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Towards constructing Tsallis' q-Coherent States

F. Pennini, Angelo Plastino, M. C. Rocca

Published Nov 27, 2015• 0 citations
OpenAlex

Abstract

In this manuscript we define q-coherent states corresponding to the Tsallis' statistics. For them and for the ordinary coherent states we calculate the respective distributions of probabilities. For the latter, we encounter some results that are not too well known and deserve better dissemination.

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•Oct 16, 1996•18 citations
In this paper, we investigate quantum uncertainties in a Tsallis’ nonadditive scenario. To such an end we appeal to [Formula: see text]-exponentials (qEs), that are the cornerstone of Tsallis’ theory. In this respect, it is found that some new mathematics is needed and we are led to construct a set of novel special states that are the qE equivalents of the ordinary coherent states (CS) of the harmonic oscillator (HO). We then characterize these new Tsallis’ special states by obtaining the associated (i) probability distributions (PDs) for a state of momentum [Formula: see text], (ii) mean values for some functions of space an momenta and (iii) concomitant quantum uncertainties. The latter are then compared to the usual ones.
Abstract The Hulthén potential is a short-range potential widely used in various fields of physics. In this paper, we investigate the distribution functions for the Hulthén potential by using statistical and superstatistical methods. We first review the ordinary statistics and superstatistics methods. We then consider distribution functions, such as uniform, 2-level, gamma, and log-normal and F distributions. Finally, we investigate the behavior of the Hulthén potential for statistical and superstatistical methods and compare the results with each other. We use the Tsallis statistics of the superstatistical system. We conclude that the Tsallis behavior of different distribution functions for the Hulthén potential exhibits better results than the statistical method. We examined the thermal properties of the Hulthén potential for five different distributions: Uniform, 2-level, Gamma, Log-normal, and F. We plotted the Helmholtz free energy and the entropy as functions of temperature for various values of q. It shows that the two uniform and 2-level distributions have the same results due to the universal relationship and that the F distribution does not become ordinary statistics at q = 1. It also reveals that the curves of the Helmholtz free energy and the entropy change their order and behavior as q increases and that some distributions disappear or coincide at certain values of q. One can discuss the physical implications of our results and their applications in nuclear and atomic physics in the future.
Following the relationship between probability distribution and coherent states, for example the well known Poisson distribution and the ordinary coherent states and relatively less known one of the binomial distribution and the su(2)su(2)su(2) coherent states, we propose ``interpretation'' of su(1,1)su(1,1)su(1,1) and su(r,1)su(r,1)su(r,1) coherent states ``in terms of probability theory''. They will be called the ``negative binomial'' (``multinomial'') ``states'' which correspond to the ``negative'' binomial (multinomial) distribution, the non-compact counterpart of the well known binomial (multinomial) distribution. Explicit forms of the negative binomial (multinomial) states are given in terms of various boson representations which are naturally related to the probability theory interpretation. Here we show fruitful interplay of probability theory, group theory and quantum theory.