Narrow 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.

A lot of problems of atomic and nuclear physics depend on Coulomb potential with high accuracy. Therefore, it is very important to carefully and accurately calculate the Coulomb potential. In this study, a new analytical expression was obtained for calculating the Coulomb potential by choosing the Fermi distribution function, which is suitable for charge distribution in nuclei. The proposed formula guarantees an accurate and simple calculation of the Coulomb potential of nuclei. Using the analytical expression obtained, the Coulomb potentials for several spherical nuclei were calculated for all values of the parameters. It is shown that the results obtained for arbitrary values of the radius are consistent with the literature data. In this study, the accepted values in the literature of the two parameters (Coulomb radius $R_c$ and diffuseness parameter $a_c)$ which are important for the Coulomb potential are also discussed.

The Klein–Gordon equation with equal scalar and vector potentials $s(\widehat{r})=v(\widehat{r})=V(\widehat{r})/2$ describing the dynamics of a three-dimensional under the modified Coulomb plus inverse-square potential is considered, in the symmetries of noncommutative quantum mechanics (NCQM), using Bopp’s shift method. The new energy of $n$th excited state $E_{nc-n}(a,b,n,j,l,m)$ is obtained as a function of the shift energy $\mathrm{\Delta }{E}_{is}(n,j,l,s,m)$ and $E_n$ is obtained via first-order perturbation theory in the three-dimensional noncommutative real space (NC: 3D-RS) symmetries instead of solving modified Klein–Gordon equation (MKGE) with the Weyl–Moyal star product. It is found that the perturbative solutions of discrete spectrum depended by the Gamma function, the discreet atomic quantum numbers $(j,l,s,m)$ and the potential parameters (A and B), in addition to noncommutativity parameters ($𝜃$ and $\sigma$), which are induced with the effect of (space–space) noncommutativity properties.

We introduce the Dunkl–Klein–Gordon (DKG) equation in 2D by changing the standard partial derivatives by the Dunkl derivatives in the standard Klein–Gordon (KG) equation. We show that the generalization with Dunkl derivative of the z-component of the angular momentum is what allows the separation of variables of the DKG equation. Then, we compute the energy spectrum and eigenfunctions of the DKG equations for the 2D Coulomb potential and the Klein–Gordon oscillator analytically and from an su(1, 1) algebraic point of view. Finally, we show that if the parameters of the Dunkl derivative vanish, the obtained results suitably reduce to those reported in the literature for these 2D problems.

Similar to how general relativity emerged, quantum mechanics resulted from experimental observations that made us radically rethink our previous assumptions. Scientists who study atoms need to solve these three equations. They are for a system with N electrons affected by the nucleus’s attractive Coulomb force and the repulsive force between each pair of electrons and other potentials. While general relativity has upended our large-scale conception of the universe, quantum mechanics calls into question all our intuition regarding particle physics. The Klein–Gordon (KG) resolution helps determine some physical systems and their properties by finding their bound states and eigenfunctions. These are resolved using analytical and numerical methods, as well as Coulomb and Yukawa potentials. In order to extract some of the first eigenvalues of the quantum mechanical system, we applied the Nikiforov–Uvarov (NU) method to the KG equation. The findings are significant for understanding nuclear charge radius, spin, nuclear diffusion and other topics in numerous theoretical physics and quantum chemistry fields because they are more generic and helpful.

We consider QED in five dimensions in a configuration where matter is localized on a 3-brane while foton propagates in the bulk. The idea is to investigate the effects of the Kaluza–Klein modes of the photon in the relativistic regime, but in low energy, and in the nonrelativistic regime. In the relativistic regime, we calculate the cross-section for the reaction e⁺+e^-→μ⁺+μ^-. We compare our theoretical result with a precise measurement of this cross-section at . As result, we extract a lower bound on the size of the extra dimension. In the nonrelativistic regime, we derive the contribution for the Coulomb potential due to the whole tower of the Kaluza–Klein excited modes of the photon. We use the modified potential to calculate the Rutherford scattering differential cross-section.

We study the covariant Dirac equation in the space–time generated by a cosmic string in presence of vector and scalar potentials of electromagnetic field. We obtain the solution of the radial part of Dirac equation. We consider the scattering states under the Coulomb potential and obtain the phase shifts.

The relation between certain Hamiltonians, known as dual, or partner Hamiltonians, under the transformation $x\to {\bar{x}}^{\bar{\alpha }}$ has long been used as a method of simplifying spectral problems in quantum mechanics. This paper seeks to examine this further by expressing such Hamiltonians in terms of the generators of sl(2) algebra, which provides another method of solving spectral problems. It appears that doing so greatly restricts the set of allowable potentials, with the only nontrivial potentials allowed being the Coulomb $\frac{1}{r}$ potential and the harmonic oscillator $r^2$ potential, for both of which the sl(2) expression is already known. It also appears that, by utilizing both the partner potential transformation and the formalism of the Lie-algebraic construction of quantum mechanics, it may be possible to construct part of a Hamiltonian’s spectrum from the quasi-solvability of its partner Hamiltonian.

In this paper, we study the relativistic scalar particle described by the Klein–Gordon equation that interacts with the uniform magnetic field in the context of the Som–Raychaudhuri space–time. Based on the property of the biconfluent Heun function equation, the corresponding Klein–Gordon oscillator and generalized Klein–Gordon oscillator under considering the Coulomb potential are separately investigated, and the analogue of the Aharonov–Bohm effect is analyzed in this scenario. On this basis, we also give the influence of different parameters including parameter $\alpha$ and oscillator frequency $\omega$, and the potential parameter ${\xi }_2$ on the energy eigenvalues of the considered systems.

The Thomas–Fermi (TF) equation is solved numerically for an electron gas system that interacts via the Coulomb potential. An emphasis is placed on how certain physical properties, such as the chemical potential and the total energy, change with the shape of the confinement at finite temperatures. By comparing these results with the results calculated for the noninteracting case, we are able to analyze how the inter-particle forces affect the thermodynamic properties of electrons. It is shown that the total energy and other properties of an electron gas is very sensitive to the particle interactions and the shape of the confining potential, even at high temperatures. The results are also applicable to nanostructures like two-dimensional quantum dot systems, wires.

Several parameters that affect the optical second-harmonic generation (SHG) coefficients in the two-dimensional Coulomb system have been discussed. Due to the changes of the Coulomb potential coefficients and geometric factor, it is shown that the optical SHG coefficients' changes are effectively controlled. Moreover, the SHG coefficients depend strongly on the system size and relaxation time.

Plasmon modes in monolayer graphene on substrate are analyzed taking into account the thickness of graphene and substrate material layer in the evaluation of Coulomb potential. It is shown that plasmon mode in graphene monolayer has linear dispersion in contrast to multilayer graphene in long-wavelength limit. The slope of plasmon spectrum is determined by the thickness and dielectric constant of substrate. Obtained results are in good agreement with experimental data and other theoretical considerations.

The solutions of the Klein–Gordon equation with a Coulomb potential in D dimensions are obtained exactly, and the energy levels E(n,l,D) are analytically presented. The dependence of the energy difference ΔE(n,l,D) for D and D-1 on the dimension D is demonstrated as three different kinds of rules. The dependence of the energy E(n,l,D) on the dimension D is also analyzed. It is shown that the energy E(n,l,D) (l≠0) is almost independent of the quantum number l, while E(n,0,D) first decreases and then increases as the dimension D increases. There is no bound state of the S-wave for D=2.

The solutions of the Klein–Gordon equation with a Coulomb plus scalar potential in D dimensions are exactly obtained. The energy E(n,l,D) is analytically presented and the dependence of the energy E(n,l,D) on the dimension D is analyzed in some detail. The positive energy E(n,0,D) first decreases and then increases with increasing dimension D. The positive energy E(n,l D)(l≠0) increases with increasing dimension D. The dependences of the negative energies E(n,0,D) and E(n,l,D)(l≠0) on the dimension D are opposite to those of the corresponding positive energies E(n,0,D) and E(n,l,D)(l≠0). It is found that the energy E(n,0,D) is symmetric with respect to D=2 for D∈(0,4). It is also found that the energy E(n,l,D)(l≠0) is almost independent of the angular momentum quantum number l for large D and is completely independent of the angular momentum quantum number l if the Coulomb potential is equal to the scalar one. The energy E(n,l D) is almost overlapping for large D.

We obtain analytical solutions of the two-body spinless Salpeter (SS) equation with the Yukawa potential within the conventional approximation scheme to the centrifugal term for any l-state. The semirelativistic bound-state energy spectra and the corresponding normalized wave functions are calculated by means of the Nikiforov–Uvarov (NU) method. We also obtain the numerical energy spectrum of the SS equation without any approximation to centrifugal term for the same potential and compare them with the approximated numerical ones obtained from the analytical expressions. It is found that the exact numerical results are in good agreement with the approximated ones for the lower energy states. Special cases are treated like the nonrelativistic limit and the solution for the Coulomb problem.

We show that additional solutions must be ignored (in differences of the Schrödinger and Klein–Gordon equations) in the Dirac equation, where usually the second-order radial equation is passed, called the reduced equation, instead of a system. Analogously to the Schrödinger equation, in this process, the Dirac’s delta function appears, which was unnoted during the full history of quantum mechanics. This unphysical term we remove by a boundary condition at the origin. However, the distribution theory imposes on the radial function strong restriction and by this reason practically for all potentials, even regular, use of these reduced equations is not permissible. At the end, we include consideration in the framework of two-dimensional Dirac equation. We show that even here the additional solution does not survive as a result of usual physical requirements.

This paper is devoted to the calculations of alpha decay half-lives of ${}^{186-218}$Po $(Z=84)$ isotopes and actinide nuclei between the atomic numbers Z = 90 and Z = 102, using an energy-dependent potential model. The $\alpha$-nucleus effective potential employed in this study includes a Coulomb potential, a centrifugal barrier potential, and a nuclear potential. The nuclear potential is chosen to be the energy-dependent singularized harmonic potential, typical of nuclear interactions. The role of the energy-dependent potential is very significant in this study. Indeed, when compared to the experimental data, our results are found to be improved when energy-dependent interaction potential is employed compared to when energy-independent interactions are used, especially for small values of the parameter $\gamma$ as evidenced by the values of the root mean square deviation.

The radial part of the wave function of an electron in a Coulomb potential is the product of a Laguerre polynomial and an exponential with the variable scaled by a factor depending on the degree. This note presents an elementary proof of the orthogonality of wave functions with differing energy levels. It is also shown that this is the only other natural orthogonality for Laguerre polynomials. By expanding in terms of the usual Laguerre polynomial basis, an analogous strange orthogonality is obtained for Meixner polynomials.

The problem of a spineless charged particle with a time-dependent decaying mass interacting with a Coulomb and an inverse quadratic potentials is considered. The Green’s function is explicitly evaluated. The energy levels as well as the wave functions for the bound states are exactly determined.

In this study, the analytical solutions of the Klein–Gordon equation for any $l$ states of the modified effective mass potential under the modified unequal scalar and vector Coulomb–Hulthén potential (MUSVCH-P) are derived by using an approximation method to the centrifugal potential term in the symmetries of relativistic noncommutative three-dimensional real space (RNC: 3D-RS). The new analytical expressions for eigenvalues of the energy spectrum and the new mass of mesons, such as charmonium and bottomonium that have the quark and antiquark flavor, have been estimated by using Bopp’s shift method, and perturbation theory. The energy state equation depends on the global parameters characterizing the noncommutativity space and the potential parameter $(v_0,s_0,v_1,s_1,m_0,m_1,\delta )$ in addition to the Gamma function and the discreet atomic quantum numbers $(j,l,s,m)$. The expression for the new energy spectra is applied to obtain the new mass spectra of heavy quarkonium systems (charmonium and bottomonium) in the symmetries of (RNC: 3D-RS). The comparisons show that our theoretical results are in very good agreement with the reported works.