Soft Coulomb Potential Research Articles

Transmission of a position-dependent mass system through a soft Coulomb potential

This research delves into the fascinating transmission characteristics of a system with variable mass, confined by a Coulomb-like potential. By utilizing an analytic approach to transmission probability, we were able to unveil the intriguing features that are observed experimentally in a specially designed two-dimensional lattice. This lattice was engineered to manipulate itinerant electrons through entanglement, resulting in variable masses as they traverse the lattice. To build our model system, we utilized the displacement operator approach to the position-dependent effective mass theory. This allowed us to investigate the effects of different configurations of leads, system geometries and lattice deformations on the transmission characteristics of the system. The resulting data showed that the model system was highly sensitive to these parameters, which led to various interesting features. First, we observed Andreev-like reflections, which occur when a particle is reflected as a hole, resulting in the transfer of both energy and charge. Another feature was reflectionless transmission, in which the transmission probability of a particle is close to unity, and almost no reflection occurs. We also observed complete localization of charge carriers, where the transmission probability is effectively zero, indicating that the charge carriers are completely confined within the lattice. Our findings demonstrate the remarkable versatility of the effective mass approach in the modeling of physical systems. By unraveling the intricate relationships between system parameters and transmission probability, we have gained new insights into the behavior of itinerant charge carriers in lattice structures. These insights may guide the design and development of novel electronic devices with enhanced performance and functionality.

The envelope theory as a pedagogical tool

The envelope theory (ET) is a reliable and easy to implement method to solve time independent Schrödinger-like equations (eigenvalues and eigenvectors). It is particularly useful to solve many-body systems since the computational cost is independent from the number of particles. The purpose of this paper is twofold. First, we want to make known a method that is probably too little used. Second, we also want to show that this method can be used as a pedagogical tool, thanks to its simplicity and the reliable results that can be obtained. To reach these goals, the ET is applied to a simple problem in one dimension, the soft-Coulomb potential , characterised by a bias distance d. Such interaction is used for the study of excitons, electron–hole bound pairs where the two charges are kept separated in two different one-dimensional regions (quantum wires). In addition to its physical interest, this system has never been treated with the ET.

Is there a better way of representing stationary wave functions than basis expansion?

We extend our previously developed approach for finding the exact analytical form of ground state wave function of single-electron Schrödinger equations to target arbitrary excited states of 1D model chemical systems of hydrogen atom and H 2 + $$ {\mathrm{H}}_2^{+} $$ with soft Coulomb potential. Our representation of the wave functions is formally unified by a factorized formula involving a polynomial prefactor encoding the nodal information, a non-integer power prefactor, and an exponentially decaying term modulated by a modulator function. We show that our method leads to faster energy convergence than the basis expansion method, and hence is a better way of representing the wave function. This conclusion is also validated in the real-world H 2 + $$ {\mathrm{H}}_2^{+} $$ .

Laser-induced dynamic alignment of the HD molecule without the Born-Oppenheimer approximation.

Laser-induced molecular alignment is well understood within the framework of the Born-Oppenheimer (BO) approximation. Without the BO approximation, however, the concept of molecular structure is lost, making it hard to precisely define alignment. In this work, we demonstrate the emergence of alignment from the first-ever non-BO quantum dynamics simulations, using the HD molecule exposed to ultrashort laser pulses as a few-body test case. We extract the degree of alignment from the non-BO wave function by means of an operator expressed in terms of pseudo-proton coordinates that mimics the BO-based definition of alignment. The only essential approximation, in addition to the semiclassical electric-dipole approximation for the matter-field interaction, is the choice of time-independent explicitly correlated Gaussian basis functions. We use a variational, electric-field-dependent basis-set construction procedure, which allows us to keep the basis-set dimension low while capturing the main effects of electric polarization on the nuclear and electronic degrees of freedom. The basis-set construction procedure is validated by comparing with virtually exact grid-based simulations for two one-dimensional model systems: laser-driven electron dynamics in a soft attractive Coulomb potential and nuclear rovibrational dynamics in a Morse potential.

Exact analytical ground state solution of 1D H$$_2^+$$ with soft Coulomb potential

Exact analytical ground state solution of 1D H$$_2^+$$ with soft Coulomb potential

Exact Analytical Solution of the Ground-State Hydrogenic Problem with Soft Coulomb Potential.

We provide the exact analytical solution of the ground-state hydrogenic problem with soft Coulomb potential in 1-, 2- and 3-D. We show that the wave function is an analytical function of the inverse of the soft Coulomb potential and identify a power term, an exponentially decaying term and a mildly varying modulator function on the exponential. In approaching the bare Coulomb limit, only the exponentially decaying term survives in 2D and 3D and converges to the well-known result. This is in contrast with the 1D case, where the wave function shrinks to a delta function with a total energy of minus infinity. The asymptotic behavior of the energy in such limit has been analyzed. Moreover, by analyzing the solution in different dimensions, we find that the total energy increases with dimension and scales linearly rather than quadratically with the nuclear charge Z in the large Z limit.

A colloquium on the variational method applied to excitons in 2D materials

In this colloquium, we review the research on excitons in van der Waals heterostructures from the point of view of variational calculations. We first make a presentation of the current and past literature, followed by a discussion on the connections between experimental and theoretical results. In particular, we focus our review of the literature on the absorption spectrum and polarizability, as well as the Stark shift and the dissociation rate. Afterwards, we begin the discussion of the use of variational methods in the study of excitons. We initially model the electron-hole interaction as a soft-Coulomb potential, which can be used to describe interlayer excitons. Using an \emph{ansatz}, based on the solution for the two-dimensional quantum harmonic oscillator, we study the Rytova-Keldysh potential, which is appropriate to describe intralayer excitons in two-dimensional (2D) materials. These variational energies are then recalculated with a different \emph{ansatz}, based on the exact wavefunction of the 2D hydrogen atom, and the obtained energy curves are compared. Afterwards, we discuss the Wannier-Mott exciton model, reviewing it briefly before focusing on an application of this model to obtain both the exciton absorption spectrum and the binding energies for certain values of the physical parameters of the materials. Finally, we briefly discuss an approximation of the electron-hole interaction in interlayer excitons as an harmonic potential and the comparison of the obtained results with the existing values from both first--principles calculations and experimental measurements.

One-dimensional Lieb-Oxford bounds.

We investigate and prove Lieb-Oxford bounds in one dimension by studying convex potentials that approximate the ill-defined Coulomb potential. A Lieb-Oxford inequality establishes a bound of the indirect interaction energy for electrons in terms of the one-body particle density ρψ of a wave function ψ. Our results include modified soft Coulomb potential and regularized Coulomb potential. For these potentials, we establish Lieb-Oxford-type bounds utilizing logarithmic expressions of the particle density. Furthermore, a previous conjectured form Ixc(ψ)≥-C1∫Rρψ(x)2dx is discussed for different convex potentials.

Simulation of a hydrogen atom in a laser field using the time-dependent variational principle.

The time-dependent variational principle is used to optimize the linear and nonlinear parameters of Gaussian basis functions to solve the time-dependent Schrödinger equation in one and three dimensions for a one-body soft Coulomb potential in a laser field. The accuracy is tested comparing the solution to finite difference grid calculations using several examples. The approach is not limited to one particle systems and the example presented for two electrons demonstrates the potential to tackle larger systems using correlated basis functions.

Variational approach to the soft-Coulomb potential in low-dimensional quantum systems

The variational method is used to obtain the ground- and first-excited states for the soft-Coulomb central potential, 1/r2+d2, characterized by a bias distance d, taken into account as a fixed parameter. Applications are presented for spatially indirect excitons, i.e., photo-generated electron-hole (e-h) bound pairs, where the two charges are kept separated in two different regions of a heterostructure. We consider one- or two-dimensional systems, namely, quantum wires or wells, respectively, and compare the results with numerical calculations obtained by finite-difference diagonalization of the Hamiltonian. An explicit example is given for GaAs-based heterostructures.

Exact long-wavelength plasmon dispersion on a ring with soft Coulomb interactions

We obtain the analytical dispersion of 1D plasmonic modes on a ring from the exact solution of the hydrodynamical model with soft Coulomb potential. We compare our results with the exact plasmon dispersion in straight 1D systems and find a set of formal correspondences between the two. In light of our results, we discuss recent experiments (Schmidt et al 2014 Nat. Commun. 5 3604) where ring-confined modes in nanodiscs are found to almost coincide with plasmonic excitations in 1D metallic nanostructures. We trace the similarity to the scaling properties of the plasmonic dispersion.

Influence of atomic potential on the generation of high harmonic generation from the atoms irradiated by mid-infrared laser pulses

By numerically solving the time dependent Schrodinger equation, the harmonic spectra generated from the atoms are obtained. The atomic potentials are modeled by a short-range potential and a long-range soft Coulomb potential, respectively. It is found that using the same laser parameters, the intensity of harmonic spectrum from the short-range atom is lower than the one from the long-range atom. However, in a high energy (near the cutoff) region of harmonic spectra, their conversion efficiencies are almost the same. The differences in emission intensity among harmonic spectra decrease as the harmonic energy increases. We calculate the time dependent probabilities of the ground state and ionization. It is found that the ionization probability of the long-range potential is larger than that of the short-range potential. There is no large difference in ground probability between the potentials of two models. The high harmonic generation is a stimulated process, and its intensity is proportional to the product between the amplitude for ground state and the amplitude of the continuum state. Thus the product of the long-range atom is larger than that of the short-range atom, and the emission spectrum presents a similar character. In order to analyze the mechanism of the intensity difference between two models, we perform a time-frequency analysis of the harmonic emission spectrum. The analysis is selected of the wavelet of the time dependent dipole moment. From the emission profile of the harmonic analysis, we find that the harmonic generated from long orbit plays a dominant role for the short-range atom. The amplitudes of electric field are large for the long orbit harmonic emission, thus the ionization mechanism of the atom is the tunnel ionization. For the short orbit, the instant field for the ionization is weak. Thus the short orbit plays a small role in the harmonic emission from the short-range atom. Using this feature of the short-range atom, we generate an isolated attosecond pulse. The short model atom is widely used to study the ionization of the plasma. Thus this work will contribute to the research on the high-order harmonic generation from the plasma.

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.

Three-dimensional simulation of the femto-second pulsed laser interacting with a nitrogen molecule

Interaction of femto-second pulsed laser with a nitrogen molecule at various angles is investigated numerically in the present study. We propose a three-dimensional time-dependent Schrödinger equation (TDSE) solver using finite volume method (FVM) with single-active-electron (SAE) assumption. We construct the model potential by combining the soft-Coulomb potential and Yukawa potential and fitting both the first ionization energy and orbital type of highest-occupied molecular orbital (HOMO) of a N 2 molecule. The ratio of simulated maximum ( χ = 0 ° ) to minimum ( χ = 90 ° ) ionization yield is 1.9 and it agrees more favorably with the converted experimental data (2.3–3.3) than the MO-ADK model does (10.0). This shows that the proposed model potential in the 3D TDSE solver can faithfully predict the ionization of a nitrogen molecule under strong pulsed laser incidence and is a very useful tool for future application in similar analysis of more complicated molecules.

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.

High-order harmonic generation spectrum of an excited one-dimensional Coulomb atom in an intense laser field

Response of the wave packet of a one-dimensional Coulomb atom to an intense laser field is calculated using the symmetrized split operator fast Fourier method. The high-order harmonic generation (HHG) of the initial state separately being the ground and excited states is presented. When the hardness parameter α in the soft Coulomb potential V(x) = −1/(x2+α)1/2 is chosen to be small enough, the so-called hard Coulomb potential V(x) = −1/|x| can be obtained. It is well known that the hard one-dimensional Coulomb atom has an unstable ground state with an energy eigenvalue of ~0.5 and it has no states corresponding to physical states in the true atoms, and has the first and second excited states being degenerate. The parity effects on the HHG can be seen from the first and second excited states of the hard one-dimensional Coulomb atom. The HHG spectra of the excited states from both the soft and hard Coulomb atom models are shown to have more complex structures and to be much stronger than the corresponding HHG spectrum of the ground state of the soft Coulomb model with α = 2 in the same laser field. Laser-induced non-resonant one-photon emission is also observed.

Calculation of band structures by a discrete variable representation based on Bloch functions

The propagation of waves in periodic media is restricted to allowed bands of energy or frequencies separated by forbidden gaps. We propose a discrete variable representation, based on Bloch functions, that is suitable for the calculation of band structures in one-dimensional systems. The solutions are obtained by a single diagonalization without any integration. The method only needs the values of the potential at grid points. Applications to the standard Krönig-Penney and Mathieu potentials, and to the nonanalytical case of a soft Coulomb potential, show that very good accuracy is achieved with moderate grid sizes.

Anomalously slow phase transitions in self-gravitating systems.

The kinetics of collapse and explosion transitions in microcanonical self-gravitating ensembles is analyzed. A system of point particles interacting via an attractive soft Coulomb potential and confined to a spherical container is considered. We observed that for 100-200 particles collapse takes 10(3) - 10(4) particle crossing times to complete; i.e., it is by two to three orders of magnitude slower than the velocity relaxation. In addition, it is found that the collapse time decreases rapidly with an increase of the soft-core radius. We found that such an anomalously long collapse time is caused by the slow energy exchange between a higher-temperature compact core and relatively cold diluted halo. The rate of energy exchange between the faster modes of the core particles and slower-moving particles of the halo is exponentially small in the ratio of the frequencies of these modes. As the soft-core radius increases and the typical core modes become slower, the ratio of core and halo frequencies decreases and the collapse accelerates. Implications for astrophysical systems and phase transition kinetics are discussed.

Collapses and explosions in self-gravitating systems.

Collapse and explosion (reverse to collapse) transitions in self-gravitating systems are studied by molecular dynamics simulations. A microcanonical ensemble of point particles confined to a spherical box is considered. The particles interact via an attractive soft Coulomb potential. It is observed that a collapse indeed takes place when the energy of the uniform state is set near or below the metastability-instability threshold (collapse energy) as predicted by the mean-field theory. Similarly, an explosion occurs when the energy of the core-halo state is increased above the explosion energy, where according to the mean-field predictions the core-halo state becomes unstable. For systems consisting of 125-500 particles, the collapse takes about 10(5) single-particle crossing times to complete, while a typical explosion is by an order of magnitude faster. A finite lifetime of metastable states is observed. It is also found that the mean-field description of the uniform and core-halo states is exact within the statistical uncertainty of the molecular dynamics data.

Linear-least-squares-fitting procedure for the solution of a time-dependent wave function of a model atom in a strong laser field in the Kramers-Henneberger frame

We present a linear-least-squares-fitting method to solve the time-dependent Schrodinger equation of a one-dimensional model atom in an intense laser field in the Kramers-Henneberger frame. In comparison with calculations in the laboratory frame, it is shown that in the Kramers-Henneberger frame, the method is more stable and larger time steps in the propagation can be used. The fitting procedure also allows for easy change of basis functions during the time propagation. The method is applied to study the stabilization of a model atom in an intense laser field where the electron is bound initially by a one-dimensional soft Coulomb potential.