scholarly journals New mathematics for the nonadditive Tsallis’ scenario

2017 ◽  
Vol 31 (21) ◽  
pp. 1750151 ◽  
Author(s):  
G. L. Ferri ◽  
F. Pennini ◽  
A. plastino ◽  
M. C. Rocca
Keyword(s):  
Harmonic Oscillator ◽  
Coherent States ◽  
Probability Distributions ◽  
Mean Values

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.

Probabilistic aspects of the SU(2)-harmonic oscillator

2000 ◽  
Vol 37 (1) ◽  
pp. 306-314
Author(s):  
Shunlong Luo
Keyword(s):  
Harmonic Oscillator ◽  
Coherent States ◽  
Probability Distributions ◽  
Quantum Probability ◽  
Dimensional Sphere ◽  
Large Radius ◽  
Two Dimensional ◽  
Bargmann Transform ◽  
Gaussian Distributions ◽  
Binomial Distributions

In the framework of quantum probability, we present a simple geometrical mechanism which gives rise to binomial distributions, Gaussian distributions, Poisson distributions, and their interrelation. More specifically, by virtue of coherent states and a toy analogue of the Bargmann transform, we calculate the probability distributions of the position observable and the Hamiltonian arising in the representation of the classic group SU(2). This representation may be viewed as a constrained harmonic oscillator with a two-dimensional sphere as the phase space. It turns out that both the position observable and the Hamiltonian have binomial distributions, but with different asymptotic behaviours: with large radius and high spin limit, the former tends to the Gaussian while the latter tends to the Poisson.

Probabilistic aspects of the SU(2)-harmonic oscillator

2000 ◽  
Vol 37 (01) ◽  
pp. 306-314
Author(s):  
Shunlong Luo
Keyword(s):  
Harmonic Oscillator ◽  
Coherent States ◽  
Probability Distributions ◽  
Quantum Probability ◽  
Dimensional Sphere ◽  
Large Radius ◽  
Two Dimensional ◽  
Bargmann Transform ◽  
Gaussian Distributions ◽  
Binomial Distributions

In the framework of quantum probability, we present a simple geometrical mechanism which gives rise to binomial distributions, Gaussian distributions, Poisson distributions, and their interrelation. More specifically, by virtue of coherent states and a toy analogue of the Bargmann transform, we calculate the probability distributions of the position observable and the Hamiltonian arising in the representation of the classic group SU(2). This representation may be viewed as a constrained harmonic oscillator with a two-dimensional sphere as the phase space. It turns out that both the position observable and the Hamiltonian have binomial distributions, but with different asymptotic behaviours: with large radius and high spin limit, the former tends to the Gaussian while the latter tends to the Poisson.

PERELOMOV AND BARUT–GIRARDELLO SU(1, 1) COHERENT STATES FOR HARMONIC OSCILLATOR IN ONE-DIMENSIONAL HALF SPACE

2006 ◽  
Vol 21 (12) ◽  
pp. 2635-2644 ◽  
Author(s):  
Q. H. LIU ◽  
H. ZHUO
Keyword(s):  
Coherent State ◽  
Harmonic Oscillator ◽  
Half Space ◽  
Ground State ◽  
Classical Limit ◽  
Coherent States ◽  
Mean Values ◽  
One Dimensional ◽  
The Mean

The Perelomov and the Barut–Girardello SU(1, 1) coherent states for harmonic oscillator in one-dimensional half space are constructed. Results show that the uncertainty products ΔxΔp for these two coherent states are bound from below [Formula: see text] that is the uncertainty for the ground state, and the mean values for position x and momentum p in classical limit go over to their classical quantities respectively. In classical limit, the uncertainty given by Perelomov coherent does not vanish, and the Barut–Girardello coherent state reveals a node structure when positioning closest to the boundary x = 0 which has not been observed in coherent states for other systems.

scholarly journals Studying Heisenberg-like Uncertainty Relation with Weak Values in One-dimensional Harmonic Oscillator

2022 ◽  
Vol 9 ◽  
Author(s):  
Xing-Yan Fan ◽  
Wei-Min Shang ◽  
Jie Zhou ◽  
Hui-Xian Meng ◽  
Jing-Ling Chen
Keyword(s):  
Lower Bound ◽  
Harmonic Oscillator ◽  
Coherent States ◽  
Uncertainty Relation ◽  
Quantum States ◽  
Weak Values ◽  
Mean Values ◽  
One Dimensional ◽  
The Mean ◽  
Arbitrary Quantum

As one of the fundamental traits governing the operation of quantum world, the uncertainty relation, from the perspective of Heisenberg, rules the minimum deviation of two incompatible observations for arbitrary quantum states. Notwithstanding, the original measurements appeared in Heisenberg’s principle are strong such that they may disturb the quantum system itself. Hence an intriguing question is raised: What will happen if the mean values are replaced by weak values in Heisenberg’s uncertainty relation? In this work, we investigate the question in the case of measuring position and momentum in a simple harmonic oscillator via designating one of the eigenkets thereof to the pre-selected state. Astonishingly, the original Heisenberg limit is broken for some post-selected states, designed as a superposition of the pre-selected state and another eigenkets of harmonic oscillator. Moreover, if two distinct coherent states reside in the pre- and post-selected states respectively, the variance reaches the lower bound in common uncertainty principle all the while, which is in accord with the circumstance in Heisenberg’s primitive framework.

scholarly journals NON-HERMITIAN OSCILLATOR-LIKE HAMILTONIANS AND λ-COHERENT STATES REVISITED

2001 ◽  
Vol 16 (02) ◽  
pp. 91-98 ◽  
Author(s):  
JULES BECKERS ◽  
NATHALIE DEBERGH ◽  
JOSÉ F. CARIÑENA ◽  
GIUSEPPE MARMO
Keyword(s):  
Harmonic Oscillator ◽  
Hilbert Spaces ◽  
Coherent States ◽  
Scalar Product ◽  
Points Of View ◽  
Usual Scalar ◽  
Usual Scalar Product

Previous λ-deformed non-Hermitian Hamiltonians with respect to the usual scalar product of Hilbert spaces dealing with harmonic oscillator-like developments are (re)considered with respect to a new scalar product in order to take into account their property of self-adjointness. The corresponding deformed λ-states lead to new families of coherent states according to the DOCS, AOCS and MUCS points of view.

scholarly journals Lewis-Riesenfeld quantization and SU(1, 1) coherent states for 2D damped harmonic oscillator

2018 ◽  
Vol 59 (11) ◽  
pp. 112101 ◽  
Author(s):  
Latévi M. Lawson ◽  
Gabriel Y. H. Avossevou ◽  
Laure Gouba
Keyword(s):  
Harmonic Oscillator ◽  
Coherent States ◽  
Damped Harmonic Oscillator

Comparison of Monte Carlo and Fuzzy Math Simulation Methods for Quantitative Microbial Risk Assessment

2003 ◽  
Vol 66 (10) ◽  
pp. 1900-1910 ◽  
Author(s):  
VALERIE J. DAVIDSON ◽  
JOANNE RYKS
Keyword(s):  
Risk Assessment ◽  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Interval Arithmetic ◽  
Food Systems ◽  
Probability Distributions ◽  
Economic Cost ◽  
Fuzzy Mathematics ◽  
Quantitative Microbial Risk Assessment ◽  
Mean Values

The objective of food safety risk assessment is to quantify levels of risk for consumers as well as to design improved processing, distribution, and preparation systems that reduce exposure to acceptable limits. Monte Carlo simulation tools have been used to deal with the inherent variability in food systems, but these tools require substantial data for estimates of probability distributions. The objective of this study was to evaluate the use of fuzzy values to represent uncertainty. Fuzzy mathematics and Monte Carlo simulations were compared to analyze the propagation of uncertainty through a number of sequential calculations in two different applications: estimation of biological impacts and economic cost in a general framework and survival of Campylobacter jejuni in a sequence of five poultry processing operations. Estimates of the proportion of a population requiring hospitalization were comparable, but using fuzzy values and interval arithmetic resulted in more conservative estimates of mortality and cost, in terms of the intervals of possible values and mean values, compared to Monte Carlo calculations. In the second application, the two approaches predicted the same reduction in mean concentration (−4 log CFU/ml of rinse), but the limits of the final concentration distribution were wider for the fuzzy estimate (−3.3 to 5.6 log CFU/ml of rinse) compared to the probability estimate (−2.2 to 4.3 log CFU/ml of rinse). Interval arithmetic with fuzzy values considered all possible combinations in calculations and maximum membership grade for each possible result. Consequently, fuzzy results fully included distributions estimated by Monte Carlo simulations but extended to broader limits. When limited data defines probability distributions for all inputs, fuzzy mathematics is a more conservative approach for risk assessment than Monte Carlo simulations.

scholarly journals Phase probability distributions of Roy-type even and odd nonlinear coherent states

2004 ◽  
Vol 53 (11) ◽  
pp. 3729
Author(s):  
Wang Ji-Suo ◽  
Liu Tang-Kun ◽  
Feng Jian ◽  
Sun Jin -Zuo
Keyword(s):  
Coherent States ◽  
Probability Distributions
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