Angular Wave Functions Research Articles

Photoassociation of rotating ultra-long range Rydberg molecules

We discuss the rotational structure of ultra-long range Rydberg molecules and their angular nuclear wave function. Expressing the complete molecular wave function in a laboratory fixed frame, we derive the transition matrix elements for the photoassociation of free ground state atoms. The coupling strength depends on the angular momentum coupling in the Rydberg molecule, which differs for the various types of Rydberg molecules. This work explains the different steps to calculate the wave functions and the transition matrix elements in a way, that they can be transferred to other Rydberg molecules involving different atomic species or molecular coupling cases.

Wave equations in conformal gravity

We study the wave equation governing massless fields of all spins (s = 0, [Formula: see text], 1, [Formula: see text] and 2) in the most general spherical symmetric metric of conformal gravity. The equation is separable, the solution of the angular part is a spin-weighted spherical harmonic, and the radial wave function may be expressed in terms of solutions of the Heun equation which has four regular singular points. We also consider various special cases of the metric and find that the angular wave functions are the same for all cases, the actual shape of the metric functions affects only the radial wave function. It is interesting to note that each radial equation can be transformed into a known ordinary differential equation (i.e. Heun equation, or confluent Heun equation, or hypergeometric equation). The results show that there are analytic solutions for all the wave equations of massless spin fields in the spacetimes of conformal gravity. This is amazing because exact solutions are few and far between for other spacetimes.

Structure and decays of nuclear three-body systems: The Gamow coupled-channel method in Jacobi coordinates

${\bf Background:}$ Weakly bound and unbound nuclear states appearing around particle thresholds are prototypical open quantum systems. Theories of such states must take into account configuration mixing effects in the presence of strong coupling to the particle continuum space. ${\bf Purpose:}$ To describe structure and decays of three-body systems, we developed a Gamow coupled-channel (GCC) approach in Jacobi coordinates by employing the complex-momentum formalism. We benchmarked the new framework against the complex-energy Gamow Shell Model (GSM). ${\bf Methods:}$ The GCC formalism is expressed in Jacobi coordinates, so that the center-of-mass motion is automatically eliminated. To solve the coupled-channel equations, we use hyperspherical harmonics to describe the angular wave functions while the radial wave functions are expanded in the Berggren ensemble, which includes bound, scattering and Gamow states. ${\bf Results:}$ We show that the GCC method is both accurate and robust. Its results for energies, decay widths, and nucleon-nucleon angular correlations are in good agreement with the GSM results. ${\bf Conclusions:}$ We have demonstrated that a three-body GSM formalism explicitly constructed in cluster-orbital shell model coordinates provides similar results to a GCC framework expressed in Jacobi coordinates, provided that a large configuration space is employed. Our calculations for $A=6$ systems and $^{26}$O show that nucleon-nucleon angular correlations are sensitive to the valence-neutron interaction. The new GCC technique has many attractive features when applied to bound and unbound states of three-body systems: it is precise, efficient, and can be extended by introducing a microscopic model of the core.

Analytical solution of energy eigen value, eigen function and angular wave function of Dirac equation with Rosen Morse plus Rosen Morse potential in terms of Romanovski polynomials for exact spin symmetry

<p class="Abstract">The energy eigenvalues and eigenfunctions of Dirac equation for Rosen Morse plus Rosen Morse potential are investigated numerically in terms of finite Romanovsky Polynomial. The bound state energy eigenvalues are given in a closed form and corresponding eigenfunctions are obtained in terms of Romanovski polynomials. The energi eigen value is solved by numerical method with Matlab 2011.</p>

Algebraic Construction of Spherical Harmonics

The angular wave functions for a hydrogen atom are well known to be spherical harmonics, and are obtained as the solutions of a partial differential equation. However, the differential operator is given by the Casimir operator of the $SU(2)$ algebra and its eigenvalue $l(l+1) \hbar^2$, where $l$ is non-negative integer, is easily obtained by an algebraic method. Therefore the shape of the wave function may also be obtained by extending the algebraic method. In this paper, we describe the method and show that wave functions with different quantum numbers are connected by a rotational group in the cases of $l=0$, 1 and 2.

Energy analysis of four dimensional extended hyperbolic Scarf I plus three dimensional separable trigonometric noncentral potentials using SUSY QM approach

The non-relativistic energies and wave functions of extended hyperbolic Scarf I plus separable non-central shape invariant potential in four dimensions are investigated using Supersymmetric Quantum Mechanics (SUSY QM) Approach. The three dimensional separable non-central shape invariant angular potential consists of trigonometric Scarf II, Manning Rosen and Poschl-Teller potentials. The four dimensional Schrodinger equation with separable shape invariant non-central potential is reduced into four one dimensional Schrodinger equations through variable separation method. By using SUSY QM, the non-relativistic energies and radial wave functions are obtained from radial Schrodinger equation, the orbital quantum numbers and angular wave functions are obtained from angular Schrodinger equations. The extended potential means there is perturbation terms in potential and cause the decrease in energy spectra of Scarf I potential.

Computation of scattering of a plane wave from multiple prolate spheroids using the collocation multipole method.

The collocation multipole method is presented to solve three-dimensional acoustic scattering problems with multiple prolate spheroids subjected to a plane sound wave. To satisfy the three-dimensional Helmholtz equation in prolate spheroidal coordinates and the radiation condition at infinity, the scattered field is formulated in terms of radial and angular prolate spheroidal wave functions. Instead of using the complicated addition theorem of prolate spheroidal wave functions, the multipole method, the directional derivative, and the collocation technique are combined to solve multiple scattering problems semi-analytically. For the sound-hard or Neumann conditions, the normal derivative of the acoustic pressure with respect to a non-local prolate spheroidal coordinate system is developed without any truncation error for multiply connected domain problems. By truncating the higher order terms of the multipole expansion, a finite linear algebraic system is obtained and the scattered field is determined from the given incident acoustic wave. Once the total field is calculated as the sum of the incident field and the scattered field, the near field acoustic pressure and the far field scattering pattern are determined. Numerical experiments for convergence are performed to provide the guide lines for the proposed method. The proposed results of acoustic scattering by one, two, and three prolate spheroids are compared with those of an available analytical method and the boundary element method to validate the proposed method. Finally, the effects of the eccentricity of a prolate spheroidal scatterer, the separation between scatterers and the incident wave number on the near-field acoustic pressure and the far-field scattering pattern are investigated.

The relationship between possible hidden variables in time and space and the particle's angular momentum and wave function

The relationship between possible hidden variables in time and space and the particle's angular momentum and wave function

Relativistic Energy Analysis of Five-Dimensionalq-Deformed Radial Rosen-Morse Potential Combined withq-Deformed Trigonometric Scarf Noncentral Potential Using Asymptotic Iteration Method

We study the exact solution of Dirac equation in the hyperspherical coordinate under influence of separableq-deformed quantum potentials. Theq-deformed hyperbolic Rosen-Morse potential is perturbed byq-deformed noncentral trigonometric Scarf potentials, where all of them can be solved by using Asymptotic Iteration Method (AIM). This work is limited to spin symmetry case. The relativistic energy equation and orbital quantum number equationlD-1have been obtained using Asymptotic Iteration Method. The upper radial wave function equations and angular wave function equations are also obtained by using this method. The relativistic energy levels are numerically calculated using Matlab, and the increase of radial quantum numberncauses the increase of bound state relativistic energy level in both dimensionsD=5andD=3. The bound state relativistic energy level decreases with increasing of both deformation parameterqand orbital quantum numbernl.

Schmidt decomposition for non-collinear biphoton angular wave functions**To Vladimir and Margarita Man’ko with great respect and best wishes.

Schmidt modes of non-collinear biphoton angular wave functions are found analytically. The experimentally realizable procedure for their separation is described. Parameters of the Schmidt decomposition are used to evaluate the degree of the biphoton's angular entanglement.

The Origin and Mathematical Characteristics of the Super-Universal Associated-Legendre Polynomials

We study the mathematical characteristics of the super-universal associated-Legendre polynomials arising from a kind of double ring-shaped potentials and obtain their polar angular wave functions with certain parity. We find that there exists the even or odd parity for the polar angular wave functions when the parameter η is taken to be positive integer, while there exist both even and odd parities when η is taken as positive non-integer real values. The relations among the super-universal associated-Legendre polynomials, the hypergeometric polynomials, and the Jacobi polynomials are also established.

Exact Solution of Klein-Gordon with the Pöschl-Teller Double-Ring-Shaped Coulomb Potential

Analytical solution of the Klein Gordon equation under the equal scalar and vector Poschl Teller double-ring-shaped Coulomb potentials is obtained. We have used the Nikiforov Uvarov method in our calculations. The radial wave function in terms of the Laguerre polynomials is presented and the angular wave functions are expressed in terms of the Jacobi polynomials. We have also considered some special cases of the Poschl Teller double-ring-shaped Coulomb potential and represented the energy eigenvalues and the corresponding wave functions.

Eigenanalysis for a confocal prolate spheroidal resonator using the null-field BIEM in conjunction with degenerate kernels

In this paper, we employ the null-field boundary integral equation method (BIEM) in conjunction with degenerate kernels to solve eigenproblems of the prolate spheroidal resonators. To detect the spurious eigenvalues and the corresponding occurring mechanisms which are common issues while utilizing the boundary element method or the BIEM, we use angular prolate spheroidal wave functions and triangular functions to expand boundary densities. In this way, the boundary integral of a prolate spheroidal surface is exactly determined, and eigenequations are analytically derived. It is revealed that the spurious eigenvalues depend on the integral, representations and the shape of the inner boundary. Furthermore, it is interesting to find that some roots of the confocal prolate spheroidal resonator are double roots no matter that they are true or spurious eigenvalues. Illustrative examples include confocal prolate spheroidal resonators of various boundary conditions. To validate these findings and accuracy of the present approach, the commercial finite-element software ABAQUS is also applied to perform acoustic analyses. Good agreement is obtained between the acoustic results obtained by the null-field BIEM and those provided by the commercial finite-element software ABAQUS.

Ground motions around a semi-circular valley partially filled with an inclined alluvial layer under SH-polarized excitation

A simplified mathematical model, composed of a semi-circular valley partially filled with an inclined alluvial layer under plane SH-wave incidence, is presented. To evaluate the site response theoretically, a rigorous series solution is derived via the region-matching technique. For angular wavefunctions constrained by an inclined free surface, the original form of Graf's addition formula is recast to arbitrarily shift the local coordinate system. The valley geometry, filling material, angle of incidence, and wave frequency are taken as significant parameters in exploring the site effect on ground motions. Also included are the frequency- and time-domain computations. Two canonical cases, the semi-circular vacant canyon and the fully filled semi-circular alluvial valley, with exact analytical solutions, and the partly horizontally filled case previously studied, are taken to be particular cases of the proposed general model. Steady-state results show that the peak amplitudes of motion may increase at low frequencies when the filling layer inclines to the illuminated region. At low-grazing incidence, the phenomenon of wave focusing becomes evident on the shadow side of the filling layer. Transient-state simulations elucidate how a sequence of surface waves travel on the topmost alluvium along opposite directions and interfere with multiple reflected waves within the filling layer.

Spin-one Duffin–Kemmer–Petiau equation in the presence of Manning–Rosen potential plus a ring-shaped-like potential

We present the solution of the Duffin–Kemmer–Petiau equation for Manning–Rosen potential plus a ring-shaped-like potential in (1+3)-dimensional space–time for spin-one particles within the framework of an exponential approximation for the centrifugal term. We have used the Nikiforov–Uvarov method in our calculations. The radial wavefunction and the angular wavefunctions are expressed in terms of Jacobi polynomials. We have also represented some numerical results for the Manning–Rosen potential plus a ring-shaped-like potential.

The quantum characteristics of a class of complicated double ring-shaped non-central potential

Firstly we outline how to obtain the normalized polar angular wave functions of the Schrödinger equation with a class of complicated double ring-shaped non-central potential by introducing the super-universal associated Legendre polynomials, and present the exact energy equation and normalized radial wave functions for the given central potentials, such as the harmonic oscillator and Coulomb potential. We then discuss in detail the quantum characteristics of the polar and radial wave functions and energy equation, and their reducing problems. These features include bound state relations between the harmonic oscillator and Coulomb potential quantum systems, and degeneracy of their energy levels. We also observe that the bound state relations, including the energy levels and wave functions for the harmonic oscillator and Coulomb potential systems, can be obtained from each other by mapping the system parameters. Special cases related to reduction questions are discussed in detail.

B‐spline one‐center method for molecular Hartree–Fock calculations

We introduce one‐center method in spherical coordinates to carry out Hartree–Fock calculations. Both the radial wave function and the angular wave function are expanded by B‐splines to deal with electron‐nucleus cusps, which results in the improved convergence for several typical closed‐shell diatomic molecules with moderate basis numbers. B‐splines could represent both the bound state and continuum state wave functions properly, and the present approach has been applied to investigate the ionization dynamics for H2 in the intense laser field adopting single‐active‐electron model. © 2013 Wiley Periodicals, Inc.

Analytic solutions of the double ring-shaped Coulomb potential in quantum mechanics

The exact solutions of the Schrödinger equation with the double ring-shaped Coulomb potential are presented, including the bound states, continuous states on the “κ/2π scale", and the calculation formula of the phase shifts. The polar angular wave functions are expressed by constructing the so-called super-universal associated Legendre polynomials. Some special cases are discussed in detail.

Systematic search of exactly solvable ring-shaped potential using the transformation method

We apply the extended transformation method, comprising a coordinate transformation supplemented by a functional transformation, to generate (regenerate) exactly solvable ring-shaped potentials from already known exactly solvable central potentials for physical quantum systems in a non-relativistic regime. The angular wave functions corresponding to the generated potentials are transformed from the radial wave functions of the original central potentials in a very straightforward way and they are also shown to be normalizable.

Exact solutions of the Schrödinger equation with double ring-shaped oscillator

Abstract We present the exact solutions of the Schrodinger equation with the double ring-shaped oscillator (DRSO) potential. By introducing a new variable x = cos θ and constructing super-universal associated Legendre polynomials we express the polar angular wave functions explicitly. We observe that the present DRSO has caused the symmetry breaking from the original spherical oscillator SU ( 3 ) ⊃ SO ( 3 ) ⊃ O ( 2 ) symmetries to the present O ( 2 ) symmetry due to the surrounded two ring-shaped inversed square potentials. Some special cases are also discussed.