Quantum Implications of Non-Extensive Statistics

Exploring the analogy between quantum mechanics and statistical mechanics, we formulate an integrated version of the Quantropy functional. With this prescription, we compute the propagator associated to Boltzmann-Gibbs statistics in the semiclassical approximation as K = F(T)exp(iS(cl)/h). We determine also propagators associated to different nonadditive statistics; those are the entropies depending only on the probability S-+/- and Tsallis entropy S-q. For S-+/-, 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 the work of Nobre et al. where a modified q-Schrodinger 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 nonlinear quantum implications of nonadditive statistics. In a similar manner, the corresponding generalized wave functions associated to S-+/- 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 semiclassical approximation is exact, and also the case of a particle with an infinite potential barrier is discussed.

Automated summary

By exploring the analogy between quantum mechanics and statistical mechanics, an integrated version of the Quantropy functional was formulated to compute propagators associated with Boltzmann-Gibbs statistics and other nonadditive statistics. The work was motivated by the development of a modified q-Schrodinger equation and q-wave function for a free particle, leading to the study of q-wave functions in problems with interactions. The research also involved constructing generalized wave functions and determining corrections to the original propagator in various quantum systems.