Three-dimensional Coulomb Potential Research Articles

Out-of-time-order correlators of a Skyrmion as a baryon in holographic QCD

As the out-of-time-order correlator (OTOC) is a measure of quantum chaos and an important observable in the context of AdS/CFT, we investigate the OTOC of a holographic Skyrmion which is described by an analytical quantum mechanical system from the D4/D8 model (as the holographic QCD). By employing the OTOC defined in quantum mechanics, we derive the formulas and demonstrate the numerical calculations of the OTOC explicitly which is also available for the general case with a central force field. Our numerical evaluation illustrates the behaviors of the OTOC with large Nc; however, the expected exponential growth of OTOC is not obtained. Besides, we also take a look at the classical limit of the OTOC and analyze the associated behaviors. At the end of this work, we additionally study the OTOC with three-dimensional Coulomb potential, as another example for the central force field, to support our analyses of the general properties of quantum OTOC. Published by the American Physical Society 2024

Electrostatic potential and magnetic moment of radially insulating Corbino disk

We solve analytically a three-dimensional Poisson equation for the potential produced by two coaxial metallic contacts at the edges of the Corbino disk. It is assumed that there is a finite tangential current, while the radial current is absent. This solution is compared with a solution of the two-dimensional problem for conducting Corbino disk. We calculate the magnetic dipole moment of Corbino disk and compare our results with the case of two-dimensional Coulomb distribution of the potential which is realized in Corbino disk in the presence of the radial current. We also discuss the application of torque magnetometry to study metal–insulator transition, in particular, in the quantum Hall regime. • Radially insulating Corbino disk. • Two-dimensional vs three-dimensional Coulomb potential. • Magnetic moment.

Note on Crystallization for Alternating Particle Chains

We investigate one-dimensional periodic chains of alternate type of particles interacting through mirror symmetric potentials. The optimality of the equidistant configuration at fixed density—also called crystallization—is shown in various settings. In particular, we prove the crystallization at any scale for neutral and non-neutral systems with inverse power laws interactions, including the three-dimensional Coulomb potential. We also show the minimality of the equidistant configuration at high density for systems involving inverse power laws and repulsion at the origin. Furthermore, we derive a necessary condition for crystallization at high density based on the positivity of the Fourier transform of the interaction potentials sum.

Atoms confined by very thin layers

The Hamiltonian of an atom with N electrons and a fixed nucleus of infinite mass between two parallel planes is considered in the limit when the distance a between the planes tends to zero. We show that this Hamiltonian converges in the norm resolvent sense to a Schrödinger operator acting effectively in \documentclass[12pt]{minimal}\begin{document}$L^{2}(\mathbb {R}^{2N})$\end{document}L2(R2N) whose potential part depends on a. Moreover, we prove that after an appropriate regularization this Schrödinger operator tends, again in the norm resolvent sense, to the Hamiltonian of a two-dimensional atom (with the three-dimensional Coulomb potential-one over distance) as a → 0. This makes possible to locate the discrete spectrum of the full Hamiltonian once we know the spectrum of the latter one. Our results also provide a mathematical justification for the interest in the two-dimensional atoms with the three-dimensional Coulomb potential.

Binding energy of the ground and first few excited states of hydrogenic donor impurity in a rectangular GaAs quantum dot in the presence of electric field

Within the quasi-one-dimensional effective potential model and effective mass approximation, we calculate the ground and the first 9 excited-state binding energies of a hydrogenic donor impurity in a rectangular quantum dot (RQD) in the presence of electric field. The analytical form of the quasi-one-dimensional effective potential replacing the three-dimensional Coulomb potential in our model is derived by Fourier transforms. We discuss detailedly dependence of the binding energies on the impurity positions and electric fields. For the ground-state binding energy, our results qualitatively agree with that of Mendoza et al. (2005) in which they only calculated the ground-state binding energies in cubic quantum dots by variational method. However, for first 9 excited-state binding energies, such dependence has complex manner since there are two or three peaks in the electronic probability density distribution curves. The strengths and positions of these peaks in RQD affect the interaction potential between electron and impurity, which appears to be the critical control on the binding energies of impurity. The applied electric field pushes the positions of these peaks downwards, and the strengths of peaks located at the upper half of RQD increase while the strengths of lower peaks firstly decrease, then increase with increasing electric field. The high peak strength can lead to increase of the binding energy while the large distance between the position of peak and impurity center results in reduce of the energy, which is an interesting competition. This competition is more obvious for excited-state binding energies of off-central impurity.

Dimensional Effects on Electric Potentials and Fields in High-permittivity Thin Films and Interfaces

An efiective Lagrange density for electromagnetic flelds in the presence of a high- permittivity thin fllm or interface yields an analytic expression for electromagnetic potentials which interpolates between the two-dimensional logarithmic Coulomb potential at short distances and the three-dimensional Coulomb potential at large distances. It is shown that the analytic expression for the dimensionally hybrid electromagnetic potential is a very good approximation to the inflnite series solution from image charges.

Approximate multiplication of tensor matrices based on the individual filtering of factors

Algorithms are proposed for the approximate calculation of the matrix product \( \tilde C \) ≈ C = A · B, where the matrices A and B are given by their tensor decompositions in either canonical or Tucker format of rank r. The matrix C is not calculated as a full array; instead, it is first represented by a similar decomposition with a redundant rank and is then reapproximated (compressed) within the prescribed accuracy to reduce the rank. The available reapproximation algorithms as applied to the above problem require that an array containing r2d elements be stored, where d is the dimension of the corresponding space. Due to the memory and speed limitations, these algorithms are inapplicable even for the typical values d = 3 and r ∼ 30. In this paper, methods are proposed that approximate the mode factors of C using individually chosen accuracy criteria. As an application, the three-dimensional Coulomb potential is calculated. It is shown that the proposed methods are efficient if r can be as large as several hundreds and the reapproximation (compression) of C has low complexity compared to the preliminary calculation of the factors in the tensor decomposition of C with a redundant rank.

A linear equation for Wilson loops

The Makeenko–Migdal loop equation is non-linear and first order in the area derivative, but we show that for simple loops in QCD2 it is possible to reformulate this equation as a linear equation with second order derivatives. This equation is a bound state Schrödinger equation with a three-dimensional Coulomb potential. Thus, loop dynamics leads to a surprising new picture of confinement, where this phenomenon is due to a (bound state) localization in loop space, with the Wilson loops decaying exponentially outside a characteristic radius.

Eigenenergies of 3D-Coulomb and 3D-Harmonic Oscillator Potentials from WKB Quantization: Point Canonical Transformation

A direct application of the WKB quantization to the three-dimensional Coulomb potential does not yield the exact eigenenergies. The three-dimensional Coulomb potential is converted to a Morse potential by using the point canonical transformation. Then the WKB quantization is applied to the Morse potential to find a relationship between the eigenenergies of the Coulomb and those of the Morse potentials. From the relationship the exact eigenenergis of the Coulomb potential are determined. The same method is found to be also valid for the three-dimensional harmonic oscillator potential. And the Langer modified WKB quantization is algebraically derived.

Quantum, classical and semiclassical momentum distributions: II. Morse and Coulomb potentials

The basic properties of momentum distributions in quantum mechanics for elementary systems as well as their (semi-)classical analogue derived in a preceding paper are illustrated by a detailed analysis of the Morse oscillator and the three-dimensional Coulomb potential, where the distributions can be calculated in closed form.

Stabilization of atoms in ultrastrong laser fields: A classical approach.

We study the classical theory of the response of atoms to ultrastrong laser pulses. We argue that the stabilization of atoms in a strong laser field through bound states in a Kramers-Henneberger potential has a classical interpretation. We present results of numerical simulations based on a phase-space-averaging method for a one-dimensional smoothed potential, which agree very well with the results of quantum theory. In a three-dimensional Coulomb potential, classical simulations hardly allow for stabilization, in contrast to recently obtained quantum-mechanical results. We identify physical reasons for this discrepancy with the effect of smoothing the potential singularity. We stress the relevance of our studies in the context of the correspondence principle.

Green's functions through so(2, 1) Lie algebra in nonrelativistic quantum mechanics

We discuss an algebraic technique to construct the Green's function for systems described by the noncompact so(2, 1) Lie algebra. We show that this technique solves the one-dimensional linear oscillator and Coulomb potentials and also generates particular solutions for other one-dimensional potentials. Then we construct explicitly the Green's function for the three-dimensional oscillator and the three-dimensional Coulomb potential, which are generalizations of the one-dimensional cases, and the Coulomb plus an Aharonov-Bohm potential. We discuss the dynamical algebra involved in each case and also find their wave functions and bound state spectra. Finally we introduce a point canonical transformation in the generators of so(2, 1) Lie algebra, show that this procedure permits us to solve the one-dimensional Morse potential in addition to the previous cases, and construct its Green's function and find its energy spectrum and wave functions.

Laser effects in photoionization: numerical solution of coupled equations for a three-dimensional Coulomb potential

The time-dependent Schrodinger equation was solved numerically as a set of coupled equations for an electron moving in three spatial dimensions, bound initially (t ≤ 0) by a Coulomb potential, and acted on by a linearly polarized, single-frequency, classical electromagnetic field that is turned on abruptly at t = 0. The laser field was assumed to be spatially homogeneous, and magnetic interactions between the laser beam and the electron were ignored. Characteristic photoionization times were determined for a wide range of scaled laser intensities and scaled laser frequencies and for a range of electron initial-state quantum numbers nlm. Results are compared with those obtained previously [ Phys. Rev. A40, 683 ( 1989)] for a one-dimensional δ potential.