
in the semiclassical approximation as $K=F(T) \exp\left(i S_{cl}/\hbar\right)$.
We determine also propagators associated to different non-additive statistics;
those are the entropies depending only on the probability $S_{\pm}$ [2] and
Tsallis entropy $S_q$ [3]. For $S_{\pm}$ we obtain a power series solution for
the probability vs. the energy, which can be analytically continued to the
complex plane, and employed to obtain the propagators. Our work is motivated by
[4] where a modified q-Schr\"odinger equation is obtained; that provides the
wave function for the free particle as a q-exponential. The modified
q-propagator obtained with our method, leads to the same q-wave function for
that case. The procedure presented in this work allows to calculate q-wave
functions in problems with interactions; determining non-linear quantum
implications of non-additive statistics. In a similar manner the corresponding
generalized wave functions associated to $S_{\pm}$ can also be constructed. The
corrections to the original propagator are explicitly determined in the case of
a free particle and the harmonic oscillator for which the semi-classical
approximation is exact.
