Angular Quantum Number Research Articles

Broadening of energy levels of Rydberg states with small orbital angular momenta in ions of group IIb by ambient thermal radiation

We present the derivation of simple analytical expressions for calculating the widths of Rydberg states with small orbital angular momenta in Zn+, Cd+, Hg+ ions of group IIb of the periodic system of elements induced by blackbody radiation (BBR) at ambient temperatures from 100 to 3000 K. The probabilities of radiative transitions from excited nS, nP, nD and nF states to all dipole-accessible states of ions are calculated. The wave functions of the quantum defect (QD) method for the initial and final states of the Rydberg electron are used to calculate the amplitudes of transitions between bound states. The dependences of the probabilities of induced decays and excitations on the BBR temperature, the principal and orbital quantum numbers of the Rydberg ion are determined. Analytical expressions are derived for numerical estimates of the contributions of the probabilities of thermally induced decays and excitations to the width of the Rydberg energy level. The numerical values of the coefficients of interpolation polynomials representing asymptotic expansions in powers of the principal quantum number n for the relative probabilities of decays and excitations of Rydberg states with large values of n and small orbital angular momenta l = 0, 1, 2, 3 are calculated.

Investigations on transition properties of X2Σ+, A2Π, B2Σ+, C2Π, and a4Σ+ states, and their Ω states of BO

The transition properties were calculated for the X2Σ+, A2Π, B2Σ+, C2Π, and a4Σ+ states, and eight Ω states of BO. The calculations were performed using the CASSCF method, followed by the icMRCI approach. Core–valence correlation and relativistic corrections were included. The radiative lifetimes of the first 16 vibrational levels were roughly 1.5–1.6 μs for the A2Π state, 96–177 ns for the B2Σ+ state, and 16.7–18.5 ns for the C2Π state. The B2Σ+ – X2Σ+ and C2Π – X2Σ+ transitions were strong, followed by the A2Π – X2Σ+, B2Σ+ – A2Π, and C2Π – A2Π transitions. The radiative lifetime of the first 16 levels was approximately 4.9–10.4 ms for the a4Σ+1/2 state, 0.21–24 ms for the a4Σ+3/2 state, 2.9–3.6 μs for the A2Π1/2 state, 3.5–3.8 μs for the A2Π3/2 state, 134–230 ns for the B2Σ+1/2 state, 23.2–25.4 ns for the C2Π1/2 state, and 23.0–25.2 ns for the C2Π3/2 state. Among all spin–forbidden transitions generated from the a4Σ+ state, the a4Σ+1/2 – X2Σ+1/2 and a4Σ+3/2 – X2Σ+1/2 transitions were relatively strong. The transition properties reported herein were compared with the experimental results. The variation of the radiative–lifetime distribution with the rotational angular quantum number was evaluated for A2Π1/2, A2Π3/2, B2Σ+1/2, C2Π1/2, and C2Π3/2 states. It is expected that the transition properties calculated in this study can provide some useful guidelines for future experimental and theoretical investigations.

Weak kaon production off the nucleon and Watson's theorem

We have improved the tree-level model of Ref. [1] for weak production of kaons off nucleons by partially restoring unitarity. This is achieved by imposing Watson's theorem to the dominant vector and axial-vector contributions in appropriate angular momentum and isospin quantum number sectors. The observable consequences of this procedure are investigated.

Bound State Solution Schrödinger Equation for Extended Cornell Potential at Finite Temperature

In this paper, we study the finite temperature-dependent Schrödinger equation by using the Nikiforov-Uvarov method. We consider the sum of the Cornell, inverse quadratic, and harmonic-type potentials as the potential part of the radial Schrödinger equation. Analytical expressions for the energy eigenvalues and the radial wave function are presented. Application of the results for the heavy quarkonia and B c meson masses are in good agreement with the current experimental data except for zero angular momentum quantum numbers. Numerical results for the temperature dependence indicates a different behaviour for different quantum numbers. Temperature-dependent results are in agreement with some QCD sum rule results from the ground states.

Variational Method for Hydrogen Atom Embedded in Non-ideal Classical Plasmas

The energies, radial moments, and radial distributions of different bound states [ $$n=1-6$$ , $$l=0-5$$ ; n and l being the principal and orbital angular momentum quantum numbers, respectively] of hydrogen atom embedded in non-ideal plasma background have been estimated under the framework of variational method. In order to study the effect of non-ideality, the same structural properties are also determined under the ideal classical plasma environment.

Maxwell quasinormal modes on a global monopole Schwarzschild-anti-de Sitter black hole with Robin boundary conditions

We generalize our previous studies on the Maxwell quasinormal modes around Schwarzschild-anti-de-Sitter black holes with Robin type vanishing energy flux boundary conditions, by adding a global monopole on the background. We first formulate the Maxwell equations both in the Regge–Wheeler–Zerilli and in the Teukolsky formalisms and derive, based on the vanishing energy flux principle, two boundary conditions in each formalism. The Maxwell equations are then solved analytically in pure anti-de Sitter spacetimes with a global monopole, and two different normal modes are obtained due to the existence of the monopole parameter. In the small black hole and low frequency approximations, the Maxwell quasinormal modes are solved perturbatively on top of normal modes by using an asymptotic matching method, while beyond the aforementioned approximation, the Maxwell quasinormal modes are obtained numerically. We analyze the Maxwell quasinormal spectrum by varying the angular momentum quantum number ell , the overtone number N, and in particular, the monopole parameter 8pi eta ^2. We show explicitly, through calculating quasinormal frequencies with both boundary conditions, that the global monopole produces the repulsive force.

A crossed focused vortex beam with application to cold molecules* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11834003 and 91836103).

We report the generation of a crossed, focused, optical vortex beam by using a pair of hybrid holograms, which combine the vortex phase and lens phase onto a spatial light modulator. We study the intensity distributions of the vortex beam in free propagation space, and the relationship of its dark spot size with the incident Gaussian beam’s waist, the lens’s focal length, and its orbital angular momentum. Our results show that the crossed, focused, vortex beam’s dark spot size can be as small as 16.3 μm and adjustable by the quantum number of the orbital angular momentum, and can be used to increase the density of trapped molecules. Furthermore, we calculate the optical potential of the blue-detuned, crossed vortex beam for MgF molecules. It is applicable to cool and trap neutral molecules by intensity-gradient-induced Sisyphus cooling, as the intensity gradient of such vortex beam is extremely high near the focal point.

Structural properties of Na atom under impenetrable spatial confinement

The structural properties and radial distributions of the valence electron in different excited levels of Na atom (n = 3–5, l = 0–4; n and l being the principal and orbital angular momentum quantum numbers, respectively) under impenetrable spherical confinement have been studied, where the interaction between the frozen core and the valence electron is mimicked by a model potential available in the literature. The effect of the core on the valence electron has been investigated by estimating the structural properties of Na10+ion under similar confinement. Scaled radial densities at the nucleus and related ratios are presented for a few excited states of the valence electron of Na atom and the corresponding analytic results have been tested numerically.

Understanding the structure of complex multidimensional wave functions. A case study of excited vibrational states of ammonia

Generalization of an earlier reduced-density-matrix-based vibrational assignment algorithm is given, applicable for systems exhibiting both large-amplitude motions, including tunneling, and degenerate vibrational modes. The algorithm developed is used to study the structure of the excited vibrational wave functions of the ammonia molecule, 14NH3. Characterization of the complex dynamics of systems with several degenerate vibrations requires reconsidering the traditional degenerate-mode description given by vibrational angular momentum quantum numbers and switching to a symmetry-based approach that directly predicts state degeneracy and uncovers relations between degenerate modes. Out of the 600 distinct vibrational eigenstates of ammonia obtained by a full-dimensional variational computation, the developed methodology allows for the assignment of about 500 with meaningful labels. This study confirms that vibrationally excited states truly have modal character recognizable up to very high energies even for the non-trivial case of ammonia, a molecule which exhibits a tunneling motion and has two two-dimensional normal modes. The modal characteristics of the excited states and the interplay of the vibrational modes can be easily visualized by the reduced-density matrices, giving an insight into the complex modal behavior directed by symmetry.

The Magnetic Hooke-Newton Transmutation in Momentum Space

The magnetic Hooke-Newton transmutation, which emerges from the transformation design of the quadratic conformal mapping for the system of charged particles moving in the uniform magnetic field, is investigated in the momentum space. It is shown that there are two ways to turn the linear interaction force of the system into the inverse square interaction. The first one, which involves simply applying the mapping to the system, has the spectrum with the Landau levels of even angular momentum quantum number. The second one considers the geometric structure of the mapping as an effective potential which leads us to the transmuted Coulomb system with the novel quantum spectrum. The wave functions of momentum for the bound and scattering states of the transmutation system are given. It is also shown that the effective potential due to the geometric structure can be generalized to a general 2D surface, and the Schrödinger equation of a particle moving on the surface while under the action of the potential can be solved by the form-invariant Schrödinger equation of the free particle. The solution of a particle moving on the hyperbolic surface under the potential is given with the conclusion. The presentation manifests the transformation design of the quantum state, depending mainly on the geometric structure of the representation space, not on the action of the specific potential field. This characteristic makes it possible for us to use the geometric structure of different representation spaces to explore some novel behaviors of quantum particles.

Analytical Solution of the Klein–Fock–Gordon Equation for the Rosen–Morse Potential

With the help of a successful scheme for overcoming difficulties arising for l ≠ 0 in the centrifugal part of the Rosen–Morse potential with bound states, a solution of the modified Klein–Fock–Gordon equation is found. The energy eigenvalues and the corresponding radial wave functions are found for an arbitrary value of the orbital quantum number ( l ≠ 0 ).

Strong enhancement of Penning ionisation in cold Rydberg gases II: Tom and Jerry pairs for alkali-metal atoms

Penning ionisation (PI) processes involving pairs of Rydberg alkali-metal atoms, excited to different quantum states and experiencing dipole–dipole interactions, have a wide range of important properties in atomic physics. Within the framework of the semiclassical approximation, we have used both numerical and analytical approaches to examine the Penning autoionisation width dependence on the state quantum numbers in a quasi-molecule formed by the interacting partner atoms. We described the characteristics of optimal quantum numbers that lead to enhanced PI widths for the interacting Rydberg atom pairs of all alkali-metal atoms. The excited states of atoms in these pairs are asymmetric, resulting in a large atomic shell size difference: inspired by Efimov et al (2016 J. Phys. B: At. Mol. Opt. Phys. 49 125302), we call such a pair ‘Tom’ and ‘Jerry’ (for ‘big’ and ‘small’). Compared to symmetric pairs, the optimal asymmetric pairs display a significant (by several orders of magnitude) increase in the PI rate. This property makes PI a relevant source for producing charged particles in cold Rydberg systems that spontaneously evolve into cold plasma. Contrary to hydrogen atoms examined in (Efimov et al 2016 J. Phys. B: At. Mol. Opt. Phys. 49 125302), the difference in quantum defects in alkali-metal atoms results in a strong Penning width dependence on the orbital quantum numbers l of the quasi-molecule. In particular, alkali-metal atoms exhibit two PI channels associated with bound–bound optical transitions showing Δl = ±1—individual and closely spaced (doublet-like) configurations of optimal pairs. Furthermore, we demonstrate that the presence of Förster resonances can lead to a notable (up to five times) increase in the PI efficiency.

Classical calculation of radiative decay rates of hydrogenic Stark states

The Kepler–Coulomb problem is solved in parabolic coordinates and the Larmor radiation problem is analyzed to complement a previous study performed for the usual representation in spherical polar coordinates. The problem is then extended to include a weak electric field and a solution in terms of action-angle variables is provided. A comparison with quantum spontaneous decay rates shows that for azimuthal quantum number m = 0 states only transitions to nearby n −Δn principal quantum number states are described properly by the Wentzel–Kramers–Brillouin (WKB) quantized classical motions, but that for m > 0 reasonable results emerge for many values of Δn. A simple approximate expression for the lifetime of m ≠ 0 states emerges from the semi-classical analysis.

Deformation Potentials: Towards a Systematic Way beyond the Atomic Fragment Approach in Orbital-Free Density Functional Theory.

This work presents a method to move beyond the recently introduced atomic fragment approximation. Like the bare atomic fragment approach, the new method is an ab initio, parameter-free, orbital-free implementation of density functional theory based on the bifunctional formalism that treats the potential and the electron density as two separate variables, and provides access to the Kohn–Sham Pauli kinetic energy for an appropriately chosen Pauli potential. In the present ansatz, the molecular Pauli potential is approximated by the sum of the bare atomic fragment approach, and a so-called deformation potential that takes the interaction between the atoms into account. It is shown that this model can reproduce the bond-length contraction due to multiple bonding within the list of second-row homonuclear dimers. The present model only relies on the electron densities of the participating atoms, which themselves are represented by a simple monopole expansion. Thus, the bond-length contraction can be rationalized without referring to the angular quantum numbers of the participating atoms.

Direct time delay computation applied to the O + O2 exchange reaction at low energy: Lifetime spectrum of O3 * species.

We report full quantum dynamical calculations for lifetimes of scattering resonances, among which are true metastable states, of the intermediate heavy ozone complex 50O3 * of the 18O + 16O16O reaction, for any value of the total angular momentum quantum number J. We show that computations for nonzero values of J are mandatory in order to properly analyze resonances and time delays, with a view to establish a somewhat comprehensive eigenlife spectrum of the complex O3 *. Calculations have been performed in a given low to moderate energy range, including the interval between zero-point energies (ZPEs) of reagents and product species. Quasi-bound states tend to be more numerous, and eigenlifetimes themselves are seen to increase with J, reaching unusually large values for J = 30. A very dense forest of O3 * species is pictured already for J greater than 20, especially at the highest energies considered, leading to a quasi-continuum of metastable states. On the contrary, they appear as rather sparse and isolated at J = 0 and lower energies, including the domain between 18O16O and 16O16O ZPEs, embedded among many overlapping resonances that turn out to be not long-lived enough to be associated with genuine metastable states.

A program for simplifying summation of Wigner 3[formula omitted]-symbols

The program presented in this paper aims to simplify formulas involving products of a large amount of Wigner 3j-symbols summed over various magnetic quantum numbers. The algorithm used in our program is based on the graphical techniques originally developed by Yutsis, Levinson, and Vanagas, and later revised by some others. The output of our program is expressed as a weighted sum of products of 3j-, 6j- and 9j-symbols, with appropriate phase factors embedded. Compared with the previous literature, the present program is more flexible in a sense that it can deal with not only closed diagrams, such as the angular momentum recoupling coefficients, but also open diagrams. Our program is particularly useful to high precision atomic and molecular structure calculations using correlated electron–electron coordinates. Program summaryProgram Title: Reduce3jCPC Library link to program files:https://doi.org/10.17632/9zmx4b5sdd.1Licensing provisions: BSD 3-clauseProgramming language: Python 3.6.5Nature of problem: In atomic and molecular structure calculations, one often encounters expressions containing products of a large amount of Wigner 3j-symbols summed over various magnetic quantum numbers. It is desirable to simplify these expressions to make the final results compact, which is the goal of this paper.Solution method: The program uses the graphical techniques of quantum mechanical angular momentum theory, originally developed by Yutsis, Levinson, and Vanagas, and later modified by Brink and Satchler. The program is implemented in Python language, with its package “Networkx” extensively used. A friendly graphical input and output interface is developed. The input can be either a recoupling coefficient or an arbitrary expression containing products of the 3j-symbols summed over various magnetic quantum numbers. A strategically designed optimization method is also developed to help achieve our goal. Our program has significantly expanded the scope of usefulness of the existing programs so that it can now meet most of the demands from high precision atomic and molecular structure calculations.Additional comments including restrictions and unusual features: Reduce3j can effectively handle any open or closed graph, as long as the graph has more than one free line, and all the magnetic angular momentum quantum numbers from internal lines are summed over.

VPA: Computer program for the computation of the phase shift in atom–atom potential scattering using the Variable Phase Approach

A computer code for the computation of the phase shift in atom–atom and electron–atom potential scattering is presented. The phase shift is the central quantity required for the calculation of all types of scattering cross sections. The program uses the Variable Phase Approach (VPA). This is the only exact method for the direct calculation of the scattering phase shift. All other methods are based on examining the large distance behavior of the exact solution of the Schrödinger equation. Such methods yield the phase shift only modulo π. The absolute value of the phase shift and its variation with scattering energy is, however, needed for a full understanding of the scattering process, such as for instance in the study of shape resonances and Glory oscillations. The VPA has been sparingly used owing to the instability of the underlying equations and the consequent difficulty of writing computer code to solve them. We present a computer code for the efficient implementation of the VPA method for atom–atom scattering problems over a wide range of scattering energies. The code works for potentials which are singular and for those that are non-singular at the origin. An example of the implementation of the code is given for both an interaction potential with an attractive well and for a purely repulsive potential. Program summaryProgram title: VPA:Variable Phase ApproachCPC Library link to program files: https://dx.doi.org/10.17632/zh248d2362.1Code Ocean capsule: https://dx.doi.org/10.24433/CO.8146850.v1Licensing provisions: GNU General Public License 3 (GPL)Programming language: Fortran 90Nature of problem: To compute the quantum mechanical phase shift in an atom–atom or an electron–atom potential scattering problem. The interaction potential, dependent on a single radial coordinate, is defined by the user.Solution method: The Variable Phase Approach is used. This is the only available method for the direct calculation of the phase shift. The initial internuclear separation for the integration is chosen with the help of the analytical solution for a hard sphere core for all orbital angular momentum quantum numbers L of interest. The first-order nonlinear equation is integrated numerically with a modified LSODA19,20 program. The analytic formula for calculation of the Jacobian of the equation is also provided.Additional comments including restrictions and unusual features: Computer codes for the atom–atom interaction potential, POTENT1, and for its derivative with respect to the inter nuclear separation, DPOTENT1, must be supplied by a user. The calculation of the spherical Bessel and Neumann functions required is performed explicitly for large l angular momentum quantum numbers where the standard upward iterative generation of the functions becomes unstable.

Fisher Information of Free-Electron Landau States.

An electron in a constant magnetic field has energy levels, known as the Landau levels. One can obtain the corresponding radial wavefunction of free-electron Landau states in cylindrical polar coordinates. However, this system has not been explored so far in terms of an information-theoretical viewpoint. Here, we focus on Fisher information associated with these Landau states specified by the two quantum numbers. Fisher information provides a useful measure of the electronic structure in quantum systems, such as hydrogen-like atoms and under some potentials. By numerically evaluating the generalized Laguerre polynomials in the radial densities, we report that Fisher information increases linearly with the principal quantum number that specifies energy levels, but decreases monotonically with the azimuthal quantum number m. We also present relative Fisher information of the Landau states against the reference density with , which is proportional to the principal quantum number. We compare it with the case when the lowest Landau level state is set as the reference.

Structural properties of Li atom under quantum and classical plasmas: A composite variational framework

AbstractStructural properties of 1s2nl (2L) [n = 2–5, l = 0–4; where, n and l are the principal quantum number and orbital angular momentum quantum number, respectively] states of Li atom embedded in classical weakly coupled plasma (WCP) and dense quantum plasma (DQP) have been discussed. The Debye‐Hckel potential or the screened‐Coulomb potential (SCP) and exponential‐cosine‐screened Coulomb potential (ECSCP) have been used to mimic the WCP and DQP, respectively. Li atom has been treated as a composite system with a frozen core Li+ ion and a chemically active valence electron. The Rayleigh‐Ritz variational method with Hylleraas‐type basis set has been used to estimate the energy eigenvalue of 1s2 (1S) state of Li+ ion core and a pure exponential basis has been considered to compute the energy of nl (2L) states of the valence electron of Li atom. The influence of ECSCP and SCP on the radial probability distribution of the valence electron of the Li atom has also been studied.

Resonance ionisation spectrometry measurement of atomic calcium Rydberg isotope shifts

Rydberg levels in atomic calcium are of interest due to their use in highly isotope selective multiple-step resonance ionisation spectrometry, and also for nuclear physics investigations, but gaps remain in the data for higher lying levels in calcium isotopes other than . We report novel isotope shift data on four isotopes of calcium (, , , ), relative to the majority abundance isotope , in six Rydberg transitions for each P (n = 25–30) and F (n = 23–28) angular momentum quantum number designated series.