Q-deformation Parameter Research Articles

Intermediate statistics: Addressing the thermoelectric properties of solids.

We study the thermodynamics of a crystalline solid by applying intermediate statistics obtained by deforming known solid state models using the mathematics of qanalogs. We apply the resulting qdeformation to both the Einstein and Debye models and study the deformed thermal and electrical conductivities and the deformed Debye specific heat. We find that the qdeformation acts in two different ways-but not necessarily as independent mechanisms. First, it acts as an effective factor of disorder or impurity, modifying the characteristics of a crystalline structure, which are phenomena described by qbosons. Second, it also manifests intermediate statistics, namely, the Banyons (or B-type systems). For the latter case we have identified the Schottky effect, normally associated with high-T_{c} superconductors in the presence of rare-earth-ion impurities. We also find that it increases the specific heat of the solids beyond the Dulong-Petit limit at high temperature. Such an effect is usually related to anharmonicity of interatomic interactions. Alternatively, since in the q-boson's case the statistics are in principle maintained, the effect of the deformation acts more slowly due to a small change in the crystal lattice. On the other hand, Banyons that belong to modified statistics are more sensitive to the deformation. The results reported here may be verified experimentally, for instance, in experimental samples by inserting impurities, or changes in pressure or temperature if one assumes these tuning quantities are related with the q-deformation parameter.

Solving the time fractional q-deformed tanh-Gordon equation: A theoretical analysis using controlled Picard's transform method

<p>This paper presented the formulation and solution of the time fractional q-deformed tanh-Gordon equation, a new extension to the traditional tanh-Gordon equation using fractional calculus, and a q-deformation parameter. This extension aimed to better model physical systems with violated symmetries. The approach taken involved the controlled Picard method combined with the Laplace transform technique and the Caputo fractional derivative to find solutions to this equation. Our results indicated that the method was effective and highlighted our approach in addressing this equation. We explored both the existence and the uniqueness of the solution, and included various 2D and 3D graphs to illustrate how different parameters affect the solution's behavior. This work aimed to contribute to the theoretical framework of mathematical physics and has potential applications across multiple interdisciplinary fields.</p>

Chaotic behavior and construction of a variety of wave structures related to a new form of generalized q-Deformed sinh-Gordon model using couple of integration norms

<abstract><p>The generalized q-deformed sinh Gordon equation (GDSGE) serves as a significant nonlinear partial differential equation with profound applications in physics. This study investigates the GDSGE's mathematical and physical properties, examining its solutions and clarifying the essence of the q-deformation parameter. The Sardar sub-equation method (SSEM) and sine-Gordon expansion method (SGEM) are employed to solve this GDSGE. The synergistic application of these techniques improves our knowledge of the GDSGE and provides a thorough foundation for investigating different evolution models arising in various branches of mathematics and physics. A positive aspect of the proposed methods is that they offer a wide variety of solitons, including bright, singular, dark, combination dark-singular, combined dark-bright, and periodic singular solitons. Obtained solutions demonstrate the method's high degree of reliability, simplicity, and functionalization for various nonlinear equations. To better describe the physical characterization of solutions, a few 2D and 3D visualizations are generated by taking precise values for parameters using mathematical software, Mathematica.</p></abstract>

Solution of Q-Deformed D-Dimensional Klein-Gordon Equation Kratzer Potential using Hypergeometric Method

The Q-deformed D-dimensional Klein Gordon equation with Kratzer potential is solved by using Hypergeometric method in the case of exact spin symmetry. The linear radial momentum of D-dimensional Klein Gordon equation is disturbed by the presence of the quadratic radial posisiton. The Klein-Gordon D-dimensional equation is reduced to one-dimensional Schrodinger like equation with variable substitution. The solution of the D-dimensional Klein-Gordon equation is determined in the form of a general equation of the Hypergeometry function using the Kratzer potential variable and the quantum deformation variable. From this equation, relativistic energy and wave function are determined. In addition, the relativistic energy equation can be used to calculate numerical energy levels for diatomic particles (CO, NO, O2) using Matlab R2013a software. The results obtained show that the q-deformed quantum parameters, quantum numbers and dimensions affect the value of relativistic energy for zero-pin particles. The value of energy increases with increasing value of quantum number n, q-deformed parameters, and d-dimensional parameters. Of the three parameters, q-deformed parameter is the most dominant to give change in energy value; the increasing q-deformed parameter causes the energy value increases significantly compared to the d-dimensional parameter and quantum numbers n.

Model-independent astrophysical constraints on leptophilic Dark Matter in the framework of Tsallis statistics

We derive model-independent astrophysical constraints on leptophilic dark matter (DM), considering its thermal production in a supernova core and taking into account core temperature fluctuations within the framework of q-deformed Tsallis statistics. In an effective field theory approach, where the DM fermions interact with the Standard Model via dimension-six operators of either scalar-pseudoscalar, vector-axial vector, or tensor-axial tensor type, we obtain bounds on the effective cut-off scale Λ from supernova cooling and free-streaming of DM from supernova core, and from thermal relic density considerations, depending on the DM mass and the q-deformation parameter. Using Raffelt's criterion on the energy loss rate from SN1987A, we obtain a lower bound on Λ ≳ 3 (12) TeV corresponding to q = 1.0 (1.1) and an average supernova core temperature of 0TSN=3 MeV . From the optical depth criterion on the free-streaming of DM fermions from the outer 10% of the SN1987A core, the cooling bound is restricted to Λ ≳ 1 TeV . Both cooling and free-streaming bounds are insensitive to the DM mass mχ for mχ≲ TSN, whereas for mχ≫ TSN, the bounds weaken significantly due to the Boltzmann-suppression of the DM number density. We also calculate the thermal relic density of the DM particles in this setup and find that it imposes an upper bound on Λ4/mχ2, which together with the cooling/free-streaming bound significantly constrains light leptophilic DM.

Topological basis realization associated with Hermitian and non-Hermitian Heisenberg XXZ model

We investigate the topological basis realization associated with the Hermitian and non-Hermitian Heisenberg XXZ model based on the matrix representation of the Temperley-Lieb algebra. The results show that the topological space is spanned by the topological basis associated with the Temperley-Lieb algebra. When the q-deformed parameter q is a complex number and , the corresponding model is the non-Hermitian XXZ model. The real q corresponds to the Hermitan case. The topological basis realization for these two cases is constructed by means of matrix representation of the Temperley-Lieb algebra. Then the properties of the topological basis and ground state are studied. Particularly, the four-qubit Heisenberg XXZ model is investigated in detail. We also give some discussions on the case in which q is a root of unity.

A novel SUSY energy bound-states treatment of the Klein-Gordon equation with PT-symmetric and q-deformed parameter Hulthén potential

In this study, we focus on investigating the exact relativistic bound-state spectra for supersymmetric, PT-supersymmetric and non-Hermitian versions of the q-deformed parameter Hulthén potential. The Hamiltonian hierarchy mechanism, namely the factorization method, is adopted within the framework of SUSYQM. This algebraic approach is used in solving the Klein-Gordon equation with the potential cases. The results obtained analytically by executing the straightforward calculations are in consistent forms for certain values of q. Achieving the results may have a particular interest for such applications. That is, they can be involved in determining the quantum structural properties of molecules for ro-vibrational states, and optical spectra characteristics of semiconductor devices with regard to the lattice dynamics. They are also employed to construct the broken or unbroken case of the supersymmetric particle model concerning the interaction between the elementary particles.

Exact analytic solutions of the Schrödinger equations for some modified q-deformed potentials

In this paper, we introduce the exact solution for the wave function in the presence of potential energy, consisting of combination between q-deformed hyperbolic and exponential function with different argument. The functions we have used in the present communication can be regarded as a generalization of the Arai q-deformed function (modified q-deformed Morse potential). In this context, we have restricted our discussion for some particular cases of the q-deformed hyperbolic functions. This is due to the difficulty for dealing with most of the arguments included in the potential functions. For the most particular cases, the energy eigenfunctions are obtained, and the behavior is also discussed. It has been shown that the wave functions are sensitive to the variation in the value of q-deformed parameter as well as the strength of the potential parameter λ. Furthermore, the energy eigenvalues are also considered for some particular cases where the argument of the exponential function plays a strong role effecting its value.

Some electronic properties of metals through [formula omitted]-deformed algebras

We study the thermodynamics of metals by applying q-deformed algebras. We shall mainly focus our attention on q-deformed Sommerfeld parameter as a function of q-deformed electronic specific heat. The results revealed that q-deformation acts as a factor of disorder or impurity, modifying the characteristics of a crystalline structure and thereby controlling the number of electrons per unit volume.

Entanglement generation with deformed Barut-Girardello coherent states as input states in an unitary beam splitter

Using linear entropy as a measure of entanglement, we investigate the entanglement generated via a beam splitter using deformed Barut-Girardello coherent states. We show that the degree of entanglement depends strongly on the q-deformation parameter and amplitude Z of the states. We compute the Mandel Q parameter to examine the quantum statistical properties of these coherent states and make a comparison with the Glauber coherent states. It is shown that these states are useful in describing the states of real and ideal lasers by a proper choice of their characterizing parameters, using an alteration of the Holstein-Primakoff realization.

Entanglement generation from deformed spin coherent states using a beam splitter

Using the linear entropy as a measure of entanglement, we investigate the effect of a beam splitter on the Perelomov coherent states for the q-deformed Uq(su(2)) algebra. We distinguish two cases: in the classical q → 1 limit, we find that the states become Glauber coherent states as the spin tends to infinity; whereas for q ≠ 1, the states, contrary to the earlier case, become entangled as they pass through a beam splitter. The entanglement strongly depends on the q-deformation parameter and the amplitude Z of the state.

Bogoliubov’s Hamiltonian as a derivative of Dirac’s Hamiltonian via a braid relation

In this paper we discuss a new type of four-dimensional representation of the braid group. The matrices of braid operations are constructed by $q$ deformation of Hamiltonians. One is Dirac's Hamiltonian for a free electron with mass $m$, the other, which we find, is related to Bogoliubov's Hamiltonian for quasiparticles in $^{3}\text{H}\text{e}\ensuremath{-}\text{B}$ with the same free energy and mass being $m/2$. In the process, we choose the free $q$-deformation parameter as a special value in order to be consistent with the anyon description for fractional quantum Hall effect with $\ensuremath{\nu}=1/2$.

Exact solution and finite size properties of the [formula omitted] vertex models

We have diagonalized the transfer matrix of the U q [ osp ( 2 | 2 m ) ] vertex model by means of the algebraic Bethe ansatz method for a variety of grading possibilities. This allowed us to investigate the thermodynamic limit as well as the finite size properties of the corresponding spin chain in the massless regime. The leading behaviour of the finite size corrections to the spectrum is conjectured for arbitrary m. For m = 1 we find a critical line with central charge c = − 1 whose exponents vary continuously with the q-deformation parameter. For m ⩾ 2 the finite size term related to the conformal anomaly depends on the anisotropy which indicates a multicritical behaviour typical of loop models.

Q-DEFORMED DYNAMICS AND JOSEPHSON JUNCTION

We define a generalized rate equation for an observable in quantum mechanics, that involves a parameter q and whose limit q→1 gives the standard Heisenberg equation. The generalized rate equation is used to study dynamics of current-biased Josephson junction. It is observed that this toy model incorporates diffraction-like effects in the critical current. Physical interpretation for q is provided which is also shown to be a q-deformation parameter.

Uq(2)⊃Uq(1) DESCRIPTION OF VIBRATIONAL SPECTRA OF DIATOMIC MOLECULES

q-deformation of the U(1) dynamical symmetry of U(2) group is introduced for the first time for vibrations of diatomic molecules. The question of pure real and pure imaginary value of q-deformation parameter, often raised in the literature, is found to be related to the particular choice of the constants of the model. The complex q-deformation parameter is allowed, with pure real and pure imaginary values as its special cases. Compared to the other dynamical symmetry Oq(2) of Uq(2), the result of Uq(1) of Uq(2) remains better even when the vibron number in Oq(2) is varied. The role of the vibron number in the two dynamical symmetries of U(2) is shown to be different. In Uq(1) symmetry, the vibron number is fixed by the number of observed states of the system since Uq(1) represents the q-deformation of a truncated anharmonic oscillator. A possible physical meaning to the q-deformation parameter is suggested in terms of higher order anharmonicities of the truncated anharmonic oscillator.

QUANTUM GROUP SUq(2) FOR COMPLEX q-DEFORMATION

q-deformed harmonic oscillator is established for any arbitrary complex q-deformation parameter (with real and imaginary q as its special cases) and the formalism so developed is used to construct the quantum group SUq(2). The eigenvalues of the Casimir of SUq(2) are now real, even for complex q-deformation (q=ea+ib). This happens for (a,b)≪1. Thus, the use of real part of energies in the earlier works of Gupta and collaborators7–14 is mathematically rigorous, since only small values of (a, b) are involved in all the nuclear and molecular physics problems studied7–14,18–20 so far.

Q-deformed Morse oscillator.

Using the exact realization of the Morse oscillator problem in terms of SO(2,1) symmetry and its bosonization, the corresponding q deformation is performed. The ground-state energy is found to remain constant for all arbitrary (zero, real, imaginary, or complex) values of the q-deformation parameter. By fitting the resulting q-deformed energy-level spectrum to data for the hydrogen molecule in its ground electronic state, it is shown that, depending upon the value of the scaling constant, a considerable improvement results for the real and/or imaginary q deformation over the corresponding result for the classical Morse oscillator, and that the best fit is obtained in the q-deformed case with a complex value of the deformation parameter.

U q (4)⊇O q(4) description of diatomic molecules: Vibrational spectra

Quantum-deformed algebra (q-DA) of the O(4) dynamical symmetry of the U(4) group is studied in respect of the vibrational spectra of diatomic molecules. It is shown that, as in the case of the U(6) group for nuclei, the q-deformation parameter in Uq(4)⊇Oq(4) must also be complex for any meaningful gain in going from the classical to the q-deformed algebra. Application is made to the ground electronic state (X 1Σ+g) of the H2 molecule.

The interacting boson model and the quantum groups

The one-dimensional limiting (dynamical) symmetries of the quantum group SU q (2), the SUq(2) limiting symmetry of the two-dimensional SUq(3) and the three-dimensional Oq(6) limiting symmetry of q-deformed U(6) group of the s,d interacting boson model (IBM) are studied for looking at the q-deformation parameter in the algebra of quantum groups as the dynamical symmetry breaking (or mixing) parameter in the IBM. For complex q-values, the q-deformed Hamiltonian of any one of the dynamical symmetries of the IBM is shown to give the results of its general Hamiltonian.

QUANTUM GROUPS AND VON NEUMANN THEOREM

We discuss the q-deformation of Weyl-Heisenberg algebra in connection with the von Neumann theorem in quantum mechanics. We show that the q-deformation parameter labels the Weyl systems in quantum mechanics and the unitarily inequivalent representations of the canonical commutation relations in quantum field theory.