arXiv subjects

Omar Mustafa

Publications and source records attributed to Omar Mustafa.

At least 55 records · Page 3Linked to original sources

arXiv2023

Behavior of the Feshbach-Villars oscillator (FVO) in Gürses space-time under Coulomb-type potential

Our research aims to investigate how the gravitational field influences the spectroscopic structure of the Feshbach-Villars oscillator in Gürses space-time. To achieve this, we utilize the first-order Feshbach-Villars version of the Klein-Gordon equation, which is a relativistic wave equation for spinless particles. We examine the oscillator's quantum mechanical behavior in the presence of a Coulomb-type potential, and calculate the resulting wave functions and energy levels for both free and interacting scenarios. In addition, we study the interaction between the Coulomb-type potential and Gürses space-time affects the Feshbach-Villars oscillator's behavior, specifically with regard to its spectroscopic structure. This study has important implications for our understanding of the interplay between quantum mechanics, relativity, and gravitational fields at the microscopic level.

hep-th↗

arXiv2023

Massless KG-oscillators in Som-Raychaudhuri cosmic string spacetime in a fine tuned rainbow gravity

A fine tuned rainbow gravity describes both relativistic quantum particles and anti-particles alike. That is, the ratio $y=E/E_{P}$ in the rainbow functions $g_{_{0}}\left( y\right) $ and $% g_{_{1}}\left( y\right) $ should be fine tuned into $0\leq y=E/E_{P}\leq 1\Rightarrow y=\left\vert E\right\vert /E_{P}$, otherwise rainbow gravity will only secure Planck's energy scale $E_p$ invariance for relativistic particles and the anti-particles are left unfortunate (in the sense that their energies will be indefinitely unbounded). Using this fine tuning we discuss the rainbow gravity effect on Klein-Gordon (KG) oscillators in Som-Raychaudhuri cosmic string rainbow gravity spacetime background. We use the rainbow functions: (i) $g_{_{0}}\left( y\right) =1$, $% g_{_{1}}\left( y\right) =\sqrt{1-εy^{n}}, n=0,1$, loop quantum gravity motivated pairs, (ii) $% g_{_{0}}\left( y\right) =g_{_{1}}\left( y\right) =\left( 1-εy\right) ^{-1}$, a horizon problem motivated pair, and (iii) $g_{_{0}}\left( y\right) =\left( e^{εy}-1\right) /εy$, $g_{_{1}}\left( y\right) =1$, a gamma-ray bursts motivated pair. We show that the energies obtained using the first two rainbow functions in (i) completely comply with the rainbow gravity model (on the invariance of the Planck's energy scale $E_{p}$). The rainbow function pair in (ii) has no effect on massless KG-oscillators. Whereas, the one in (iii) does not show any eminent tendency towards the invariance of the Planck's energy scale. Yet, we suggest a new rainbow function pair $(g_0(y)=(1-εy)^{-1}, g_1 (y)=1)$, and show that it secures invariance of the Planck's energy scale $E_p$. Moreover, similar performance is observed when this new pair is used for KG-oscillators and KG-Coulombic particles in cosmic string rainbow gravity spacetime and magnetic fields.

gr-qc↗

arXiv2023

PDM KG-oscillators in cosmic string rainbow gravity spacetime in a non-uniform magnetic field

We consider position-dependent mass (PDM) Klein-Gordon (KG) particles in cosmic string rainbow gravity spacetime in a non-uniform magnetic field. The corresponding KG-equation is reduced into the one-dimensional form of the two-dimensional radial Schrödinger-oscillator like equation (hence the notion KG-oscillator). We first report on the effects of rainbow gravity on the energy levels of KG-oscillators with constant mass. Next, we include the PDM settings so that KG-oscillators like interaction are introduced. The effects of PDM on the spectra of KG-oscillators in cosmic string rainbow gravity spacetime are also reported. In both cases four pairs of rainbow functions are considered: (a) $% g_{_{0}}\left( y\right) =1$, $g_{_{1}}\left( y\right) =\sqrt{1-εy^{2}% }$, (b) $g_{_{0}}\left( y\right) =1$, $g_{_{1}}\left( y\right) =\sqrt{% 1-εy}$, (c) $g_{_{0}}\left( y\right) =g_{_{1}}\left( y\right) =\left( 1-εy\right) ^{-1}$, and (d) $g_{_{0}}\left( y\right) =\left( e^{εy}-1\right) /εy$, $g_{_{1}}\left( y\right) =1$, where $y=E/E_p$ and $ε$ is the rainbow parameter. We report that, as long as KG-particles are in point, the rainbow functions pair in (a) is the only pair to emphasis that the Planck energy is the maximum possible energy the KG-particles may have.

gr-qc↗

arXiv2023

Fine tuning of rainbow gravity functions and Klein-Gordon particles in cosmic string rainbow gravity spacetime

We argue that, as long as relativistic quantum particles are in point, the variable $y=E/E_p$ of the rainbow functions pair $g_{_{0}} (y)$ and $g_{_{1}} (y)$ should be fine tuned into $y=|E|/E_p$, where $E_p$ is the Planck's energy scale. Otherwise, the rainbow functions will be only successful to describe the rainbow gravity effect on relativistic quantum particles and the anti-particles will be left unfortunate. Under such fine tuning, we consider Klein-Gordon (KG) particles in cosmic string rainbow gravity spacetime in a non-uniform magnetic field (i.e., $\mathbf{B}=\mathbf{\nabla }\times \mathbf{A}=\frac{3}{2}B_{\circ }r\,\hat{z}$ ). Then we consider KG-particles in cosmic string rainbow gravity spacetime in a uniform magnetic field (i.e., $\mathbf{B}=\mathbf{\nabla }\times \mathbf{A}=\frac{1}{2}B_{\circ }\,\hat{z}$ ). Whilst the former effectively yields KG-oscillators, the later effectively yields KG-Coulombic particles. We report on the effects of rainbow gravity on both KG-oscillators and Coulombic particles using four pairs of rainbow functions: (i) $% g_{_{0}}\left( y\right) =1$, $g_{_{1}}\left( y\right) =\sqrt{1-εy^{2}% }$, (ii) $g_{_{0}}\left( y\right) =1$, $g_{_{1}}\left( y\right) =\sqrt{% 1-εy}$, (iii) $g_{_{0}}\left( y\right) =g_{_{1}}\left( y\right) =\left( 1-εy\right) ^{-1}$, and (iv) $g_{_{0}}\left( y\right) =\left( e^{εy}-1\right) /εy$, $g_{_{1}}\left( y\right) =1$, where $y=|E|/E_p$ and $ε$ is the rainbow parameter. It is interesting to report that, all KG particles' and anti-particles' energies are symmetric about $E=0$ value (a natural relativistic quantum mechanical tendency), and a phenomenon of energy states to fly away and disappear from the spectrum is observed for the rainbow functions pair (iii) at $γ=εm/E_p=1$.

gr-qc↗

arXiv2023

PDM Klein-Gordon particles in Som-Raychaudhuri cosmic string spacetime with space-like dislocation and a non-uniform magnetic field: spacetime associated degeneracies, vorticity-energy, and charge-energy correlations

We explicitly study and report the effects of the gravitational fields on the spectroscopic structure of Klein-Gordon (KG) particles in Som-Raychaudhuri cosmic string spacetime with space-like dislocation and a non-uniform magnetic field. We extend this study to include the effects of the gravitational fields on some position-dependent mass (PDM) KG-particles. As effects of the gravitational fields, we report new spacetime associated degeneracies, vorticity-energy correlations and charge-energy correlations. Such degeneracies and correlations are observed in the spectroscopic structure of both constant mass KG-particles and PDM KG-particles at hand.

gr-qc↗

arXiv2023

PDM KG-Coulombic particles in cosmic string rainbow gravity spacetime and a uniform magnetic field

Klein-Gordon (KG) particles in cosmic string rainbow gravity spacetime and a uniform magnetic field are studied in the context of the so called, metaphorically speaking, position-dependent mass (PDM) settings. We show that the corresponding KG-equation collapses into a two-dimensional radial Schrödinger-Coulomb like model. The exact textbook solution of which is used to find the energies and wave functions of KG-Coulombic particles (both constant mass and PDM ones). In so doing, we consider, with $y=E/E_{P}$, four pairs of rainbow functions: (a) $% g_{_{0}}\left( y\right) =1$, $g_{_{1}}\left( y\right) =\sqrt{1-εy^{2}% }$, (b) $g_{_{0}}\left( y\right) =1$, $g_{_{1}}\left( y\right) =\sqrt{% 1-εy}$, (c) $g_{_{0}}\left( y\right) =g_{_{1}}\left( y\right) =\left( 1-εy\right) ^{-1}$, and (d) $g_{_{0}}( y) =(e^{εy}-1) /εy$, $g_{_{1}}\left( y\right) =1$. Interestingly, we observe that the first pair in (a) introduces the Planck energy $E_{p}$ as a maximum possible KG-particle/antiparticle energy value.

gr-qc↗

arXiv2022

On the Klein-Gordon Gürses-oscillators and pseudo-Gürses-oscillators: vorticity-energy correlations and spacetime associated degeneracies

We discuss KG-oscillators in the (1+2)-dimensional Gürses spacetime and under position-dependent mass (PDM) settings. We observe that the KG-Gürses oscillators are introduced as a byproduct of the very nature of the Gürses spacetime structure. We report that the energy levels of such KG-Gürses oscillators admit vorticity-energy correlations as well as spacetime associated degeneracies (STAD). We discuss KG-Gürses oscillators' results reported by Ahmed \cite{Ahmed1 2019} and pinpoint his improper treatment of this model so that his results should be redirected to those reported in this study. Moreover, we introduce a new set of KG pseudo-Gürses oscillators that admits isospectrality and invariance with the KG-Gürses oscillators and inherits the same vorticity-energy correlations as well as STADs.

gr-qc↗

arXiv2022

Position-dependent mass Schrödinger particles in space-like screw dislocation: associated degeneracies and magnetic and Aharonov-Bohm flux fields effects

We consider non-relativistic position-dependent mass (PDM) Schrodinger particles moving in an elastic medium with space-like screw dislocation. Within the cylindrical coordinates, we study and report the effects of screw dislocation as well as PDM settings on the energy levels of some PDM-quantum mechanical systems. In so doing, we use a power-law type positive-valued dimensionless scalar multiplier f(r) (Which manifestly introduces the metaphoric notion of PDM-Schrodinger particles). Next, we subject such PDM particles to magnetic and Aharonov-Bohm flux fields. We report exact or conditionally exact eigenvalues and eigenfunctions for V (r) = 0 and V (r) = a + b r + c r^2

quant-ph↗

arXiv2022

Confined Klein-Gordon oscillators in Minkowski spacetime and a pseudo-Minkowski spacetime with a space-like dislocation: PDM KG-oscillators, isospectrality and invariance

We revisit the a confined (in a Cornell-type Lorentz scalar potential) KG-oscillator in Minkowski spacetime with space-like dislocation background. We show that the effect of space-like dislocation is to shift the energy levels along the dislocation parameter axis, and consequently energy levels crossings are unavoidable. We report some KG-particles in a pseudo-Minkowski spacetime with space-like dislocation that admit isospectrality and invariance with the confined KG-oscillator in Minkowski spacetime with space-like dislocation. An alternative PDM setting for the KG-particles (relativistic particles in general) is introduced. We discuss the effects of space-like dislocation and PDM settings on the confined KG-oscillators in Minkowski spacetime with space-like dislocation. Three confined PDM KG-oscillators are discussed as illustrative examples, (i) a PDM KG-oscillator from a dimensionless scalar multiplier $g\left( r\right) =\, exp(2αr^2)\geq0,\, α\,\geq 0$, (ii) a PDM KG-oscillator from a power law type dimensionless scalar multiplier $g\left( r\right) =Ar^{σ}\geq0$, and (iii) a PDM KG-oscillator in a Cornnell-type confinement with a dimensionless scalar multiplier $g\left( r\right) =\exp \left( ξr\right)\geq0$

gr-qc↗

arXiv2022

Klein-Gordon particles in Gödel-type Som-Raychaudhuri cosmic string spacetime and the phenomenon of spacetime associated degeneracies

We argue that only exact, comprehensive, and explicit solutions for the fundamental models, the Klein-Gordon (KG) oscillators and the KG-Coulomb, would help to understand the effects of gravitational fields on the dynamics of such quantum mechanical systems. In the current methodical proposal, the effects of the gravitational fields generated by a Gödel-type Som-Raychaudhuri (SR) cosmic string spacetime on KG-oscillators (KG-particles in general) are studied and reported. In so doing, we revisit the KG-oscillators in a topologically trivial Gödel-type spacetime background and use textbook procedures to report its exact solution that covers even and odd parities. Next, we discuss the drawbacks associated with the power series expansion approach that implies the biconfluent Heun functions/polynomials solution. We, therefore, recollect the so called pseudo perturbative shifted L expansion technique (PSLET) as an alternative and more sophisticated method/technique. Illustrative examples are used: (i) a KG-oscillator in a topologically trivial Gödel-type spacetime, (ii) a quasi-free KG-oscillator in Gödel SR-type cosmic string spacetime, (iii) a KG-Coulombic particle in Gödel SR-type cosmic string spacetime at zero vorticity, and (iv) a massless KG-particle in Gödel SR-type cosmic string spacetime in a Cornell-type Lorentz scalar potential. The corresponding exact energies are obtained from the zeroth (leading) order correction of PSLET, where all higher order correction identically vanish. The comprehensive exactness of the reported solutions manifestly suggest degeneracies associated with spacetime (STAD) phenomenon.

gr-qc↗

arXiv2021

PDM Klein-Gordon oscillators in cosmic string spacetime in magnetic and Aharonov-Bohm flux fields within the Kaluza-Klein theory

In the cosmic string spacetime and within Kaluza-Klein theory (KKT) backgrounds (indulging magnetic and Aharonov-Bohm flux fields), we introduce and study position-dependent mass (PDM) Klein-Gordon (KG) oscillators. The effective PDM is introduced as a deformation/defect in the momentum operator. We show that there are four different ways to obtain KG-oscillator. Two of which are readily known and the other two are obtained as byproducts of PDM settings. Next, we provide a thorough analysis on the corresponding spectra under different parametric effects, including the curvature parameter's effect. Such analysis is used as a reference/lead model which is used in the discussion of different PDM KG-oscillators models: a mixed power-law and exponential type PDM model that yields a pseudo-confined PDM KG-oscillator in cosmic string spacetime within KKT (i.e., the PDM KG-oscillators are confined in their own PDM manifested Cornell-type confinement), and a PDM KG-oscillator confined in a Cornell-type potential. Moreover, we extend our study and discuss a non-Hermitian $\mathcal{PT}$-symmetric PDM-Coulombic-type KG-particle model in cosmic string spacetime within KKT14

gr-qc↗

arXiv2021

Confined Klein-Gordon oscillator from a (2+1)-dimensional Gürses to a Gürses or a pseudo-Gürses space-time backgrounds: Invariance and isospectrality

We study the Klein-Gordon (KG) oscillator with a Cornell-type scalar confinement in (2+1)-dimensional Gürses space-time backgrounds and report their exact solutions. The effect of the vorticity parameter $Ω$ on the energy levels is found to yield some interesting features like; energy levels-crossings, partial clustering of positive and negative energy levels, and shifting the energy gap upwards or downwards. Such confined KG-oscillators are also studied in a general deformed Gürses space-time background. Moreover, we consider the confined-deformed KG-oscillator from a (2+1)-dimensional Gürses to Gürses and pseudo-Gürses space-time backgrounds. The resulting confined-deformed KG-oscillators are found to admit invariance and isospectrality with each other

gr-qc↗

arXiv2021

Effect of nonlocal transformations on the linearizability and exact solvability of the nonlinear generalized modified Emden type equations

The nonlinear generalized modified Emden type equations (GMEE) are known to be linearizable into simple harmonic oscillator (HO) or damped harmonic oscillators (DHO) via some nonlocal transformations. Hereby, we show that the structure of the nonlocal transformation and the linearizability into HO or DHO determine the nature/structure of the dynamical forces involved (hence, determine the structure of the dynamical equation). Yet, a reverse engineering strategy is used so that the exact solutions of the emerging GMEE are nonlocally transformed to find the exact solutions of the HO and DHO dynamical equations. Consequently, whilst the exact solution for the HO remains a textbook one, the exact solution for the DHO (never reported elsewhere, to the best of our knowledge) turns out to be manifestly the most explicit and general solution that offers consistency and comprehensive coverage for the associated under-damping, critical-damping, and over-damping cases (i.e., no complex settings for the coordinates and/or the velocities are eminent/feasible). Moreover, for all emerging dynamical system, we report illustrative figures for each solution as well as the corresponding phase-space trajectories as they evolve in time.

nlin.SI↗

arXiv2021

PDM damped-driven oscillators: exact solvability, classical states crossings, and self-crossings

Within the standard Lagrangian and Hamiltonian setting, we consider a position-dependent mass (PDM) classical particle performing a damped driven oscillatory (DDO) motion under the influence of a conservative harmonic oscillator force field $V\left( x\right) =\frac{1}{2}ω^{2}Q\left( x\right) x^{2}$ and subjected to a Rayleigh dissipative force field $\mathcal{R}\left( x,\dot{x}\right) =\frac{1}{2}b\,m\left( x\right) \dot{x}^{2}$ in the presence of an external periodic (non-autonomous) force $F\left( t\right) =F_{\circ }\,\cos \left( Ωt\right) $. Where, the correlation between the coordinate deformation $\sqrt{Q(x)}$ and the velocity deformation $\sqrt{m(x)}$ is governed by a point canonical transformation $q\left( x\right) =\int \sqrt{m\left( x\right) }dx=\sqrt{% Q\left( x\right) }x$. Two illustrative examples are used: a non-singular PDM-DDO, and a power-law PDM-DDO models. Classical-states $\{x(t),p(t)\}$ crossings are analysed and reported. Yet, we observed/reported that as a classical state $\{x_{i}(t),p_{i}(t)\}$ evolves in time it may cross itself at an earlier and/or a latter time/s.

physics.gen-ph↗

arXiv2021

$n$-dimensional PDM-damped harmonic oscillators: Linearizability, and exact solvability

We consider position-dependent mass (PDM) Lagrangians/Hamiltonians in their standard textbook form, where the long-standing \emph{gain-loss balance} between the kinetic and potential energies is kept intact to allow conservation of total energy (i.e., $L=T-V$, $H=T+V$, and $dH/dt=dE/dt=0$). Under such standard settings, we discuss and report on $n$-dimensional PDM damped harmonic oscillators (DHO). We use some $n$-dimensional point canonical transformation to facilitate the linearizability of their $n$-PDM dynamical equations into some $n$-linear DHOs' dynamical equations for constant mass setting. Consequently, the well know exact solutions for the linear DHOs are mapped, with ease, onto the exact solutions for PDM DHOs. A set of one-dimensional and a set of $n$-dimensional PDM-DHO illustrative examples are reported along with their phase-space trajectories.

quant-ph↗

arXiv2021

Isochronous $n$-dimensional nonlinear PDM-oscillators: linearizability, invariance and exact solvability

Within the standard Lagrangian settings (i.e., the difference between kinetic and potential energies), we discuss and report isochronicity, linearizability and exact solubility of some $n$-dimensional nonlinear position-dependent mass (PDM) oscillators. In the process, negative the gradient of the PDM-potential force field is shown to be no longer related to the time derivative of the canonical momentum, $\mathbf{p}% =m\left( r\right) \mathbf{\dot{r}}$, but it is rather related to the time derivative of the pseudo-momentum, $\mathbf{π}\left( r\right) =\sqrt{% m\left( r\right) }\mathbf{\dot{r}}$ (i.e., Noether momentum). Moreover, using some point transformation recipe, we show that the linearizability of the $n$-dimensional nonlinear PDM-oscillators is only possible for $n=1$ but not for $n\geq 2$. The Euler-Lagrange invariance falls short/incomplete for $n\geq 2$ under PDM settings. Alternative invariances are sought, therefore. Such invariances, like \emph{Newtonian invariance} of Mustafa \cite{42}, effectively authorize the use of the exact solutions of one system to find the solutions of the other. A sample of isochronous $n$-dimensional nonlinear PDM-oscillators examples are reported.

nlin.SI↗

arXiv2020

PDM-charged particles in PD-magnetic plus Aharonov-Bohm flux fields: unconfined "almost-quasi-free" and confined in a Yukawa plus Kratzer exact solvability

Using azimuthally symmetrized cylindrical coordinates, we consider some position-dependent mass (PDM) charged particles moving in position-dependent (PD) magnetic and Aharonov-Bohm flux fields. We focus our attention on PDM-charged particles (i.e., the PDM is only radially dependent) moving in an inverse power-law-type radial PD-magnetic fields. Under such settings, we consider two almost-quasi-free PDM-charged particles (i.e., no interaction potential). Both yield exactly solvable Schrödinger equations of Coulombic nature but with different spectroscopic structures. Moreover, we consider a Yukawa-type PDM-charged particle moving not only in the vicinity of the PD-magnetic and Aharonov-Bohm flux fields but also in the vicinity of a Yukawa plus a Kratzer type potential force field. For this particular case, we use the Nikiforov-Uvarov (NU) method to come out with exact analytical eigenvalues and eigenfunctions. Which, in turn, recover those of the almost-quasi-free PDM-charged particle. Energy levels crossings are also reported.

math-ph↗

arXiv2020

n-dimensional PDM non-linear oscillators: Linearizability and Euler-Lagrange or Newtonian invariance

We argue that, under multidimensional position-dependent mass (PDM) settings, the Euler-Lagrange textbook invariance falls short and turned out to be vividly incomplete and/or insecure for a set of PDM-Lagrangians. We show that the transition from Euler-Lagrange component presentation to Newtonian vector presentation is necessary and vital to guarantee invariance. The totality of the Newtonian vector equations of motion is shown to be more comprehensive and instructive than the Euler-Lagrange component equations of motion (they do not run into conflict with each other though). We have successfully used the Newtonian invariance amendment, along with some nonlocal space-time point transformation recipe, to extract exact solutions for a set of n-dimensional nonlinear PDM-oscillators. They are, Mathews-Lakshmanan type-I PDM-oscillators, power-law type-I PDM-oscillators, the Mathews-Lakshmanan type-II PDM-oscillators, the power-law type-II PDM-oscillators, and some nonlinear shifted Mathews-Lakshmanan type-I PDM-oscillators.

math-ph↗