A new time-dependent quantum theory based on Tsallis' distribution

TL;DR

The paper develops a new time-dependent quantum framework based on Tsallis statistics by performing an inverse Wick rotation on the Tsallis $q$-deformed Boltzmann factor, yielding a unitary $q$-evolution operator and an effective Hamiltonian $\hat{H}_{\rm eff}(t)= \hat{H}/[1+ \varepsilon^2 t^2 \hat{H}^2]$ with $\varepsilon=1-q$. It applies this to two prototypical systems: a free Gaussian wave packet and a Gaussian wave packet in a harmonic potential, deriving approximate analytic expressions at small deformation. The results show that the Tsallis deformation slows the spreading of the free packet and induces asymmetric, oscillatory spreading in the oscillator case, revealing observable consequences of the $q$-deformation in quantum dynamics while highlighting the need for further study in more complex systems. Eigenstates evolve with a phase factor $\exp\{- i\frac{\arctan[\varepsilon t E]}{\varepsilon}\}$, which preserves probability density, underscoring that observable effects arise from superpositions of energy eigenstates.

Abstract

In this paper, inspired by Tsallis' probability distribution based on a $q$-deformed Boltzmann factor, we stipulate a new $q$-deformed quantum dynamics by applying the inverse Wick rotation $ β\rightarrow i t$ to the Tsallis-deformed Boltzmann factor. We obtain a new time-dependent $q$-deformed Schrödinger equation. The free time-evolution of a Gaussian wave packet and that induced by an harmonic interaction are studied within this $q$-deformed quantum mechanical framework.

Figures (2)

• Figure 1: Plot of the spreading \ref{['sigmat']} of a freely evolving Gaussian wave packet with initial width $\sigma =1$ and time scale $\tau=1$, for $\varepsilon^2=0$ (pink), $\varepsilon^2=0.0005$ (purple) and $\varepsilon^2=0.001$ (brown).
• Figure 2: Plot of the probability density \ref{['PHO']} for the oscillating wave packet at $t = \pi/4\omega$ (left panel) and $\omega t = \pi/2\omega$ (right panel) with parameters $a =1$ and $\alpha=1$, and for $\varepsilon^2=0$ (pink), $\varepsilon^2=0.01$ (purple), and $\varepsilon^2=0.02$ (brown).