Coulomb-like Potentials Research Articles

The properties of radially excited charmonia in the light front quark model

Investigating the properties of excited charmonia is important to clarify its internal structure. In this paper, we present the mass spectra (MS) and decay constants (DC) for charmonia up to 3S states calculated by means of the light-front quark model based on a variational approach. In particular, we consider the QCD-motivated effective Hamiltonian, which includes both confinement (linear and screened) and Coulomb-like potentials. Furthermore, since the existence of the nature of heavy quark symmetry, we treat hyperfine interactions perturbatively. We developed the harmonic oscillator expansion method to approximate the wave function (WF) for excited states. We found that the results of our theoretical calculations, using screened potentials rather than linear ones, are in good agreement with experimental data. By looking at the mass and decay constant result, we found that our result on the ψ(3S) state matched the properties of the ψ(4040) resonance.

Eliminating Artificial Boundary Conditions in Time-Dependent Density Functional Theory Using Fourier Contour Deformation.

We present an efficient method for propagating the time-dependent Kohn-Sham equations in free space, based on the recently introduced Fourier contour deformation (FCD) approach. For potentials which are constant outside a bounded domain, FCD yields a high-order accurate numerical solution of the time-dependent Schrödinger equation directly in free space, without the need for artificial boundary conditions. Of the many existing artificial boundary condition schemes, FCD is most similar to an exact nonlocal transparent boundary condition, but it works directly on Cartesian grids in any dimension, and runs on top of the fast Fourier transform rather than fast algorithms for the application of nonlocal history integral operators. We adapt FCD to time-dependent density functional theory (TDDFT), and describe a simple algorithm to smoothly and automatically truncate long-range Coulomb-like potentials to a time-dependent constant outside of a bounded domain of interest, so that FCD can be used. This approach eliminates errors originating from the use of artificial boundary conditions, leaving only the error of the potential truncation, which is controlled and can be systematically reduced. The method enables accurate simulations of ultrastrong nonlinear electronic processes in molecular complexes in which the interference between bound and continuum states is of paramount importance. We demonstrate results for many-electron TDDFT calculations of absorption and strong field photoelectron spectra for one and two-dimensional models, and observe a significant reduction in the size of the computational domain required to achieve high quality results, as compared with the popular method of complex absorbing potentials.

Solutions of the Dirac equation with position-dependent mass for q-deformed Pöschl–Teller and Coulomb-like tensor potentials

We consider the Dirac equation with position-dependent mass for a q-deformed Pöschl–Teller potential plus a Coulomb-like tensor interaction in the limits of spin and pseudospin symmetries. Under the condition of spin symmetry, the bound state energy eigenvalue equation and the two radial components of the Dirac wave function in terms of the Jacobi polynomials are obtained approximately for an arbitrary shifted spin-orbit quantum number λκ = κ + H and for any value of the deformation parameter q ≥ 1. In the case of the pseudospin symmetry limit, there are no bound states. The Dirac equation describes a free physical system and the two components of the wave function are expressed in terms of the spherical Bessel functions.

Fourier Transform for Locally Integrable Functions with Rotational and Dilation Symmetry

The Fourier transform for slowly increasing functions is defined by the Parseval equation for tempered distributions. This definition was supplemented by a novel method of performing practical calculations by computing the Fourier transform for a suitably tempered function and then by integration by parts. The application of this method is illustrated both for the toy case, in which the function is integrable, so its Fourier transform can also be computed using the standard formula, and for the case of Coulomb-like potentials, which are only locally integrable functions. All of them have spherical symmetry, and two of them additionally have dilation symmetry. The proposed novel method does not violate these symmetries at any stage of the calculation.

Angular-radial integrability of Coulomb-like potentials in Dirac equations

We consider the Dirac equation, written in polar formalism, in the presence of general Coulomb-like potentials, that is, potentials arising from the time component of the vector potential and depending only on the radial coordinate, in order to study the conditions of integrability, given as some specific form for the solution: we find that the angular dependence can always be integrated, while the radial dependence is reduced to finding the solution of a Riccati equation so that it is always possible, at least in principle. We exhibit the known case of the Coulomb potential and one special generalization as examples to show the versatility of the method.

Spatial Non-locality in Confined Quantum Systems: A Liaison with Quantum Correlations

Using advanced stochastic methods (time-dependent quantum Monte Carlo, TDQMC) we explore the ground state of 1D and 2D artificial atoms with up to six bosons in harmonic trap where these interact by long-range and short-range Coulomb-like potentials (bosonic quantum dots). It is shown that the optimized value of the key variational parameter in TDQMC named nonlocal correlation length is close to the standard deviation of the Monte Carlo sample for one boson and it is slightly dependent on the range of the interaction potential. Also it is almost independent on the number of bosons for the 2D system thus confirming that the spatial quantum non-locality experienced by each particle is close to the spatial uncertainty exhibited by the rest of the particles. The intimate connection between spatial non-locality and quantum correlations is clearly evidenced.

A parabolic quasi-Sturmian approach to quantum scattering by a Coulomb-like potential

A computational method in parabolic coordinates is proposed to treat the scattering of a charged particle from both spherically and axially symmetric Coulomb-like potentials. Specifically, the long-range part of the Hamiltonian is represented in parabolic quasi-Sturmian basis functions, while the short-range part is approximated by a Sturmian $$L^2$$ -basis-set truncated expansion. We establish an integral representation of the Coulomb Green’s function in parabolic coordinates from which we derive a convenient closed form for its matrix elements in the chosen $$L^2$$ basis set. From the Green’s function, we build quasi-Sturmian functions that are also given in closed form. Taking advantage of their adequate built-in Coulomb asymptotic behavior, scattering amplitudes are extracted as simple analytical sums that can be easily computed. The scheme, based on the proposed quasi-Sturmian approach, proves to be numerically efficient and robust as illustrated with converged results for three different scattering potentials, one of spherical and two of axial symmetry.

Periodic solutions for the Lorentz force equation with singular potentials

We provide sufficient conditions for the existence of periodic solutions of the Lorentz force equation, which models the motion of a charged particle under the action of an electromagnetic fields. The basic assumptions cover relevant models with singularities like Coulomb-like electric potentials or the magnetic dipole.

Existence of propagators for time-dependent Coulomb-like potentials

We prove existence of propagators for a time-dependent Schrodinger equation with a new class of softened Coulomb potentials, which we allow to be time dependent, in the context of time-dependent density functional theory. We compute explicitly the Fourier transform of these new potentials, and provide an alternative proof for the Fourier transform of the Coulomb potential using distribution theory. Finally, we show the new potentials are dilatation analytic, and discuss various properties and applications related to the essential spectrum of the Hamiltonian associated to these potentials.

1D Schrödinger operators with Coulomb-like potentials

We study the convergence of 1D Schrödinger operators Hε with the potentials which are regularizations of a class of pseudopotentials having, in particular, the form αδ′(x) + βδ(x) + γ/|x| or αδ′(x) + βδ(x) + γ/x. The limit behavior of Hε in the norm resolvent topology, as ε → 0, essentially depends on a way of regularization of the Coulomb potential and the existence of zero-energy resonances for δ′-like potential. All possible limits are described in terms of point interactions at the origin. As a consequence of the convergence results, different kinds of L∞(R)-approximations to the even and odd Coulomb potentials, both penetrable and impenetrable in the limit, are constructed.

Spin (Pseudospin) doublet in view of energy-dependent potential

Behavior of Dirac particles in the presence of scalar, vector, and tensor potentials respectively represented by energy-dependent Morse and Coulomb-like potentials is examined by working out the Dirac equation under the condition of spin (pseudospin) symmetry. The closed form of the energy eigenvalue equation and corresponding wave functions in terms of hypergeometric functions are acquired by making use of the asymptotic iteration method. We investigate the effect of energy-dependent potential on both bound states and energy splitting between members of spin (pseudospin) doublets.

Comparison of O- and C-models on pseudosphere

We present a mathematically rigorous quantum-mechanical treatment of a two-dimensional nonrelativistic quantum dual theories (with oscillator- and Coulomb-like potentials) on pseudosphere. We construct all self-adjoint Schrödinger operators for these theories and present solutions of the corresponding spectral problems. We show, that there is one-to-one correspondence between spectral points of the energy-coupling constant planes not only for discrete, but also for continuous spectra and corresponding eigenfunctions forming the complete orthonormalized systems.

Bound and scattering states of modified Yukawa potential under relativistic spin and pseudospin symmetries with three tensor interactions

Within the framework of the spin and pseudospin symmetry limit, we approximately present the analytical solutions of bound and scattering states of the Dirac equation with modified Yukawa potential including three-tensor interactions comprising Coulomb-like, Yukawa-like and Hulthen-type potentials. The energy equation, phase shifts and normalized constant are obtained in a closed form. We also calculated the numerical results of the energy eigenvalues in the presence of tensor interactions. We discuss in detail the effect of the tensor interactions and the degeneracies removing role in the energy splitting of the pseudospin and spin doublets. Finally, special cases of the potential are also discussed.

Hulthén and Coulomb-Like Potentials as a Tensor Interaction within the Relativistic Symmetries of the Manning-Rosen Potential

The bound-state solutions of the Dirac equation for the Manning-Rosen potential are presented approximately for arbitrary spin-orbit quantum numberκwith the Hulthén and Coulomb-like potentials as a tensor interaction. The generalized parametric Nikiforov-Uvarov (NU) method is used to obtain energy eigenvalues and corresponding two-component spinors of the two Dirac particles and these are obtained in the closed form by using the framework of the spin symmetry and p-spin symmetry concept. We have also shown that tensor interaction removes degeneracies between spin and p-spin doublets. Some numerical results are also given.

Off-shell Jost solutions for Coulomb and Coulomb-like interactions in all partial waves

By exploiting the theory of ordinary differential equations, with judicious use of boundary conditions, interacting Green's functions and their integral transforms together with certain properties of higher transcendental functions, useful analytical expressions for the off-shell Jost solutions for motion in Coulomb and Coulomb-nuclear potentials are derived in maximal reduced form through different approaches to the problem in the representation space. The exact analytical expressions for the off-shell Jost solutions for Coulomb and Coulomb-like potentials are believed to be useful for the description of the charged particle scattering/reaction processes.

Variational method using basis functions

The Gaussian, exponential and Laguerre basis functions are examined in a variational calculation of energies and wavefunctions. The Laguerre basis set is already orthonormal and complete, but the Gaussian and exponential basis sets are not orthonormal. We used the linear and Coulomb potentials to test these basis functions. Calculations are performed in both position and momentum space. We also present the results with relativistic kinematics in the momentum space calculation. The Gram–Schmidt procedure is used to orthonormalize the Gaussian and exponential basis sets before using them in the calculations. We show that in the case of a pure linear potential, the orthonormal basis constructed from the Gaussian functions performs much better than the exponential and Laguerre basis for the same number of orthogonal functions. For Coulomb-like potentials, the exponential basis performs better than the other two for the same number of basis functions. The advantage of using these simple basis functions is that for the potentials that we examined, one can approach the lower bound of the low-lying states with very few basis functions.

Exact Klein-Gordon equation with spatially dependent masses for unequal scalar-vector Coulomb-like potentials

We study the effect of spatially dependent mass functions over the solution of the Klein-Gordon equation in the (3 + 1 -dimensions for spinless bosonic particles where the mixed scalar-vector Coulomb-like field potentials and masses are directly proportional and inversely proportional to the distance from the force center. The exact bound-state energy eigenvalues and the corresponding wave functions of the Klein-Gordon equation for mixed scalar-vector and pure scalar Coulomb-like field potentials are obtained by means of the Nikiforov-Uvarov (NU) method. The energy spectrum is discussed for different scalar-vector potential mixing cases and also for the constant-mass case.

Effective Dynamics for Particles Coupled to a Quantized Scalar Field

We consider a system of N non-relativistic spinless quantum particles (“electrons”) interacting with a quantized scalar Bose field (whose excitations we call “photons”). We examine the case when the velocity v of the electrons is small with respect to the one of the photons, denoted by c (v/c = e ≪ 1). We show that dressed particle states exist (particles surrounded by “virtual photons”), which, up to terms of order (v/c)3, follow Hamiltonian dynamics. The effective N-particle Hamiltonian contains the kinetic energies of the particles and Coulomb-like pair potentials at order (v/c)0 and the velocity dependent Darwin interaction and a mass renormalization at order (v/c)2. Beyond that order the effective dynamics are expected to be dissipative. The main mathematical tool we use is adiabatic perturbation theory. However, in the present case there is no eigenvalue which is separated by a gap from the rest of the spectrum, but its role is taken by the bottom of the absolutely continuous spectrum, which is not an eigenvalue. Nevertheless we construct approximate dressed electron subspaces, which are adiabatically invariant for the dynamics up to order $${(v/c)\sqrt{\ln [(v/c)^{-1}]}}$$ . We also give an explicit expression for the non-adiabatic transitions corresponding to emission of free photons. For the radiated energy we obtain the quantum analogue of the Larmor formula of classical electrodynamics.

Simple model of scalar-vector interaction for the relativistic two-center problem

The ground-state wave function and the energy term of a relativistic electron moving in the field of two fixed centers, when interaction of this particle with centers is described by two Coulomb and two Coulomb-like scalar potentials are calculated analytically by the linear combination of atomic orbitals method. By means of perturbation theory the asymptotic expansions (at small and large internuclear distances $R$) of the eigenvalues $E(R)$ of the system for arbitrary states are obtained. The dependence of critical charge ${Z}_{\text{cr}}$ and critical distance ${R}_{\text{cr}}$ from the scalar coupling constant was investigated.

TWO-CENTER PROBLEM FOR THE DIRAC EQUATION WITH A COULOMB AND SCALAR POTENTIAL

The ground state wave function and the energy term of a relativistic electron moving in the field of two fixed centers, when interaction of this particle with centers is described by two Coulomb and two Coulomb-like scalar potentials are calculated analytically by the LCAO method. Dependence of electron binding energy from value of scalar coupling constant was investigated using obtained analytic results.