One-dimensional Coulomb Potential Research Articles

Solving a singular integral equation for the one-dimensional Coulomb problem

A new integral equation that describes the behavior of the momentum space wave function for the one-dimensional Coulomb potential is proposed. The obtained result turned out to be a homogeneous Fredholm integral equation of the second kind and a singular integral equation, because its kernel has a singularity at some point in the momentum space. A nontriviality of the method of solving this singular integral equation lies in the application of the integral representation for its integral kernel. The technique applied in this paper made it possible to show that the wave function in the momentum representation is simultaneously a solution of the homogeneous Fredholm integral equation of the second kind and of the linear Volterra integral equation of the second kind. Since a linear Volterra integral equation of the second kind was easily transformed into a second order linear inhomogeneous differential equation with constant coefficients, the eigenfunctions and eigenvalues in the one-dimensional Coulomb problem were found without any difficulties. Such a circumstance may indicate the validity of the new integral equation and the proposed method of its solving.

The bound-state solutions of the one-dimensional hydrogen atom

The one-dimensional hydrogen atom is an intriguing quantum mechanics problem that exhibits several properties which have been continually debated. In particular, there has been variance as to whether or not even-parity solutions exist, and specifically whether or not the ground state is an even-parity state with infinite negative energy. We study a “regularized” version of this system, where the potential is a constant in the vicinity of the origin, and we discuss the even- and odd-parity solutions for this regularized one-dimensional hydrogen atom. We show how the even-parity states, with the exception of the ground state, converge to the same functional form and become degenerate for x > 0 with the odd-parity solutions as the cutoff approaches zero. This differs with conclusions derived from analysis of the singular (i.e., without regularization) one-dimensional Coulomb potential, where even-parity solutions are absent from the spectrum.

Numerical solution of the schrödinger equation with periodic coulomb potential

This paper investigates the energy spectrum of a periodic one-dimensional Coulomb potential. The Schrödinger equation is solved numerically by our newly developed filter method (Phys. Rev E 96(3), 033302 (2017)). We observe that the energy spectrum can be obtained with a limited number of lattice. The results show the presence of an energy band structure as a function of lattice width and edge width. The comparison with the other potential model is also discussed.

Плотность энергетического спектра электрона в поле изображения и запирающем электрическом поле

In the semiclassical approximation, the density of the electron energy spectrum near the metal surface is described, when electron is bound by the image field and the blocking electrostatic field. In the system under consideration, the confinement mechanism is realized, and the energy spectrum for the motion of an electron in the direction perpendicular to the metal surface is completely discrete. The density of states of the energy spectrum is expressed in terms of elliptic integrals, the argument of which is a sigmoidal function. When the field is turned off, it becomes the Heaviside step function. A dimensionless energy parameter is introduced, which determines the intervals with qualitatively different changes in the width of the classically accessible region of motion. For large positive values of the energy parameter, the spectrum density asymptotically tends to the density in the triangular potential with the addition of the Coulomb logarithmic correction, and for negative values of the energy parameter, the spectrum density tends to dependence for a one-dimensional Coulomb potential. Approximate expressions are obtained for the spectrum density in terms of elementary functions in a wide range of electron energies and electric field strength.

Quantum Passage of a One-Dimensioal Coulomb Potential Well

It is shown that the one-dimensional Coulomb potential well impenetrable in the regularization method becomes partially permeable if the quantum fluctuations are taken into account.

Exact ground-state properties of the one-dimensional electron gas at high density

The dynamical response theory is used to obtain an analytical expression for the exchange energy of a quantum wire for arbitrary polarization and width. It reproduces the known form of exchange energy for 1D electron gas in the limit of infinitely thin cylindrical and harmonic wires. The structure factor for these wires are also obtained analytically in the high-density or small $r_s$ limit. This structure factor enables us to get the {\it exact} correlation energy for both the wires and demonstrates that there are at least two methods to get the ideal Coulomb limit in one dimension. The structure factor and the correlation energy are found to be independent of the way the one-dimensional Coulomb potential is regularized. The analytical expression for the pair correlation function is also presented for small distances and provides a justification for the small $r_s$ expansion as long as $r_s< \frac{3}{2} \left(\frac{\pi^2}{\pi^2+3} \right)=1.15$.

WKB-approach for the 1D hydrogen atom

Taking into account results of WKB-approximation, we derive exact quantum energies and wave functions of even and odd states in the one-dimensional Coulomb potential V(x)=-1/x,-∞<x<∞.

Eigenstates and dynamics of Hooke’s atom: Exact results and path integral simulations

The system of two interacting electrons in one-dimensional harmonic potential or Hooke’s atom is considered, again. On one hand, it appears as a model for quantum dots in a strong confinement regime, and on the other hand, it provides us with a hard test bench for new methods with the “space splitting” arising from the one-dimensional Coulomb potential. Here, we complete the numerous previous studies of the ground state of Hooke’s atom by including the excited states and dynamics, not considered earlier. With the perturbation theory, we reach essentially exact eigenstate energies and wave functions for the strong confinement regime as novel results. We also consider external perturbation induced quantum dynamics in a simple separable case. Finally, we test our novel numerical approach based on real-time path integrals (RTPIs) in reproducing the above. The RTPI turns out to be a straightforward approach with exact account of electronic correlations for solving the eigenstates and dynamics without the conventional restrictions of electronic structure methods.

An improved Lobatto discrete variable representation by a phase optimisation and variable mapping method

The Lobatto discrete variable representation (LDVR) proposed by Manoloupolos and Wyatt (1988) has unique features but has not been generally applied in the field of chemical dynamics. Instead, it has popular application in solving atomic physics problems, in combining with the finite element method (FE-DVR), due to its inherent abilities for treating the Coulomb singularity in spherical coordinates. In this work, an efficient phase optimisation and variable mapping procedure is proposed to improve the grid efficiency of the LDVR/FE-DVR method, which makes it not only be competing with the popular DVR methods, such as the Sinc-DVR, but also keep its advantages for treating with the Coulomb singularity. The method is illustrated by calculations for one-dimensional Coulomb potential, and the vibrational states of one-dimensional Morse potential, two-dimensional Morse potential and two-dimensional Henon–Heiles potential, which prove the efficiency of the proposed scheme and promise more general applications of the LDVR/FE-DVR method.

One-dimensional Coulomb problem in Dirac materials

We investigate the one-dimensional Coulomb potential with application to a class of quasirelativistic systems, so-called Dirac-Weyl materials, described by matrix Hamiltonians. We obtain the exact solution of the shifted and truncated Coulomb problems, with the wavefunctions expressed in terms of special functions (namely Whittaker functions), whilst the energy spectrum must be determined via solutions to transcendental equations. Most notably, there are critical bandgaps below which certain low-lying quantum states are missing in a manifestation of atomic collapse.

Calculations for the one-dimensional soft Coulomb problem and the hard Coulomb limit.

An efficient way of evolving a solution to an ordinary differential equation is presented. A finite element method is used where we expand in a convenient local basis set of functions that enforce both function and first derivative continuity across the boundaries of each element. We also implement an adaptive step-size choice for each element that is based on a Taylor series expansion. This algorithm is used to solve for the eigenpairs corresponding to the one-dimensional soft Coulomb potential, 1/sqrt[x(2)+β(2)], which becomes numerically intractable (because of extreme stiffness) as the softening parameter (β) approaches zero. We are able to maintain near machine accuracy for β as low as β = 10(-8) using 16-digit precision calculations. Our numerical results provide insight into the controversial one-dimensional hydrogen atom, which is a limiting case of the soft Coulomb problem as β → 0.

Mathematical predominance of Dirichlet condition for the one-dimensional Coulomb potential

We restrict a quantum particle under a Coulombian potential (i.e., the Schrödinger operator with inverse of the distance potential) to three-dimensional tubes along the x axis and diameter ɛ, and study the confining limit ɛ → 0. In the repulsive case we prove a strong resolvent convergence to a one-dimensional limit operator, which presents Dirichlet boundary condition at the origin. Due to the possibility of the falling of the particle in the center of force, in the attractive case we need to regularize the potential and also prove a norm resolvent convergence to the Dirichlet operator at the origin. Thus, it is argued that, among the infinitely many self-adjoint realizations of the corresponding problem in one dimension, the Dirichlet boundary condition at the origin is the reasonable one-dimensional limit.

Quantum solution for the one-dimensional Coulomb problem

The one-dimensional hydrogen atom has been a much studied system with a wide range of applications. Since the pioneering work of Loudon [R. Loudon, Am. J. Phys. 27, 649 (1959).], a number of different features related to the nature of the eigenfunctions have been found. However, many of the claims made throughout the years in this regard are not correct---such as the existence of only odd eigenstates or of an infinite binding-energy ground state. We explicitly show that the one-dimensional hydrogen atom does not admit a ground state of infinite binding energy and that the one-dimensional Coulomb potential is not its own supersymmetric partner. Furthermore, we argue that at the root of many such false claims lies the omission of a superselection rule that effectively separates the right side from the left side of the singularity of the Coulomb potential.

On Solvable Potentials, Supersymmetry, and the One-Dimensional Hydrogen Atom

The ways for improving on techniques for finding new solvable potentials based on supersymmetry and shape invariance has been discussed by Morales et al. [1] In doing so they address the peculiar system known as the one-dimensional hydrogen atom. In this paper we show that their remarks on such problem are mistaken. We do this by explicitly constructing both the one-dimensional Coulomb potential and the superpotential associated with the problem, objects whose existence are denied in the mentioned paper.

Information-theoretic properties of the half-line Coulomb potential

The half-line one-dimensional Coulomb potential is possibly the simplest D-dimensional model with physical solutions which has been proved to be successful to describe the behaviour of Rydberg atoms in external fields and the dynamics of surface-state electrons in liquid helium, with potential applications in constructing analog quantum computers and other fields. Here, we investigate the spreading and uncertaintylike properties for the ground and excited states of this system by means of the logarithmic measure and the information-theoretic lengths of Renyi, Shannon and Fisher types; so, far beyond the Heisenberg measure. In particular, the Fisher length (which is a local quantity of internal disorder) is shown to be the proper measure of uncertainty for our system in both position and momentum spaces. Moreover the position Fisher length of a given physical state turns out to be not only directly proportional to the number of nodes of its associated wavefunction, but also it follows a square-root energy law.

Fine splits of photon emission spectrum of hydrogen atom caused by transitions between different dressed states in intense high frequency laser field

The photon emission spectrum of the hydrogen atoms in an intense high-frequency laser pulse is simulated by using one-dimensional soft Coulomb potential. Regular fine structures appear on the two sides of both the odd and even multiples of photon energy of the laser field besides the ordinary odd harmonic peaks. It is proved that the splits of the fine structures are responsible for hyper-Raman lines and the energy spacing between the odd harmonic lines is equal to the direrence in energy between the eigenstates with the same parity of the time averaged Krameters–Henneberger (KH) potential. By analysing the features of the fine structures, we also verify that the so-called even order harmonics under the stabilization condition are indeed hyper-Raman lines caused by the transitions between the dressed atomic states with different values of parity.

Almost sudden perturbation of a quantum system with ultrashort electric pulses

We present an alternative approach to analyze atomic behavior when an external field perturbation is not sudden for a number of states of the field-free system. It is shown that the probability amplitudes for the system to be in these states can be accurately estimated from the closed set of their integral equations. Numerical examples for an electron in a one-dimensional Coulomb potential interacting with (i) laser and (ii) half-cycle pulses are provided. Comparison with exact calculations indicates the strength of the approach.

Fast electrons from electron-ion collisions in strong laser fields

Electron-ion collisions in the presence of a strong laser field lead to a distribution of fast electrons with maximum energy Emax=(k0+2v0)2∕2(a.u.), where k0 is the impact and v0 the quiver velocity of the electron. The energy spectrum is calculated by two approaches: (1) The time-dependent Schrödinger equation is numerically solved for wave packet scattering from a one-dimensional softcore Coulomb potential. Multiphoton energy spectra are obtained demonstrating a separation of the energy spectrum into an exponential distribution for transmission and a plateau distribution for reflection. (2) The energy spectrum is analytically calculated in the framework of classical instantaneous Coulomb collisions with random impact parameters and random phases of the laser field. An exact solution for the energy spectrum is obtained from which the fraction of fast electrons in the plateau region can be estimated.

Solvable rotating potentials and their generalizations

An alternative approach to find new solvable symmetric or non-symmetric radial potentials as well as their generalizations is presented. This is based on the application of the supersymmetry and shape invariance methods along with the generalized and standard Darboux transforms to Sturm–Liouville equations including the effective potential. As a useful application of the proposed approach, the procedure is used to obtain the well known isotonic and hydrogen-like potentials as well as to find the solvable rotating Morse and the Hulthen potentials that come from the corresponding Witten superpotentials leading to the Morse and the Hulthen models for s bounded states. In the hydrogen-like case, the results give rise to a justification for which the one-dimensional Coulomb potential should not exist strictly. In the Morse case, the resulting solvable radial potential for ℓ≠0 corresponds to a linear combination of a Morse, a specific Yukawa and a specific hydrogenic potential. Similarly, the solvable rotating Hulthen potential found is given as a linear combination of a specific hydrogenic potential, with a term proportional to the potential 1/( r(exp(− Ar)−1)). For these new potentials the eigenfunctions, the partner isospectral potentials as well as the operators that factorize the associated Hamiltonians are also presented. The proposed method, can be advantageously applied for finding other new solvable rotating potentials that can be useful in the treatment of outstanding quantum mechanical applications.

Electron near a helium liquid surface

The energies of an electron above a liquid-helium surface are analyzed in terms of an electron-atom polarizability potential with an infinite barrier. An integration over a layer, and a summation over different layers, leads to a one-dimensional Coulomb potential with a 1/x 2 correction. This potential provides an accurate description of the observed one-dimensional Rydberg energies with a quantum defect.