AUTHOR=Cabo Bizet Nana , Damián César , Obregón Octavio , Santos-Silva Roberto 

TITLE=Quantum Implications of Non-Extensive Statistics

JOURNAL=Frontiers in Physics

VOLUME=9

YEAR=2021

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2021.634547

DOI=10.3389/fphy.2021.634547

ISSN=2296-424X

ABSTRACT=<p>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 <inline-formula id="inf1"><mml:math id="minf1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>K</mml:mi><mml:mo>=</mml:mo><mml:mi>F</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>T</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mtext>exp</mml:mtext><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo>/</mml:mo><mml:mi>ℏ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula>. We determine also propagators associated to different nonadditive statistics; those are the entropies depending only on the probability <inline-formula id="inf2"><mml:math id="minf2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mo>±</mml:mo></mml:msub></mml:mrow></mml:math></inline-formula> and Tsallis entropy <inline-formula id="inf3"><mml:math id="minf3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mi>q</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>. For <inline-formula id="inf4"><mml:math id="minf4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mo>±</mml:mo></mml:msub></mml:mrow></mml:math></inline-formula>, 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-Schrödinger 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 <inline-formula id="inf5"><mml:math id="minf5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mo>±</mml:mo></mml:msub></mml:mrow></mml:math></inline-formula> 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.</p>