Coulomb Oscillator Research Articles

Inverse localization and global approximation for some Schrödinger operators on hyperbolic spaces

We consider the question of whether the high-energy eigenfunctions of certain Schrödinger operators on the d-dimensional hyperbolic space of constant curvature −κ2 are flexible enough to approximate an arbitrary solution of the Helmholtz equation Δh + h = 0 on Rd, over the natural length scale O(λ−1/2) determined by the eigenvalue λ ≫ 1. This problem is motivated by the fact that, by the asymptotics of the local Weyl law, approximate Laplace eigenfunctions do have this approximation property on any compact Riemannian manifold. In this paper we are specifically interested in the Coulomb and harmonic oscillator operators on the hyperbolic spaces Hd(κ). As the dimension of the space of bound states of these operators tends to infinity as κ ↘ 0, one can hope to approximate solutions to the Helmholtz equation by eigenfunctions for some κ > 0 that is not fixed a priori. Our main result shows that this is indeed the case, under suitable hypotheses. We also prove a global approximation theorem with decay for the Helmholtz equation on manifolds that are isometric to the hyperbolic space outside a compact set, and consider an application to the study of the heat equation on Hd(κ). Although global approximation and inverse approximation results are heuristically related in that both theorems explore flexibility properties of solutions to elliptic equations on hyperbolic spaces, we will see that the underlying ideas behind these theorems are very different.

The total absorption coefficient and refractive index change of a spherical quantum dot in the confinement radial potential

We have systematically investigated the linear, third-order nonlinear, total absorption coefficients and refractive index change of the spherical quantum dot in the confinement potential family V(r)=Ar2−Br+Crκ by solving the radial Schrödinger equation with the Numerov method in case of the effective mass approximation for the allowed transition of the ground state (1s) to first excited state (2p). We examine the dependencies of the total absorption coefficient and refractive index change on the confinement potential parameters for varying incident photon energy and relaxation times. Our results demonstrate that the absorption coefficients and the change in refractive index strongly depend on the confining potential parameters non-linearly representing various intersubband transitions of quantum dots. We find that increasing the parameter B in the Coulomb plus harmonic oscillator potential or decreasing the parameter A in the Kratzer plus harmonic oscillator potential, the total absorption coefficient and refractive index change shift to higher photon energies.

A family of analytically solvable Schrödinger equations related by Levi-Civita transformation

Some two-dimensional problems in non-relativistic quantum mechanics can connect to each other by certain spatial transformations such as Levi-Civita transformation. This property allows forming a series of two-dimensional problems into an interrelated family. Starting from two related problems namely Coulomb plus harmonic oscillator and sextic double-well anharmonic oscillator potentials, such family is constructed via repeatedly applying Levi-Civita transformations. Obviously, this family contains various of exactly analytically solvable problems. The quasi-exact solution for each unknown member of this family is also obtained and systematically investigated.

On the motion of non-relativistic particle in doubled ring-shaped Coulomb oscillator potential within cosmic string framework

The spacetime metric of cosmic string has been used to get the new form of Laplacian in Schrodinger equation for doubled ring-shaped Coulomb oscillator (DRSCO) potential. We use separation variable method to get three of one dimensional Schrodinger equations. Radial form is the main idea of this problem, so we just point out the solution of one dimensional Schrodinger equation for the radial part. Supersymmetric quantum mechanic method is used to get the energy spectrum and the wave function and we have used the approximation in the calculation of superpartner potential to get the same variable function as sextic-like potential in the radial form.

Exact Solutions of Schrodinger Equation in Cylindrical Coordinates for Double Ring-Shaped Coulomb Oscillator Potential Using SUSY QM Method

Schrodinger equation for a doubled ring-shaped coulomb oscillator potential is investigated using supersymmetric quantum mechanics approach. The three dimensional Schrodinger equation for a doubled ring-shaped coulomb oscillator potential in cylindrical coordinates is separated into three parts that contains one dimensional Schrodinger type equation which are solved using supersymmetric operator and the idea of shape invariant potential. The energy spectrum and the total wave functions are obtained.

Electron propagator with vector and scalar energy-dependent potentials in (2+1)-dimensional space–time

We have studied the effect of energy-dependent potentials for the relativistic spinning particle using the formalism for supersymmetric path integrals. That leaves behind a new normalization of wave function, which is examined via the Dirac equation and can be confirmed by Feynman’s path integral method. Based on two important examples, Coulomb and Harmonic oscillator potentials, we find that the frequency and the Coulomb’s constant are dependent on spectral parameters. The propagator is calculated and the energy eigenvalues with their corresponding eigenfunctions are deduced.

Quantum mechanics on phase space: The hydrogen atom and its Wigner functions

Symplectic quantum mechanics (SQM) considers a non-commutative algebra of functions on a phase space Γ and an associated Hilbert space HΓ, to construct a unitary representation for the Galilei group. From this unitary representation the Schrödinger equation is rewritten in phase space variables and the Wigner function can be derived without the use of the Liouville–von Neumann equation. In this article the Coulomb potential in three dimensions (3D) is resolved completely by using the phase space Schrödinger equation. The Kustaanheimo–Stiefel(KS) transformation is applied and the Coulomb and harmonic oscillator potentials are connected. In this context we determine the energy levels, the amplitude of probability in phase space and correspondent Wigner quasi-distribution functions of the 3D-hydrogen atom described by Schrödinger equation in phase space.

Superintegrable and shape invariant systems with position dependent mass

Second order integrals of motion for 3d quantum mechanical systems with position dependent masses (PDM) are classified. Namely, all PDM systems are specified which, in addition to their rotation invariance, admit at least 1 second order integral of motion. All such systems appear to be also shape invariant and exactly solvable. Moreover, some of them possess the property of double shape invariance and can be solved using two different superpotentials. Among them there are systems with double shape invariance which present nice bridges between the Coulomb and isotropic oscillator systems. A simple algorithm for calculating the discrete spectrum and the corresponding state vectors for the considered PDM systems is presented and applied to solve five of the found systems.

Dynamical Symmetries of Two-Dimensional Dirac Equation with Screened Coulomb and Isotropic Harmonic Oscillator Potentials

In this paper, we study symmetrical properties of two-dimensional (2D) screened Dirac Hydrogen atom and isotropic harmonic oscillator with scalar and vector potentials of equal magnitude (SVPEM). We find that it is possible for both cases to preserve so(3) and su(2) dynamical symmetries provided certain conditions are satisfied. Interestingly, the conditions for preserving these dynamical symmetries are exactly the same as non-relativistic screened Hydrogen atom and screened isotropic oscillator preserving their dynamical symmetries. Some intuitive explanations are proposed.

INFLUENCE OF EXTERNAL FIELDS ON THE KILLINGBECK POTENTIAL: QUASI EXACT SOLUTION

The Killingbeck potential consists of oscillator potential plus Cornell potential, i.e. ar2+ br - c/r, that it has received a great deal of attention in particle physics. In this paper, we study the energy levels and wave function for arbitrary m-state in two-dimensional (2D) Schrödinger equation (SE) with a Killingbeck potential under the influence of strong external uniform magnetic and Aharonov–Bohm (AB) flux fields perpendicular to the plane where the interacting particles are confined. We use the wave function ansatz method to solve the radial problem of the Schrödinger equation with Killingbeck potential. We obtain the energy levels in the absence of external fields and also find the energy levels of the familiar Coulomb and harmonic oscillator potentials.

Relativistic effect of external magnetic and Aharonov-Bohm fields on the unequal scalar and vector Cornell model

The Cornellpotential consists of linear and Coulomb potentials, i.e. −a/r+br, and has attracted a great deal of attention in particle physics. In this article, we study the energy levels and the wave function for an arbitrary m-state in the two-dimensional (2D) Klein-Gordon (KG) equation with the unequal scalar-vector Cornell potentials under the influence of strong external uniform magnetic and Aharonov-Bohm (AB) flux fields perpendicular to the plane where the interacting particles are confined. We use the wave function ansatz method to solve the radial problem of the KG equation with the Cornell potential. We obtain the energy levels in the absence of external fields and also find the energy levels of the familiar Coulomb and harmonic oscillator potentials.

Hyperspherical three-body calculation for muonic atoms

Ground state energies of exotic three-body atomic systems consisting two muons and a positively charged nucleus like: 1H+μ−μ−, 4He2+μ−μ−, 3He2+μ−μ−, 7Li3+μ−μ−, 6Li3+μ−μ−, 9Be4+μ−μ−, 12C6+μ−μ−, 16O8+μ−μ−, 20Ne10+μ−μ−, 28Si14+μ−μ− and 40Ar18+μ−μ− have been calculated using hyperspherical harmonics expansion (HHE) method. Calculation of matrix elements of two body interactions involved in the HHE method for a three body system is greatly simplified by expanding the bra- and ket- vector states in the hyperspherical harmonics basis states appropriate for the partition corresponding to the interacting pair. This involves the Raynal-Revai coefficients (RRC), which are the transformation coefficients between the hyperspherical harmonics bases corresponding to the two partitions. Use of these coefficients found to be very useful for the numerical solution of three-body Schrodinger equation where the two-body potentials are other than Coulomb or harmonic oscillator type. However, in this work the interaction potentials involved are purely Coulomb. The calculated energies have been compared with (i) those obtained by straight forward manner; and (ii) with those found in the literature (wherever available). The calculated binding energies agree within the computational error.

Boundary contributions to the hypervirial theorem

It is shown that under certain boundary conditions the virial theorem has to be modified. We analyze the origin of the extra term and compute it in particular examples. The Coulomb and harmonic oscillator with point interaction have been studied in the light of this generalization of the virial theorem.

Spin-One DKP Equation in the Presence of Coulomb and Harmonic Oscillator Interactions in (1+3)-Dimension

In this work, we study Duffin-Kemmer-Petiau equation in the presence of coulomb and harmonic oscillator potentials in (1+3)-dimension for spin-one particles and we obtain energy eigenvalues and corresponding eigenfunctions.

Effect of the Velocity-Dependent Potentials on the Bound State Energy Eigenvalues

We investigate the effect of isotropic velocity-dependent potentials on the bound state energy eigenvalues for the first time for any quantum states of the Coulomb and harmonic oscillator potentials within the framework of the asymptotic iteration method. When the velocity-dependent term is selected as a constant parameter ρ0, we present that the energy eigenvalues can be obtained analytically for both Coulomb and harmonic oscillator potentials. However, when the velocity-dependent term is considered as a harmonic oscillator type ρ0r2, taking the velocity-dependent term as a perturbation, we present how to obtain the energy eigenvalues of the Coulomb and harmonic oscillator potentials for any n and ℓ quantum states by using perturbation expansion and numerical calculations in the asymptotic iteration method procedure.

SYMMETRY OF DIRAC EQUATION AND CORRESPONDING PHENOMENOLOGY

It has been suggested that the high symmetries in the Schrödinger equation with the Coulomb or harmonic oscillator potentials may remain in the corresponding relativistic Dirac equation. If the principle is correct, in the Dirac equation the potential should have a form as [Formula: see text] where V(r) is [Formula: see text] for hydrogen atom and κr2 for harmonic oscillator. However, in the case of hydrogen atom, by this combination the spin–orbit coupling term would not exist and it is inconsistent with the observational spectra of hydrogen atom, so that the symmetry of SO(4) must reduce into SU(2). The governing mechanisms QED and QCD which induce potential are vector-like theories, so at the leading order only vector potential exists. However, the higher-order effects may cause a scalar fraction. In this work, we show that for QED, the symmetry restoration is very small and some discussions on the symmetry breaking are made. At the end, we briefly discuss the QCD case and indicate that the situation for QCD is much more complicated and interesting.

Connection between Coulomb and harmonic oscillator potentials in relativistic quantum mechanics

The Levi-Civita transformation is applied in the two-dimensional (2D) Dirac and Klein–Gordon (KG) equations with equal external scalar and vector potentials. The Coulomb and harmonic oscillator problems are connected via the Levi-Civita transformation. These connections lead to an approach to solve the Coulomb problems using the results of the harmonic oscillator potential in the relativistic systems mentioned above.

ODPEVP: A program for computing eigenvalues and eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm–Liouville problem

article i nfo abstract A FORTRAN 77 program is presented for calculating with the given accuracy eigenvalues, eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm- Liouville problem with the parametric third type boundary conditions on the finite interval. The program calculates also potential matrix elements - integrals of the eigenfunctions multiplied by their first derivatives with respect to the parameter. Eigenvalues and matrix elements computed by the ODPEVP program can be used for solving the bound state and multi-channel scattering problems for

Large degeneracy of excited hadrons and quark models

The pattern of a large approximate degeneracy of the excited hadron spectra (larger than the chiral restoration degeneracy) is present in the recent experimental report of Bugg. Here we try to model this degeneracy with state of the art quark models. We review how the Coulomb Gauge chiral invariant and confining Bethe-Salpeter equation simplifies in the case of very excited quark-antiquark mesons, including angular or radial excitations, to a Salpeter equation with an ultrarelativistic kinetic energy with the spin-independent part of the potential. The resulting meson spectrum is solved, and the excited chiral restoration is recovered, for all mesons with $J>0$. Applying the ultrarelativistic simplification to a linear equal-time potential, linear Regge trajectories are obtained, for both angular and radial excitations. The spectrum is also compared with the semiclassical Bohr-Sommerfeld quantization relation. However, the excited angular and radial spectra do not coincide exactly. We then search, with the classical Bertrand theorem, for central potentials producing always classical closed orbits with the ultrarelativistic kinetic energy. We find that no such potential exists, and this implies that no exact larger degeneracy can be obtained in our equal-time framework, with a single principal quantum number comparable to the nonrelativistic Coulomb or harmonic oscillator potentials. Nevertheless we find it plausible that the large experimental approximate degeneracy will be modeled in the future by quark models beyond the present state of the art.

Polynomial Solution of Non-Central Potentials

We show that the exact energy eigenvalues and eigenfunctions of the Schrodinger equation for charged particles moving in certain class of non-central potentials can be easily calculated analytically in a simple and elegant manner by using Nikiforov and Uvarov (NU) method. We discuss the generalized Coulomb and harmonic oscillator systems. We study the Hartmann Coulomb and the ring-shaped and compound Coulomb plus Aharanov-Bohm potentials as special cases. The results are in exact agreement with other methods.