Narrow Results

In this paper, we investigate the approximate analytical solutions of the relativistic Duffin–Kemmer–Petiau equation in the presence of a Mie-Type potential in noncommutative space by using the Nikiforov–Uvarov method. We determine the energy eigenvalues and eigenfunctions of the DKP equation. Furthermore, energy shift due to noncommutativity space–time is obtained via the perturbation theory. We observe that this feature is comparable to the Zeeman effect and the degeneracy of the classical spectral line is fractured in the transition from commutative to noncommutative space by the splitting of the states.

An algebraic formulation is given for the embedded noncommutative spaces over the Moyal algebra developed in a geometric framework in [8]. We explicitly construct the projective modules corresponding to the tangent bundles of the embedded noncommutative spaces, and recover from this algebraic formulation the metric, Levi–Civita connection and related curvatures, which were introduced geometrically in [8]. Transformation rules for connections and curvatures under general coordinate changes are given. A bar involution on the Moyal algebra is discovered, and its consequences on the noncommutative differential geometry are described.

We propose a new method to derive energy-level gap for Hamiltonians in the context of noncommutative quantum mechanics (NCQM). This method relies on finding invariant eigen-operators whose commutators with Hamiltonian are still the operators themselves but with some eigenvalue-like coefficients, which correspond to the energy-level gaps of the systems. Based on this method, only after some simple algebra, we derive the energy-level gaps for several important systems in NCQM, and most of these results have not been reported in literature so far.

Atom-field interaction and absorption, emission of photons in noncommutative space are investigated. We calculate the rate and the directional distribution of the radiation. The modified selection rules which govern allowed and forbidden transitions in noncommutative space are derived.

Deformation quantization is a powerful tool to quantize some classical systems especially in noncommutative space. In this work we first show that for a class of special Hamiltonian one can easily find relevant time evolution functions and Wigner functions, which are intrinsic important quantities in the deformation quantization theory. Then based on this observation we investigate a two-coupled harmonic oscillators system on the general noncommutative phase space by requiring both spatial and momentum coordinates do not commute each other. We derive all the Wigner functions and the corresponding energy spectra for this system, and consider several interesting special cases, which lead to some significant results.

In this paper the stationary Klein–Gordon equation is considered for the Coulomb potential in noncommutative space. The energy shift due to noncommutativity is obtained via the perturbation theory. Furthermore, we show that the degeneracy of the initial spectral line is broken in transition from commutative space to noncommutative space.

We study the two-dimensional harmonic oscillator on a noncommutative plane. We show that by introducing appropriate Bopp shifts, one can obtain the Hamiltonian of a two-dimensional harmonic oscillator on a sphere according to the Higgs model. By calculating the commutation relations, we show that this noncommutativity is strictly dependent on the curvature of the background space. In other words, we introduce a kind of duality between noncommutativity and curvature by introducing noncommutativity parameters as functions of curvature. Also, it is shown that the physical realization of such model is a charged harmonic oscillator in the presence of electromagnetic field.

The free energy of an imperfect gas of nonrelativistic fermions in noncommutative space is calculated to 2-loop order and its zero and low temperature limits are derived. Implications on the degeneracy pressure of the system is discussed.

Noncommutative space which is rotationally invariant is considered. The hydrogen atom is studied in this space. We exactly find the leading term in the asymptotic expansion of the corrections to the ns energy levels over the small parameter of noncommutativity.

We consider the motion of a particle in a uniform field in noncommutative space which is rotationally invariant. On the basis of exact calculations, it is shown that there is an effect of coordinate noncommutativity on the mass of a particle. A particular case of motion of a particle in a uniform gravitational field is considered and the equivalence principle is studied. We propose the way to solve the problem of violation of the equivalence principle in the rotationally invariant noncommutative space.

In noncommutative quantum mechanics, the energy-dependent harmonic oscillator problem is studied by solving the Schrödinger equation in polar coordinates. The presence of the noncommutativity in space coordinates and the dependence on energy for the potential yield energy-dependent mass and potential. The correction of normalization condition is calculated and the parameter-dependences of the results are studied graphically.

In this paper, we define the Pauli Hamiltonian function for the twist-deformed N-enlarged Newton–Hooke spacetime provided by M. Daszkiewicz [Mod. Phys. Lett. A27, 1250083 (2012)]. Further, we derive its energy spectrum, i.e. we find the corresponding eigenvalues as well as the proper eigenfunctions.

The Klein–Gordon equation is considered for the Kratzer potential in the spherical polar coordinate in laboratory frame in noncommutative space. The energy shift due to noncommutativity is obtained via the perturbation theory. After rather cumbersome algebra, we found the eigenfunctions and eigenvalues of the system for a noncommutative phase space.

In this paper, by using the Dirac derivatives the Klein–Gordon (K-G) equation is determined in a $\kappa$-Minkowski spacetime. The dispersion relation and the first-order approximation case are deduced. The Feshbach–Villars (FV) equation is derived by applying the new linearization process to the time. We then study the effect of magnetic interaction on energies spectrum in a $\kappa$-Minkowski spacetime as an application, as a result we found that the energies spectrum are not symmetrical. We also study the case of hydrogen atom in non-relativistic limit by using perturbation theory. The upper bound of the $\kappa$-deformation parameter is evaluate, on the basis of the experimental data for $1s-2s$ transition frequency.

We introduce quantum heat engines that perform quantum Otto cycle and the quantum Stirling cycle by using a coupled pair of harmonic oscillator as its working substance. In the quantum regime, different working medium is considered for the analysis of the engine models to boost the efficiency of the cycles. In this work, we present Otto and Stirling cycle in the quantum realm where the phase space is non-commutative in nature. By using the notion of quantum thermodynamics, we develop the thermodynamic variables in non-commutative phase space. We encounter a catalytic effect (boost) on the efficiency of the engine in non-commutative space (i.e. we encounter that the Stirling cycle reaches near to the efficiency of the ideal cycle) when compared with the commutative space. Moreover, we obtained a notion that the working medium is much more effective for the analysis of the Stirling cycle than that of the Otto cycle.

In this study, we investigate the Pauli oscillator in a noncommutative space. In other words, we derive wave function and energy spectrum of a spin half non-relativistic charged particle that is moving under a constant magnetic field with an oscillator potential in noncommutative space. We obtain critical values of the deformation parameter and the magnetic field, which they counteract the normal and anomalous Zeeman effects. Moreover, we find that the deformation parameter has to be smaller than $2.57\times 10^{-26}{\mathrm{\text{m}}}^2$. Then, we derive the Helmholtz free energy, internal energy, specific heat and entropy functions of the Pauli oscillator in the non commutative space. With graphical methods, at first, we compare these functions with the ordinary ones, and then, we demonstrate the effects of magnetic field on these thermodynamic functions in the commutative and noncommutative space, respectively.

Under the influence of the deformation space-space symmetries, the improved Mobius square plus generalized Yukawa potentials (IMSGYPs) have been employed to solve the deformed Klien–Gordon equation in three-dimensional noncommutative relativistic quantum space (3D-RNCQS) symmetries. Combined with the approximation approach suggested by Greene and Aldrich, we also employ the parametric Bopp’s shift approach and standard perturbation theory to derive novel relativistic energy eigenvalues. The new relativistic energy eigenvalues of (N₂, K₂, NI, ScI, and RbH) diatomic molecules under the IMSGYPs were shown to be sensitive to the atomic quantum numbers ($j,l,s,m$), the mixed potential depths ($V_0,A,B,C,D,V_1$), the screening parameter’s inverse $\alpha$ and non-commutativity parameters ($\mathrm{\Theta }$, $\beta$, $\zeta$). In addition, we analyzed the new non-relativistic energy values in three-dimensional noncommutative non-relativistic quantum space (3D-NRNCQS) symmetries, by applying the well-known mapping in the literature. Furthermore, we studied many special cases useful to researchers in the framework of the new extended symmetries, such as the newly generalized Mobius square potential, the newly generalized Yukawa potential, and the newly generalized Deng-Fan potential. The study is further extended to calculate the mass spectra of mesons of the heavy quarkonium system, such as $c\overline{c}$, bottomonium $b\overline{b}$, $\overline{b}c$ and light mesons $c\overline{s}$ and $b\overline{s}$, that have the quark and antiquark flavors within the framework of the IMSGYPs model in 3D-NRNCQS symmetries.

In this paper, the modified Dirac equation (MDE) has been solved approximately for improved Manning–Rosen plus quasi-Hellman potentials (IMRQHPs), including the improved Yukawa tensor interaction under new spin symmetry and pseudospin symmetry for arbitrary spin–orbit number $k$, an appropriate approximation employed on the centrifugal terms in the symmetries of three-dimensional extended relativistic quantum mechanics (3D-ERQM). This potential is a superposition of Manning–Rosen plus quasi-Hellman potentials and some other exponential terms. Through the application of both new approximations to deal with the centrifugal term, Bopp’s shift method and the independent-time perturbation theory method, the recent modified approximate energy corrections under IMRQHPs are obtained for nuclei ${}^{17}$O and ${}^{17}$F and (O₂, I₂, N₂, H₂, LiH, CO and NO) diatomic molecules in 3D-ERQM and three-dimensional extended non-relativistic quantum mechanics (3D-ENRQM) symmetries. The recent values that we get appear sensitive to $(j,k,l∕\tilde{l},s∕\tilde{s},m∕\tilde{m})$, which are known as the discrete atomic quantum numbers, the mixed potential depths ($C∕A$, $D∕B$) and $(V_0∕a,V_1∕b)$, the range of the potential $\alpha$ and non-commutativity parameters $(\mathrm{\Theta },\sigma ,\chi )$. In both 3D-ERQM and 3D-ENRQM symmetries, we show that the recent corrections obtained on the new bound states under IMRQHPs are infinitesimal to the principal part of energy in the ordinary cases of 3D-RQM and 3D-NRQM symmetries. In the new symmetries of 3D-ERQM symmetries, it is not possible to get the exact analytical solutions $k=1$ and $k\ne -1$, only the approximate solutions are possible. In addition, we discussed in detail the non-relativistic study of the studied mixed potentials. Four special cases, i.e. we investigated some special cases in the context of modified Schrödinger theories. In the framework of known relativistic quantum mechanics, we have clearly demonstrated that the modified Schrödinger equation can be represented as the Dirac equation under the influence of IMRQHPs.

In this paper, we investigate the quantum entanglement induced by phase-space noncommutativity. Both the position–position and momentum–momentum noncommutativity are incorporated to study the entanglement properties of coordinate and momentum degrees of freedom under the shade of oscillators in noncommutative space. Exact solutions for the systems are obtained after the model is re-expressed in terms of canonical variables, by performing a particular Bopp’s shift to the noncommuting degrees of freedom. It is shown that the bipartite Gaussian state for an isotropic oscillator is always separable. To extend our study for the time-dependent system, we allow arbitrary time dependency on parameters. The time-dependent isotropic oscillator is solved with the Lewis–Riesenfeld invariant method. It turns out that even for arbitrary time-dependent scenarios, the separability property does not alter.

We extend our study to the anisotropic oscillator, which provides an entangled state even for time-independent parameters. The Wigner quasi-probability distribution is constructed for a bipartite Gaussian state. The noise matrix (covariance matrix) is explicitly studied with the help of Wigner distribution. Simon’s separability criterion (generalized Peres–Horodecki criterion) has been employed to find the unique function of the (mass and frequency) parameters, for which the bipartite states are separable. In particular, we show that the mere inclusion of non-commutativity of phase-space is not sufficient to generate the entanglement, rather anisotropy is important at the same footing.

We explore the experimental viability of our result through the computation of extractable work for the current situation.

We discuss the different possibilities of constructing the various energy–momentum tensors for noncommutative gauge field models. We use Jackiw's method in order to get symmetric and gauge invariant stress tensors — at least for commutative gauge field theories. The noncommutative counterparts are analyzed with the same methods. The issues for the noncommutative cases are worked out.