Abstract:We address this work to investigate symbolic sequences with long-range correlations by using computational simulation. We analyze sequences with two, three and four symbols that could be repeated l times, with the probability distribution p(l) ∝ 1/l µ . For these sequences, we verified that the usual entropy increases more slowly when the symbols are correlated and the Tsallis entropy exhibits, for a suitable choice of q, a linear behavior. We also study the chain as a random walk-like process and observe a no… Show more
“…and the numerical solutions for ψ(x 1 , x 2 , t) are shown in Fig. (7). due to the interaction with the potential, the binding behavior appears in different times, see for example t = 50.…”
Section: Evolution Of the Wave Function For Q=2mentioning
confidence: 99%
“…where k is a constant and q is a real parameter, has shown to be useful in the description of several systems where the additivity fails such as systems characterized by non-Markovian processes [3,4,6,7], nonergodic dynamics and long-range many-body interactions [5]. The nonadditive characteristic of S q [8,9,10] has also brought new insights to several works in the area of complex systems [9,10] and contributed to diverse problems of quantum mechanics [11,12,13,14,15,16,17,18,19,20,21].…”
We report on the time dependent solutions of the q−generalized Schrödinger equation proposed by Nobre et al. [Phys. Rev. Lett. 106, 140601 (2011)]. Here we investigate the case of two free particles and also the case where two particles were subjected to a Moshinsky-like potential with time dependent coefficients. We work out analytical and numerical solutions for different values of the parameter q and also show that the usual Schrödinger equation is recovered in the limit of q → 1. An intriguing behavior was observed for q = 2, where the wave function displays a ring-like shape, indicating a bind behavior of the particles.Differently from the results previously reported for the case of one particle, frozen states appear only for special combinations of the wave function parameters in case of q = 3.
Highlights
Solutions of a nonlinear Schrödinger equation; Time dependent wave functions;Free particles and Moshinsky-like potential.
“…and the numerical solutions for ψ(x 1 , x 2 , t) are shown in Fig. (7). due to the interaction with the potential, the binding behavior appears in different times, see for example t = 50.…”
Section: Evolution Of the Wave Function For Q=2mentioning
confidence: 99%
“…where k is a constant and q is a real parameter, has shown to be useful in the description of several systems where the additivity fails such as systems characterized by non-Markovian processes [3,4,6,7], nonergodic dynamics and long-range many-body interactions [5]. The nonadditive characteristic of S q [8,9,10] has also brought new insights to several works in the area of complex systems [9,10] and contributed to diverse problems of quantum mechanics [11,12,13,14,15,16,17,18,19,20,21].…”
We report on the time dependent solutions of the q−generalized Schrödinger equation proposed by Nobre et al. [Phys. Rev. Lett. 106, 140601 (2011)]. Here we investigate the case of two free particles and also the case where two particles were subjected to a Moshinsky-like potential with time dependent coefficients. We work out analytical and numerical solutions for different values of the parameter q and also show that the usual Schrödinger equation is recovered in the limit of q → 1. An intriguing behavior was observed for q = 2, where the wave function displays a ring-like shape, indicating a bind behavior of the particles.Differently from the results previously reported for the case of one particle, frozen states appear only for special combinations of the wave function parameters in case of q = 3.
Highlights
Solutions of a nonlinear Schrödinger equation; Time dependent wave functions;Free particles and Moshinsky-like potential.
“…These entropies depend upon a real-valued parameter a which has different meanings depending upon the considered phenomenon under investigation, and very often it is more or less related to fractal dimension. Shannon's entropy has been useful in coding theory [9] and all these entropies have been successfully applied to many fields [10,40,45,52,54], mainly via Jaynes' maximum entropy principle [13,14], like in synergetics (organization and self-organization), see for instance [11]. The reference [24] by Kapur displays a large sample of systems which are analyzed by using the maximum entropy principle.…”
Section: A Brief Summary Of Entropy Historymentioning
“…As a consequence, it is also an essential aspect of our science and technology. The related research theme, that was motivated initially mainly by gambling and led eventually to probability theory [4,5], is nowadays a crucial part of many different fields of study such as computational simulations, information theory, cryptography, statistical estimation, system identification, and many others [6,7,8,9,10,11,12].…”
The generation of pseudo-random discrete probability distributions is of paramount importance for a wide range of stochastic simulations spanning from Monte Carlo methods to the random sampling of quantum states for investigations in quantum information science. In spite of its significance, a thorough exposition of such a procedure is lacking in the literature. In this article we present relevant details concerning the numerical implementation and applicability of what we call the iid, normalization, and trigonometric methods for generating an unbiased probability vector p = (p 1 , · · · , p d ). An immediate application of these results regarding the generation of pseudo-random pure quantum states is also described.
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