arXiv subjects

Djamil Bouaziz

Publications and source records attributed to Djamil Bouaziz.

11 recordsLinked to original sources

arXiv2020

Thermostatistics in deformed space with maximal length

The method for calculating the canonical partition function with deformed Heisenberg algebra, developed by Fityo (Fityo, 2008), is adapted to the modified commutation relations including a maximum length, proposed recently in 1D by Perivolaropoulos (Perivolaropoulos, 2017). Firstly, the formalism of 1D maximum length deformed algebra is extended to arbitrary dimensions. Then, by employing the adapted semiclassical approach, the thermostatistics of an ideal gas and a system of harmonic oscillators (HOs) is investigated. For the ideal gas, the results generalize those obtained recently by us in 1D (Bensalem and Bouaziz, 2019), and show a complete agreement between the semiclassical and quantum approaches. In particular, a stiffer real-like equation of state for the ideal gas is established in 3D; it is consistent with the formal one, which we presented in the aforementioned paper. By analyzing some experimental data, we argue that the maximal length might be viewed as a macroscopic scale associated with the system under study. Finally, the thermostatistics of a system of HOs compared to that of an ideal gas reveals that the effects of the maximal length depend on the studied system. On the other hand, it is observed that the maximal length effect on some thermodynamic functions of the HOs is analogous to that of the minimal length, studied previously in the literature.

cond-mat.stat-mech↗

arXiv2017

Comment on "Cornell potential in generalized uncertainty principle formalism: the case of Schrödinger equation"

In the recent paper, Ref. 1, the l-waves Schrödinger equation for the Cornell's potential is solved in quantum mechanics with a generalized uncertainty principle by following Ref. 2. It is showed here that the approach of Ref. 2 can only be used for the s-waves, and then the solution given in Ref. 1 would be true only in the special case l=0. Furthermore, it is highlighted that the abstract and the conclusion of Ref. 1 do not accurately reflect the results of the paper.

quant-ph↗

arXiv2017

Singular inverse square potential in coordinate space with a minimal length

The problem of a particle of mass m in the field of the inverse square potential is studied in quantum mechanics with a generalized uncertainty principle, characterized by the existence of a minimal length. Using the coordinate representation, for a specific form of the generalized uncertainty relation, we solve the deformed Schrödinger equation analytically in terms of confluent Heun functions. We explicitly show the regularizing effect of the minimal length on the singularity of the potential. We discuss the problem of bound states in detail and we derive an expression for the energy spectrum in a natural way from the square integrability condition; the results are in complete agreement with the literature.

quant-ph↗

arXiv2017

Deformed Heisenberg Algebra with a minimal length: Application to some molecular potentials

We review the essentials of the formalism of quantum mechanics based on a deformed Heisenbeg algebra, leading to the existence of a minimal length scale. We compute in this context, the energy spectra of the pseudoharmonic oscillator and Kratzer potentials by using a perturbative approach. We derive the molecular constants, which characterize the vibration--rotation energy levels of diatomic molecules, and investigate the effect of the minimal length on each of these parameters for both potentials. We confront our result to experimental data for the hydrogen molecule to estimate an order of magnitude of this fundamental scale in molecular physics.

quant-ph↗

arXiv2015

Kratzer's molecular potential in quantum mechanics with a generalized uncertainty principle

The Kratzer's potential $V(r)=g_{1}/r^{2}-g_{2}/r$ is studied in quantum mechanics with a generalized uncertainty principle, which includes a minimal length $\left( ΔX\right) _{\min}=\hbar\sqrt{5β}$. In momentum representation, the Schrödinger equation is a generalized Heun's differential equation, which reduces to a hypergeometric and to a Heun's equations in special cases. We explicitly show that the presence of this finite length regularizes the potential in the range of the coupling constant $g_{1}$ where the corresponding Hamiltonian is not self-adjoint. In coordinate space, we perturbatively derive an analytical expression for the bound states spectrum in the first order of the deformation parameter $β$. We qualitatively discuss the effect of the minimal length on the vibration-rotation energy levels of diatomic molecules, through the Kratzer interaction. By comparison with an experimental result of the hydrogen molecule, an upper bound for the minimal length is found to be of about $0.01$ Å. We argue that the minimal length would have some physical importance in studying the spectra of such systems

quant-ph↗

arXiv2014

Pseudoharmonic oscillator in quantum mechanics with a minimal length

The pseudoharmonic oscillator potential is studied in non relativistic quantum mechanics with a generalized uncertainty principle characterized by the existence of a minimal length scale. By using a perturbative approach, we analytically compute the correction to the energy spectrum in the first order of the minimal length parameter \b{eta}. We investigate the effect of this fundamental length on the vibration-rotation energy levels of diatomic molecules with this potential function interaction. We explicitly show that the minimal length would have some importance in studying the spectra of diatomic molecules.

quant-ph↗

arXiv2014

Singular inverse-square potential: renormalization and self-adjoint extensions for medium to weak coupling

We study the radial Schrödinger equation for a particle of mass $m$ in the field of the inverse-square potential $α/r^{2}$ in the medium-weak-coupling region, i.e., with $-1/4\leq2mα/\hbar^{2}\leq3/4$. By using the renormalization method of Beane \textit{et} \textit{al.,}with two regularization potentials, a spherical square well and a spherical $δ$ shell, we illustrate that the procedure of renormalization is independent of the choice of the regularization counterterm. We show that, in the aforementioned range of the coupling constant $α$, there exists at most one bound state, in complete agreement with the method of self-adjoint extensions. We explicitly show that this bound state is due to the attractive square-well and delta-function counterterms present in the renormalization scheme. Our result for $2mα/\hbar^{2}=-1/4$ is in contradiction with some results in the literature.

math-ph↗

arXiv2013

Klein-Gordon Equation with Coulomb Potential in the Presence of a Minimal Length

We study the Klein-Gordon equation for Coulomb potential, V(r)=(-Ze^{2})/r, in quantum mechanics with a minimal length. The zero energy solution is obtained analytically in momentum space in terms of Heun's functions. The asymptotic behavior of the solution shows that the presence of a minimal length regularize the potential in the strong attractive regime, Z>68. The equation with nonzero energy is established in a particular case in the first order of the deformation parameter; it is a generalized Heun's equation.

quant-ph↗

arXiv2010

Singular inverse square potential in arbitrary dimensions with a minimal length: Application to the motion of a dipole in a cosmic string background

We solve analytically the Schrödinger equation for the N-dimensional inverse square potential in quantum mechanics with a minimal length in terms of Heun's functions. We apply our results to the problem of a dipole in a cosmic string background. We find that a bound state exists only if the angle between the dipole moment and the string is larger than π/4. We compare our results with recent conflicting conclusions in the literature. The minimal length may be interpreted as a radius of the cosmic string.

quant-ph↗

arXiv2010

Hydrogen atom in momentum space with a minimal length

A momentum representation treatment of the hydrogen atom problem with a generalized uncertainty relation,which leads to a minimal length (ΔX_{i})_{min}= \hbar \sqrt(3β+β'), is presented. We show that the distance squared operator can be factorized in the case β'=2β. We analytically solve the s-wave bound-state equation. The leading correction to the energy spectrum caused by the minimal length depends on \sqrtβ. An upper bound for the minimal length is found to be about 10^{-9} fm.

quant-ph↗

arXiv2007

Regularization of the Singular Inverse Square Potential in Quantum Mechanics with a Minimal length

We study the problem of the attractive inverse square potential in quantum mechanics with a generalized uncertainty relation. Using the momentum representation, we show that this potential is regular in this framework. We solve analytically the s-wave bound states equation in terms of Heun's functions. We discuss in detail the bound states spectrum for a specific form of the generalized uncertainty relation. The minimal length may be interpreted as characterizing the dimension of the system.

quant-ph↗