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The European Physical Journal Plus

Two-parameter deformed quantum mechanics based on Fibonacci calculus and Debye crystal model of two-parameter deformed quantum statistics

Abdullah Algin, Won Sang Chung

Published Feb 28, 2024• 6 citations
Publisher

Abstract

Abstract Starting on the basis of Fibonacci calculus and Fibonacci oscillator algebra, we introduce the main properties to develop a new formalism for the two-parameter $$({q}_{1},{q}_{2})$$ ( q 1 , q 2 ) -deformed quantum mechanics, where $${q}_{1}$$ q 1 and $${q}_{2}$$ q 2 are real positive independent deformation parameters. As applications of such a two-parameter deformed formalism, we investigate the behavior of a quantum particle in some different physical phenomena covering the free particle and the inverse-harmonic potential case. The effect of two deformation parameters on the wave functions for these applications is studied. Another application is carried out onto the quantum statistics of lattice oscillations through a model of the $$({q}_{1},{q}_{2})$$ ( q 1 , q 2 ) -deformed phonon gas, and it is shown that the high- and low-temperature behavior of the model specific heat differs notably from the classical theories for the interval $$0<({q}_{1},{q}_{2})<\infty $$ 0 < ( q 1 , q 2 ) < ∞ . We also construct a two-parameter deformed non-extensive entropy based on some elements of the Fibonacci calculus and discuss its possible connection with the Tsallis entropy in non-extensive statistical mechanics. Finally, other possible application areas of the present two-parameter $$({q}_{1},{q}_{2})$$ ( q 1 , q 2 ) -deformed construction on quantum mechanics are discussed.

DOI
10.1140/epjp/s13360-024-05008-2

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Abstract Starting on the basis of Fibonacci calculus and Fibonacci oscillator algebra, we introduce the main properties to develop a new formalism for the two-parameter (q1,q2)({q}_{1},{q}_{2})(q1​,q2​) ( q 1 , q 2 ) -deformed quantum mechanics, where q1{q}_{1}q1​ q 1 and q2{q}_{2}q2​ q 2 are real positive independent deformation parameters. As applications of such a two-parameter deformed formalism, we investigate the behavior of a quantum particle in some different physical phenomena covering the free particle and the inverse-harmonic potential case. The effect of two deformation parameters on the wave functions for these applications is studied. Another application is carried out onto the quantum statistics of lattice oscillations through a model of the (q1,q2)({q}_{1},{q}_{2})(q1​,q2​) ( q 1 , q 2 ) -deformed phonon gas, and it is shown that the high- and low-temperature behavior of the model specific heat differs notably from the classical theories for the interval 0<(q1,q2)<∞0<({q}_{1},{q}_{2})<\infty 0<(q1​,q2​)<∞ 0 < ( q 1 , q 2 ) < ∞ . We also construct a two-parameter deformed non-extensive entropy based on some elements of the Fibonacci calculus and discuss its possible connection with the Tsallis entropy in non-extensive statistical mechanics. Finally, other possible application areas of the present two-parameter (q1,q2)({q}_{1},{q}_{2})(q1​,q2​) ( q 1 , q 2 ) -deformed construction on quantum mechanics are discussed.
Abstract In this paper, we consider a system of the q-deformed bosonic Tamm-Dancoff oscillators, whose spectrum has some exponential cutoff factors at high energies. We first investigate the q-calculus in the Tamm-Dancoff (TD) boson algebra, and within this framework, the q-derivative, q-integral and q-exponential function are introduced. Using these properties, we construct a new formalism for the q-deformed quantum mechanics, which accordingly involve the q-adjoint operator and the q-Hermitian operator properties. We then derive the q-deformed Heisenberg relation, and develop the q-Hermitian momentum operator. The q-deformed Schrödinger equation is introduced, and as applications, we study the momentum eigenfunction and one-dimensional box problem. Another application of the TD type deformation onto lattice oscillations is also discussed through a model of the q-deformed Debye solid. Finally, other potential applications of the TD-oscillators gas model are concisely pointed out.
We consider the -deformed formalism constructed with some elements of Fibonacci calculus to study the 1D infinite potential box problem where 1-D Schrdinger equation is reframed with well-known modified Fibonacci difference operator and has been solved using q-series solution method. In order to do it, at first, all the selective but essential basics are displayed one by one and -deformed trigonometric functions have been plotted to review and understand the deformed quantum mechanical framework. Hence, on the basis of two parameter deformed algebra, wave functions associated with the particle confined inside the infinite potential box has been obtained. The normalized wave functions and energy expressions have also been obtained for . It is found that all the expressions can be reduced to their conventional form within the limit .
In this paper, the [Formula: see text]-deformed bosonic Tamm–Dancoff oscillator algebra is considered. First, the quantum algebraic and representative properties of these deformed bosons are analyzed in detail. The representations of the [Formula: see text]-fermion algebra of Tamm–Dancoff type are also studied. Second, the high-temperature thermostatistical properties of a gas of Tamm–Dancoff type [Formula: see text]-fermions are investigated. The fermionic distribution function and the other important thermodynamic functions such as the entropy and the specific heat are derived in terms of the real deformation parameter [Formula: see text]. Finally, the time evaluation of a two-level atom in a Tamm–Dancoff oscillator trap interacting with a single-mode traveling light field is concisely discussed.
Abstract Based on the framework established in a recent work (Ubriaco in Physica A 414:128, 2014), wherein a generalized bosonic model was formulated using the Dunkl operator formalism (DOF), we extend this approach to its different applications with introducing a statistical distribution—termed the Dunkl-deformed Bose–Einstein (DDBE) distribution—constructed by generalizing the usual Bose–Einstein distribution through the DOF. In this framework, we study further general thermodynamical and statistical properties of a gas model of the Dunkl-deformed bosons obeying the DDBE statistics. Emphasis is given to a careful analysis on the conditions under which the Bose–Einstein-like condensation would occur in the present Dunkl-deformed boson gas. It is shown that the critical temperature for such a system is higher than that of the usual (undeformed) Bose gas for values of the model deformation parameter θ\thetaθ θ in the range θ>0\theta > 0θ>0 θ > 0 . As another application of the DDBE statistics, we investigate the thermostatistics of the Dunkl-deformed Debye solid and the effects of the deformation parameter θ\thetaθ θ on the high- and low-temperature behavior of the model specific heat are discussed. Our results obtained in this study could provide much physical insight for approaching not only in the treatment of interacting bosons theories including collective excitations, but also in the condensation characteristics of quantum systems beyond the standard formalism.
An outstanding idea originally introduced by Greenberg is to investigate whether there is equivalence between intermediate statistics, which may be different from anyonic statistics, and q-deformed particle algebra. Also, a model to be studied for addressing such an idea could possibly provide us some new consequences about the interactions of particles as well as their internal structures. Motivated mainly by this idea, in this work, we consider a q-deformed Fermi gas model whose statistical properties enable us to effectively study interpolating statistics. Starting with a generalized Fermi-Dirac distribution function, we derive several thermostatistical functions of a gas of these deformed fermions in the thermodynamical limit. We study the high-temperature behavior of the system by analyzing the effects of q deformation on the most important thermostatistical characteristics of the system such as the entropy, specific heat, and equation of state. It is shown that such a deformed fermion model in two and three spatial dimensions exhibits the interpolating statistics in a specific interval of the model deformation parameter 0 < q < 1. In particular, for two and three spatial dimensions, it is found from the behavior of the third virial coefficient of the model that the deformation parameter q interpolates completely between attractive and repulsive systems, including the free boson and fermion cases. From the results obtained in this work, we conclude that such a model could provide much physical insight into some interacting theories of fermions, and could be useful to further study the particle systems with intermediate statistics.
This work addresses the study of the oscillator algebra, defined by four parameters ppp, qqq, ααα, and ννν. The time-independent Schrödinger equation for the induced deformed harmonic oscillator is solved; explicit analytic expressions of the energy spectrum are given. Deformed states are built and discussed with respect to the criteria of coherent state construction. Various commutators involving annihilation and creation operators and their combinatorics are computed and analyzed. Finally, the correlation functions of matrix elements of main normal and antinormal forms, pertinent for quantum optics analysis, are computed.