Angular Quantum Number Research Articles

Benchmark studies on the large errors of calculated binding energies in metallophilic interactions.

Aurophilicity is a d10-d10 closed-shell interaction, which is repulsively calculated by the Hartree-Fork (HF) method, whereas binding energies (Eb) are largely overestimated under the second-order Møller-Plesset (MP2) method, compared to the coupled cluster singles and doubles with perturbative triples [CCSD(T)] method. The unusual energy errors between different wave functional methods were also verified in other closed-shell metallophilic systems and even were taken as a label of metallophilic interaction. Here, we performed a benchmark study on a collection of structures with weak interactions, sp-sp bonds, sp-d bonds, and d-d bonds, to investigate the influence factor of the errors of HF and MP2 methods. It was found that the large energy errors of HF and MP2 methods were not specified for closed-shell interactions, and the errors could also be very large for many covalent bonds, which was strongly related to the azimuthal quantum number of interaction orbitals. Compared to the CCSD(T) method, the MP2 method weakens the s-s covalent interactions slightly, strengthens the p-p covalent interactions slightly, and overestimates the d-d covalent interactions largely (can be -170 kcal/mol for the Re-Requadruple bond). This benchmark study suggests that the special energy errors in metallophilicity may result from the participation of d orbitals. Benchmark studies on various density functional methods were also given for calculating binding energies of d-d bonds.

Topological Effects of a Spiral Dislocation on Quantum Revivals

We analyse the influence of spiral dislocation topology on the revival time for the harmonic oscillator, for a particle confined to one-dimensional quantum ring, and a two-dimensional quantum ring. We first investigate the effects of a cut-off point that stems from the topology of this defect on the harmonic oscillator. Then, we show that the influence of spiral dislocation topology on the harmonic oscillator gives rise to a non-null revival time related to the radial quantum number. In the case of the two-dimensional quantum ring, we show that the revival times related to the radial quantum number and the angular momentum quantum number are influenced by the spiral dislocation topology.

Bound-state effects on dark matter coannihilation: Pushing the boundaries of conversion-driven freeze-out

Bound-state formation can have a large impact on the dynamics of dark matter freeze-out in the early Universe, in particular for colored coannihilators. We present a general formalism to include an arbitrary number of excited bound states in terms of an effective annihilation cross section, taking bound-state formation, decay and transitions into account, and derive analytic approximations in the limiting cases of no or efficient transitions. Furthermore, we provide explicit expressions for radiative bound-state formation rates for states with arbitrary principal and angular quantum numbers $n,\ell$ for a mediator in the fundamental representation of $SU(3)_c$, as well as electromagnetic transition rates among them in the Coulomb approximation. We then assess the impact of bound states within a model with Majorana dark matter and a colored scalar $t$-channel mediator. We consider the regime of coannihilation as well as conversion-driven freeze-out (or coscattering), where the relic abundance is set by the freeze-out of conversion processes. We find that the region in parameter space where the latter occurs is considerably enhanced into the multi-TeV regime. For conversion-driven freeze-out, dark matter is very weakly coupled, evading direct and indirect detection constraints but leading to prominent signatures of long-lived particles that provide great prospects to be probed by dedicated searches at the upcoming LHC runs.

Approximate Solutions, Thermal Properties, and Superstatistics Solutions to Schrödinger Equation

In this work, we apply the parametric Nikiforov-Uvarov method to obtain eigensolutions and total normalized wave function of Schrödinger equation expressed in terms of Jacobi polynomial using Coulomb plus Screened Exponential Hyperbolic Potential (CPSEHP), where we obtained the probability density plots for the proposed potential for various orbital angular quantum number, as well as some special cases (Hellmann and Yukawa potential). The proposed potential is best suitable for smaller values of the screening parameter α . The resulting energy eigenvalue is presented in a close form and extended to study thermal properties and superstatistics expressed in terms of partition function Z and other thermodynamic properties such as vibrational mean energy U , vibrational specific heat capacity C , vibrational entropy S , and vibrational free energy F . Using the resulting energy equation and with the help of Matlab software, the numerical bound state solutions were obtained for various values of the screening parameter ( α ) as well as different expectation values via Hellmann-Feynman Theorem (HFT). The trend of the partition function and other thermodynamic properties obtained for both thermal properties and superstatistics were in excellent agreement with the existing literatures. Due to the analytical mathematical complexities, the superstatistics and thermal properties were evaluated using Mathematica 10.0 version software. The proposed potential model reduces to Hellmann potential, Yukawa potential, Screened Hyperbolic potential, and Coulomb potential as special cases.

Interferometry of black holes with Hawking radiation

We investigate the wave optical imaging of black holes with Hawking radiation. The spatial correlation function of Hawking radiation is expressed in terms of transmission and reflection coefficients for scalar wave modes and evaluated by numerically summing over angular quantum numbers for the Unruh-Hawking state of the Kerr-de Sitter black hole. Then, wave optical images of an evaporating black hole are obtained by the Fourier transformation of the spatial correlation function. For short wavelength, the image of the black hole with the outgoing mode of the Unruh-Hawking state has the appearance of a star with its surface given by the photon sphere. It is found that interference between incoming modes from the cosmological horizon and reflected modes due to the scattering of the black hole can enhance brightness of images in the vicinity of the photon sphere. For a long wavelength, the entire field of view is bright, and the emission region of Hawking radiation cannot be identified.

Exact solutions of the 2D Schrödinger equation with the inverse square root potential

The exact solutions of the 2D Schrödinger equation with the inverse square root potential (a > 0) are found as the Biconfluent Heun function . The wave functions with the variation of the potential parameter a and angular momentum quantum number m are illustrated graphically.

Investigation the three dimensional bound states in quantum dot nanowire systems

Electronic confined states of a cylindrical InP quantum dot embedded in infinite GaP nanowire is studied theoretically. To this end the Schrodinger equation is solved numerically in the effective mass approximation. The subband separation between the GaP nanowire and InP/GaP core–shell nanowire is utilized to predict the possible existence of the three dimensionally confined states of the system. Using this, the nanowire critical radius is calculated for each channel of angular momentum, the frontier of type I-Type II transition in the InP/GaP quantum dot nanowire system. It is shown that the lower limit of the continuum energy depends on the angular momentum and increases by it, causing discrete bound states of higher angular momentum to lie in the continuum of the lower ones. Surviving the energy states in the presence of external magnetic field reveals stronger confinement for electrons in states with negative angular momentum quantum numbers.

Gauge effects in bound-bound Rydberg transition matrix elements

Accurate data on electric-dipole transition matrix elements (EDTMs) for bound-bound Rydberg-atom transitions have become increasingly important in science and technology. Here we compute radial EDTMs of rubidium in length, velocity, and acceleration forms for electric-dipole-allowed transitions between states with principal and angular-momentum quantum numbers $n$ and $\ensuremath{\ell}$ ranging from 15 to 100. Wave functions are computed based upon model potentials from [Phys. Rev. A 49, 982 (1994)]. Length-gauge EDTMs of strong low-$\ensuremath{\ell}$ Rydberg transitions, often employed in research and technology, are found to deviate from the fundamentally more accurate velocity-gauge form by approximately-$n$-independent and series-dependent shifts. The shift amount peaks at about $0.34\phantom{\rule{4pt}{0ex}}e{a}_{0}$ for the Rb $nD\ensuremath{\leftrightarrow}\phantom{\rule{4pt}{0ex}}(n+1)P$ series, a particularly strong Rydberg-transition series in Rb. The shift corresponds to relative EDTM corrections of up to $\ensuremath{\approx}{10}^{\ensuremath{-}3}$, which can be of concern in high-precision applications. We discuss the physical reasons for the observed gauge differences, explain the conditions for applicability of the velocity- and length-gauge forms for different transition series, and present a decision tree of how to choose EDTMs.

Quantization of Acoustic Modes in Dumbbell Nanoparticles.

The vibrational eigenmodes of dumbbell-shaped polystyrene nanoparticles are recorded by Brillouin light spectroscopy (BLS), and the full experimental spectra are calculated theoretically. Different from spheres with a degeneracy of (2l+1), with l being the angular momentum quantum number, the eigenmodes of dumbbells are either singly or doubly degenerate owing to their axial symmetry. The BLS spectrum reveals a new, low-frequency peak, which is attributed to the out-of-phase vibration of the two lobes of the dumbbell. The quantization of acoustic modes in these molecule-shaped dumbbell particles evolves from the primary colloidal spheres as the separation between the two lobes increases.

The Information-Theoretic Treatment of Spinless Particles with the Assorted Diatomic Molecular Potential

The relativistic solutions of the Klein-Gordon equation comprising an interaction of the generalized inversely quadratic Yukawa potential mixed linearly with the hyperbolic Schiöberg molecular potential is achieved employing the idea of parametric Nikiforov-Uvarov and the Greene-Aldrich approximation scheme. The energy spectra and the corresponding normalized wave functions are derived regarding the hypergeometric function in a closed form for arbitrary ℓ -state. Numerical results of the energy eigenvalue are proposed. Moreover, special circumstances of this potential are reviewed, and their energy eigenvalues were assessed. Subsequently, the Tsallis entropy and Rényi entropy both in position and momentum spaces are defined under the desired potential. The impacts of these entropies on the angular momentum quantum number are explored in detail.

JRAF: A Julia package for computation of relativistic molecular auxiliary functions

The evaluation of relativistic molecular integrals over exponential−type spinor orbitals requires the use of relativistic auxiliary functions in prolate spheroidal coordinates, and has been recently achieved (Bağcı and Hoggan (2015) [14]). This process is used in the solution of the molecular Dirac equation for electrons moving in a Coulomb potential. A series of papers on a method for fully analytical evaluation of relativistic auxiliary functions has been published [2, 3, 4] From the perspective of computational physics, these studies demonstrate how to deal with the integrals of the product of power functions with non−integer exponents and incomplete gamma functions. The computer program package used to calculate these auxiliary functions with high accuracy is presented. It is designed using the Julia programming language and yields highly accurate results for molecular integrals over a wide range of orbital parameters and quantum numbers. Additionally, the program package facilitates the efficient calculation of the angular momentum coefficients that arise from the product of two normalized Legendre functions centered at different atomic positions, and the determination of the rotation angular functions used for both complex and real spherical harmonics. Sample calculations are performed for two−center one−electron integrals over non−integer Slater−type orbitals, and the results prove the robustness of the package. Program summaryProgram Title: JRAFCPC Library link to program files:https://doi.org/10.17632/942xsbvfdf.1Developer's repository link:https://github.com/abagciphys/JRAF.jlLicensing provisions: MITProgramming language:Julia programming languageSupplementary material: An experimental version of the computer program package written in Mathematica programming language [5].External routines/libraries:Nemo computer algebra package for the Julia programming language [6], Cuba multidimensional numerical integration using different algorithms in Julia [7].Nature of problem: Relativistic molecular auxiliary function integrals result from the expression of a two−center two−electron Coulomb energy associated with a charge density. The Coulomb energy is transformed into kinetic energy integrals using Poisson's equation and the single−center potential, considering that the Laplace expansion for the Coulomb interactions is expressed in terms of normalized non−integer Slater−type orbitals [1]. Using the resulting expression for the two−center two−electron integrals, relativistic auxiliary function integrals are derived in prolate ellipsoidal coordinates. These auxiliary functions are generalized to the entire set of physical potential operators for the Coulomb potential case.The integral of the relativistic auxiliary functions have no closed−form solutions except that their parameters are integers. As such, the analytical evaluation of these functions is challenging. They are used in the solution of the matrix form representation of the molecular Dirac−Fock self−consistent field (SCF) equation.Solution method: A criterion that considers the symmetry properties of two−center two−electron molecular integrals is initially proposed [2]. This obviates the need for the computation of incomplete and complementary incomplete gamma functions, and utilizes their sum (P+Q=1). The resulting form of the integral of the relativistic molecular auxiliary functions is expressed in terms of the convergent series representation of incomplete beta functions. Recurrence relationships are then derived for each of these sub−functions [3]. The algorithm for computation of the auxiliary functions is based on the vectorization procedure defined in [4].

Quasinormal modes of novel 4D Einstein-Gauss-Bonnet Anti-de Sitter black holes

We comprehensively analyze the Quasinormal Modes (QNMs) of the external matter field perturbation (including scalar, electromagnetic and Dirac field perturbation) and the gravitational perturbation (including odd and even parity perturbation) of novel 4D Einstein-Gauss-Bonnet (EGB) black holes (BHs) in Anti-de Sitter (AdS) spacetime. The corresponding master equation in the form of perturbations and their respective effective potential is derived. Here we focus on the influence of the cosmological constant Λ, coupling parameters α and angular quantum numbers l on the QNMs frequency of novel 4D EGB AdS BHs. The results show that with the increase of the absolute value of the coupling parameters α and angular quantum numbers l, the oscillation of the novel 4D EGB BHs is enhanced, but the decay decreases. However, as the absolute value of the cosmological constant Λ increases, the oscillation and decay of the novel 4D EGB BHs becomes faster and faster. Therefore, the analysis of QNMs by the dynamic evolution image provides direct evidence for the local stability of the novel 4D EGB AdS BHs.

Electron vortices in spin-orbit coupling system

There exists an electron vortex solution with orbital angular momentum quantum in a non-spin-orbit coupling system which has nonconservative orbital angular momentum. We discuss the system with spin-orbit coupling and nonconservative orbital angular momentum, and we can find that the electrons with the total angular momentum numbers also have vortex beam solutions. And the vortex beam is expressed as an entangled wave function of the spin wave function and the vortex wave function. Taking the electrons in the central force field for example, in this paper constructed is a spinor vortex structure which is caused by the propagation of electrons carrying a fixed quantum number of total angular momentum along the <i>z</i>-axis. The spinor vortex structure is under the condition that the orbital angular momentum caused by spin-orbit coupling is non-conserved but the total angular momentum is conserved. The corresponding electron vortex beams in spin-vortex entanglement are solved by perturbation method, and the Foldy-Wouthuysen transformation is utilized to show that the vortex solution of the four-component spinor does exist in the case of relativity, when the electron with a fixed total angular momentum quantum number propagates along the <i>z</i>-axis in the central force field. The spinor provides theoretical support for the existence of the vortex structure for the system where the orbital angular momentum is not conserved but the total angular momentum is conserved due to spin-orbit coupling.

Effects of ocean turbulence on photon orbital angular momentum quantum communication

The effect of the turbulent motion of ocean on the quantum communication based on the orbital angular momentum in an underwater quantum channel is studied in this work. Based on the power spectrum model of ocean turbulence proposed by Elamassie, the quantitative relationships of different ocean turbulence parameters with the single photon detection probability of orbital angular momentum photons, the channel capacity, the key generation rate, the concurrence of two entangled photons are proposed. The maximum entanglement distance of the orbital angular momentum entangled photon-pairs in the ocean turbulence is further studied by the universal entanglement decay of the concurrence of entangled photon-pairs in the ocean turbulence. The results show that the detection probability of single photon, the channel capacity, the key generation rate, and the concurrence of entangled photon-pairs decrease with the increase of the dissipation rate of turbulent kinetic energy and the decrease of the rate of dissipation of mean-squared temperature. The influence of the temperature and salinity balance parameter of ocean turbulence on the performance of underwater quantum communication are significantly different under the condition of whether the stable stratification of seawater is assumed or not. In the ocean turbulent environment, the increasing of the initial orbital angular momentum quantum number of signal photons can improve the key generation rate of quantum key distribution and the resistance of entangled photons to entanglement decay.

Аналитическое решение уравнения Шредингера для суммы потенциала Хюльтена и класс Юкавы

In this study, the bound state's solution of the modified Schrödinger equation is found for the new supposed combined Hultén potential and the Yukawa class potentials. Analytical expressions for the energy eigenvalue and the corresponding radial wave functions are obtained for any orbital quantum number. Obtained eigenfunctions are expressed in terms of hypergeometric functions. To overcome the potential's centrifugal part difficulties, we applied the developed approximation scheme for studying bound states. It is shown that energy levels and eigenfunctions are susceptible, depending on potential parameters. Finally, we investigate also the eigenvalue of the energy and the corresponding radial wave function under some exceptional cases.

Optical trapping force on two types of particles with a focused partially coherent Lommel-Gaussian beam

In order to explore the capture of two types of particles with different refractive-indexes with focused partially coherent Lommel-Gaussian beam (PCLGB), we numerically derived the expressions of the intensity, and its distribution is also simulated. We found that the focused PCLGB is able to capture two high refractive-index particles at different positions on the focal plane and a low refractive-index particle at the focus, respectively. when the orbital angular momentum (OAM)quantum number, lens focal length, asymmetry parameter decreased, and the spatial correlation length and beam waist increased, the transverse gradient force of the focused PCLGB increased; when the OAM quantum number, lens focal length, asymmetry parameter, spatial correlation length decreased and the beam waist increased, the axial gradient force and scattering force of the focused PCLGB increased. We can choose the optimum optical and lens parameters, a large range and stable trapping can be achieved. The obtained results have certain reference value for the focused PCLGB applied in micromanipulation technology and biotechnology.

Superatomic Rydberg State Excitation.

Superatomic molecular orbitals (SAMOs) have symmetries (angular quantum numbers) similar to those of atoms, and thus, it is possible to realize Rydberg state excitations (RSEs) in superatomic molecules. In this Letter, the feasibility of superatomic Rydberg state excitation (SRSE) is explored using gold superatoms based on first-principles calculations. The results show that the SRSE exists in the high and low excited states of the gold superatoms and their SAMOs make a major contribution to electronic transitions. The radial distribution function of electronic density shows that the main distribution of electrons in the lowest unoccupied molecular orbitals and other unoccupied superatomic molecular orbitals is extremely far from the geometric center, and thus, they can be unambiguously identified as Rydberg orbitals. We found that due to the two-dimensional ductility of the planar SAMOs, superatoms are superior in the RSE regulation. Our findings may provide a new source of superatom-based RSE and will contribute to the regulation and efficient preparation of Rydberg states.

Maxwell perturbations in a cavity with Robin boundary conditions: two branches of modes with spectrum bifurcation on Schwarzschild black holes

We perform a systematic study of the Maxwell quasinormal spectrum in a mirror-like cavity following the generic Robin type vanishing energy flux principle, by starting with the Schwarzschild black holes in this paper. It is shown that, for black holes in a cavity, the vanishing energy flux principle leads to two different sets of boundary conditions. By solving the Maxwell equations with these two boundary conditions both analytically and numerically, we observe two distinct sets of modes. This indicates that the vanishing energy flux principle may be applied not only to asymptotically anti-de Sitter (AdS) black holes but also to black holes in a cavity. In the analytic calculations, the imaginary part of the Maxwell quasinormal modes are derived analytically for both boundary conditions, which match well with the numeric results. While in the numeric calculations, we complete a thorough study on the two sets of the Maxwell spectrum by varying the mirror radius r_m, the angular momentum quantum number ell , and the overtone number N. In particular, we proclaim that the Maxwell spectrum may bifurcate for both modes when the mirror is placed around the black hole event horizon, which is analogous to the spectrum bifurcation effects found for the Maxwell fields on asymptotically AdS black holes. This observation provides another example to exhibit the similarity between black holes in a cavity and the AdS black holes.

Stark broadening of low-[formula omitted] hydrogen lines in strongly magnetized hydrogen plasmas: Influence of [formula omitted]-degeneracy removal due to the quadratic Zeeman effect

The Stark broadening of hydrogen lines with a low upper principal quantum number n is reexamined in the case of plasmas subject to a large (≳1 kT) external magnetic field. This investigation has been motivated by the observation of strong Zeeman splitting in a significant number of white dwarf spectra. The diamagnetic Zeeman effect term present in the Hamiltonian removes the degeneracy of the energy levels in orbital quantum number l, which results in an alteration of the Stark perturbation due to the microfield. In this work, we investigate this effect by applying a computer simulation method designed for Stark broadening modeling. A focus is put on the Lyman α and H-α lines. Based on line shape calculations, we show that the width of the Zeeman components is strongly dependent on the magnetic field.

Electron-LO-phonon intrasubband scattering rates in a hollow cylinder under the influence of a uniform axial applied magnetic field

Scattering rates arising from the interactions of electrons with bulk longitudinal optical (LO) phonon modes in a hollow cylinder are calculated as functions of the inner radius and the uniform axial applied magnetic field. Now, the specific nature of electron-phonon interactions mainly depends on the character of the energy spectrum of electrons. As is well known, in cylindrical quantum wires, the application of a parallel magnetic field lifts the double degeneracy of the non-zero azimuthal quantum number states; \(m \not = 0\); irrespective of all electron’s radial quantum number l states. In fact, this Zeeman splitting is such that the \(m < 0\) electron’s energy subbands initially decrease with the increase of the parallel applied magnetic field. In a solid cylinder, the lowest-order; {\(l = 1;\,m = 0\)} subband is always the ground state. In a hollow cylinder, however, as the axial applied magnetic field is increased, the electron’s energy subbands take turns at becoming the ground state; following the sequence \(\lbrace m=0,-1,-2...~ -N\rbrace\) of azimuthal quantum numbers. Furthermore, in a hollow cylinder, in general, the electron’s energy separations between any two subbands are less than the LO phonon energy except for exceptionally high magnetic fields, and some highest-order quantum number states. In view of this, the discussion of the energy relaxation here is focused mainly on intrasubband scattering of electrons and only within the lowest-order {\(l = 1;\,m = 0\)} electron’s energy subband. The intrasubband scattering rates are found to be characterized by shallow minima in their variations with the inner radius, again, for a fixed outer radius. This feature is a consequence of a balance between two seemingly conflicting effects of the electron’s confinement by the inner and outer walls of the hollow cylinder. First; increased confinement of the charge carriers generally leads to the enhancement of the rates. Second; the presence of a hole in a hollow cylinder leads to a significant suppression of the scattering rates. The intrasubband scattering rates also show a somewhat parabolic increase in their variations with the applied magnetic field; an increase which is more pronounced in a relatively thick hollow cylinder.