Two-dimensional Hydrogen Atom Research Articles

Cosmic string influence on a 2D hydrogen atom and its relationship with the Rytova–Keldysh logarithmic approximation in semiconductors

A two-dimensional hydrogen atom offers a promising alternative for describing the quantum interaction between an electron and a proton in the presence of a cosmic string. Reducing the hydrogen atom to two dimensions enhances its suited to capture the cylindrical/conical symmetry associated with the cosmic string, providing a more appropriate description of the physical system. After solving Schrd̈inger’s equation, we calculate the eigenenergies, probability distribution function, and expected values for the hydrogen atom with logarithmic potential under the influence of the topological defect. The calculations for the 2D hydrogen atom are performed for the first time using the Finite Difference Method. The results are presented through graphics, tables, and diagrams to elucidate the system’s physical properties. We have verified that our calculations agree with a linear variational method result. Our model leads to an interesting analogy with excitons in a two-dimensional monolayer semiconductor located within a specific semiconductor region. To elucidate this analogy, we present and discuss some interaction potentials and their exciton eigenstates by comparing them with the results from the literature.

Nondipole signatures in ionization and high-order harmonic generation

We analyze nondipole effects arising in ionization and high-order harmonic generation for a two-dimensional hydrogen atom irradiated with either low- or high-frequency laser pulses. In the low-frequency case, the electron wave packet dynamics is dominated by rescattering processes within the laser pulse. Here both odd- and even-order harmonics are generated in the direction of the laser field propagation and polarization, respectively. For high-frequency pulses, such rescattering processes can be neglected. We demonstrate that a significant portion of photoelectrons is detected opposite to the laser pulse propagation direction as a consequence of their postpulse wave packet spreading and interaction with the parent ion. This is accompanied by rich interference structures formed in the momentum distributions of photoelectrons. Our results follow from the numerical solution of the time-dependent Schr\"odinger equation, which is based on the Suzuki-Trotter scheme with the split-step Fourier approach. The method relies on a Hamiltonian decomposition, where except for the components depending exclusively on the momentum or on the position operators, there are also terms depending on both momentum and position operators in particular configurations. We demonstrate that, as long as the latter does not depend on noncommuting coordinates of the momentum and position operators, nondipole effects in laser-matter interactions can be studied without applying extra approximations and unitary operations.

Analogy and dissimilarity of excitons in monolayer and bilayer of MoSe2

Excitons in thin layers of semiconducting transition metal dichalcogenides are highly subject to the strongly modified Coulomb electron–hole interaction in these materials. Therefore, they do not follow the model system of a two-dimensional hydrogen atom. We investigate experimentally and theoretically excitonic properties in both the monolayer (ML) and the bilayer (BL) of MoSe2 encapsulated in hexagonal BN. The measured magnetic field evolutions of the reflectance contrast spectra of the MoSe2 ML and BL allow us to determine g-factors of intralayer A and B excitons, as well as the g-factor of the interlayer exciton. We explain the dependence of g-factors on the number of layers and excitation state using first principles calculations. Furthermore, we demonstrate that the experimentally measured ladder of excitonic s states in the ML can be reproduced using the approach with the Rytova–Keldysh potential that describes the electron–hole interaction. In contrast, the analogous calculation for the BL case requires taking into account the out-of-plane dielectric response of the MoSe2 BL.

Planar magnetic texture on the surface of a topological insulator

We study the planar magnetic textures coupled to a two-dimensional Dirac surface state, where both the magnetic texture and surface state form in the magnetic topological insulator surface layer. It is shown that the radial vortex with winding number $w=\ifmmode\pm\else\textpm\fi{}1$ leads to the confinement of Dirac states, where a mapping to the Schr\"odinger equation of a two-dimensional hydrogen atom is found. The fully spin-polarized zero-energy bound state forms a flat band that resembles the zeroth Landau level of Dirac electrons in a uniform out-of-plane magnetic field. Remarkably, the number of the zero-energy state is topologically robust. Interestingly, when such a system is proximity coupled to an $s$-wave superconductor, the existence of Majorana zero modes at the Abrikosov vortex depends only on the relative value of the magnetic exchange coupling and the pairing strength. We conclude with a brief discussion on the physical realization with such magnetic textures.

Exact Noncommutative Two-Dimensional Hydrogen Atom

In this work, we present an exact analysis of the two-dimensional noncommutative hydrogen atom. In this study, the Levi-Civita transformation was used to perform the solution of the noncommutative Schrodinger equation for Coulomb potential. As an important result, we determine the energy levels for the considered system. Using the result obtained and experimental data, a bound on the noncommutativity parameter was obtained.

Analytical solutions for the hydrogen atom in plasmas with electric, magnetic, and Aharonov-Bohm flux fields.

The combined effects of external electric, magnetic, and Aharonov-Bohm (AB) flux fields on the two-dimensional hydrogen atom embedded in both Debye and quantum plasmas modeled by the more general exponential cosine Coulomb (MGECSC) potential are investigated using the general analytic approach, namely the homotopy analysis method (HAM). The analytical convergent solutions are obtained for the ground state as well as excited states at both weak and strong intensity of the external fields. The influence of the screening parameters on the quantum levels are exhaustively explored in the presence of three external fields. It is worth emphasizing that our analytical HAM results have 4-10 digits of accuracy in comparison with the numerical results. In the framework of the HAM method, there is no any small parameter different from the perturbation. Owing to this advantage, the convergent accurate solutions always can be obtained by the HAM approach even for the strong external fields. There is no limit to the value of the parameters or the strength of the external fields. It is also observed that the combined effects of the external fields play an important role on the interaction potential profile and the applied external magnetic field is the most dominant in the hydrogen atomic system. Also note that the combined effect of the fields is stronger than individual effects in both Debye and quantum plasmas. The findings obtained by the HAM-based approach in this study shed substantial light on the more complicated problems in plasmas for the atomic systems or molecular physics.

Charge currents and induced magnetic fields in a bounded two-dimensional hydrogen atom

The present work involves solving the time-independent Schrodinger equation for a two-dimensional (2D) hydrogen atom under the combined effect of static electric field, magnetic field and circular confinement. A study of the influence of external fields as well as spatial confinement on charge currents and induced magnetic fields has been carried out. It has been found that applied magnetic field reduces Stark shifts. The presence of tight spatial confinement leads to increase in the magnitude of currents as well as induced magnetic fields. In fact, induced magnetic field of the order of 400 T has been observed in one of the cases. The results of the work may prove helpful in manipulating these currents and fields.

Energy level splitting of a 2D hydrogen atom with Rashba coupling in non-commutative space

We explore the non-commutative (NC) effects on the energy spectrum of a two-dimensional hydrogen atom. We consider a confined particle in a central potential and study the modified energy states of the hydrogen atom in both coordinates and momenta of non-commutativity spaces. By considering the Rashba interaction, we observe that the degeneracy of states can also be removed due to the spin of the particle in the presence of NC space. We obtain the upper bounds for both coordinates and momenta versions of NC parameters by the splitting of the energy levels in the hydrogen atom with Rashba coupling. Finally, we find a connection between the NC parameters and Lorentz violation parameters with the Rashba interaction.

Zeeman Effect in Phase Space

The two-dimensional hydrogen atom in an external magnetic field is considered in the context of phase space. Using the solution of the Schrödinger equation in phase space, the Wigner function related to the Zeeman effect is calculated. For this purpose, the Bohlin mapping is used to transform the Coulomb potential into a harmonic oscillator problem. Then, it is possible to solve the Schrödinger equation easier by using the perturbation theory. The negativity parameter for this system is realised.

H-TALIF measurement for wall radical quenching modelling in microscale combustion

Two-dimensional hydrogen atom (H) distribution in a CH4/air premixed flame in a narrow channel was measured using a newly-developed two-photon absorption LIF (TALIF) setup with high efficiency 205 nm laser. The wall chemical effect of SUS321 and quartz surfaces on the H distribution in a methane flame was examined. Based on the measured H atom concentration near the wall, a radical quenching model focused on the H adsorption is proposed and the initial sticking coefficient (S0) of H on SUS321 and quartz surfaces are estimated as 0.1 and 0.01, respectively. Sensitivity analysis of CO, HRR and species distributions with varying S0 of H, O, OH, CH3 were performed. It is found that the H adsorption should be dominant in the wall chemical effect on the methane flame.

Anisotropic features of two-dimensional hydrogen atom in magnetic field

The aim of the current work is the numerical research of the anisotropic characteristics of the two-dimensional hydrogen atom induced by a magnetic field. The ground state energy (GSE) of the two-dimensional hydrogen atom and the corresponding wave function have been numerically calculated in the Born-Oppenheimer approximation and with taking into account the finite mass of the proton. The non-linear dependence of GSE on the angle {\alpha} between the magnetic field vector and the normal to the plane of electron motion in a wide range of magnetic field strength has been found. The effect of a significant reduction of GSE (up to 1.9-fold) is observed with increasing the angle {\alpha} up to 90 degrees.

Excited states of two-dimensional hydrogen atom in tilted magnetic field: Quantum chaos

The aim of the current work is the research of the influence of a tilted magnetic field direction on the spectrum and the energy level spacing distribution of a two-dimensional (2D) hydrogen atom and of an exciton in GaAs/Al0.33Ga0.67As quantum well. It was discovered that the quantum chaos (QC) is initiated with an increasing angle α between the magnetic field direction and the normal to the atomic plane. It is characterized by the repulsion of levels leading to the eliminating of the shell structure and by changing the spectrum statistical properties. The statement about the initiation of chaos and its dominance over regular motion with increasing angle α is confirmed by the results of our calculations of the classical dynamics presented in this paper. The evolution of the spatial distribution of the square of the absolute value of the wave function at an increasing angle α was observed. The differences of calculated dependencies of energies for various excited states on the tilt angle at a wide range of the magnetic field strength were described.

Numerical transforms from position to momentum space via Gaussian quadrature in the complex plane

A method for the numerical evaluation of integrals involving Bessel functions of the first kind, via non-standard Gaussian quadrature, is appraised. This method uses specialized Gaussian quadratures based on weight functions which are modified Bessel functions of the third kind. We apply the method to calculate momentum space radial wave functions of the two-dimensional hydrogen atom and harmonic oscillator, via two-dimensional Fourier or Hankel transforms of the position space functions. Results show that the method performs best for the asymptotic or large p regions of the hydrogen atom. We also illustrate that accurate results can be obtained for the smaller argument regions with the use of higher-order quadratures. On the other hand, the method fails in its application to the harmonic oscillator. A discussion of the strengths and weaknesses of the method is presented.

Exact analytical solutions of a two-dimensional hydrogen atom in a constant magnetic field

A hydrogen atom in a constant magnetic field is a problem whose Schrödinger equation has been considered to be analytically unsolvable. However, it has been found for the case of two-dimensional space some specific values of the magnetic field for which the problem has analytical solutions. In this paper, we give a full theory for these analytical solutions. First, we construct an equation to calculate of exact-solution-points, the values of magnetic field which allow exact solutions for the Schrödinger equation. The explicit forms of the wave functions corresponding to these analytical solutions are also given. Second, we analyse the exact-solution-points and discover some important rules. (i) There exist (n − 1)2 different values of the magnetic field so that the Schrödinger equation of a two-dimensional hydrogen atom in such field has exact analytical solutions with the principal quantum number n. (ii) For more details, among these (n − 1)2 exact analytical solutions, there are nr=n−|m|−1 solutions with the same magnetic quantum number m. Both properties are also proved mathematically in this paper. For illustration, some concrete values are shown for the cases of quantum number n up to 8.

Quantum mechanics on phase space and the Coulomb potential

Symplectic quantum mechanics (SMQ) makes possible to derive the Wigner function without the use of the Liouville–von Neumann equation. In this formulation of the quantum theory the Galilei Lie algebra is constructed using the Weyl (or star) product with Qˆ=q⋆=q+iħ2∂p,Pˆ=p⋆=p−iħ2∂q, and the Schrödinger equation is rewritten in phase space; in consequence physical applications involving the Coulomb potential present some specific difficulties. Within this context, in order to treat the Schrödinger equation in phase space, a procedure based on the Levi-Civita (or Bohlin) transformation is presented and applied to two-dimensional (2D) hydrogen atom. Amplitudes of probability in phase space and the correspondent Wigner quasi-distribution functions are derived and discussed.

Numerical exact diagonalization of singularity in the ground state of two-dimensional hydrogen atom

With the development of computing technology, numerical exact diagonalization method plays a vital role in modern computational condensed matter physics, especially in the research area of strongly correlated electron systems: it becomes a benchmark for other numerical computational techniques, such as quantum Monte Carlo, numerical renormalization group, density matrix renormalization group, and dynamic mean field theory. In this paper, we first numerically exactly diagonalize the three-dimensional hydrogen atom with the combination of finite-difference method, and find that the numerical wave function of ground state is in good agreement with the analytical calculations. We then turn to discuss the space dimension confinement hydrogen system, two-dimensional hydrogen atom, and notice that the numerical wave function is no longer in agreement with the analytical calculation, where the ground state wave function has a numerical singularity as radius approaches to zero. Compared with the case of the three-dimensional hydrogen atom, this issue mainly comes from the nature of space dimension confinement. To resolve such an issue of numerical singularity in two-dimensional hydrogen atom, we need to construct a new discrete and normalized Bessel function as a basis to study the ground state behavior of dimension confinement system based on the framework of Lanczos-type numerical exact diagonalization. The constructed normalized Bessel basis is orthogonal and discrete, and thus becomes suitable for practical calculation. Besides, these prominent properties of such a Bessel basis greatly reduce the complexity and difficulty in practical calculation, and thus makes computing work efficient. In addition, Lanczos-type numerical exact diagonalization method can extremely speed up the process of solving the eigenvalue equation. As a result, such a high efficient calculation of our method demonstrates the consistence between numerical and analytical ground state energy value, and the corresponding wave function with enough truncated basis number. Since this kind of numerical singularity occurs in many space dimension confinement systems, our finding for constructing a new discrete Bessel basis function may be helpful in studying the quantum systems with numerical singularity behaviors in wavefunctions in future. On the other hand, it should be pointed out that the Bessel basis is incorporated into the linear augment plane wave method in the density functional theory to study the electronic band structure of the condensed material and obtain high accurate results, especially in the theoretical prediction of topological insulators and in experimental realization as well.

Confinement effects of magnetic field on two-dimensional hydrogen atom in plasmas

In this study, for the first time, the Schrödinger equation with more general exponential cosine screened Coulomb (MGECSC) potential is solved numerically in the presence and in the absence of an external magnetic field within two-dimensional formalism using the asymptotic iteration method. The MGECSC potential includes four different potential forms when considering different sets of the parameters in the potential. The plasma screening effects in the weak and strong magnetic field regimes as well as the confinement effects of magnetic field on the two-dimensional hydrogen atom in Debye and quantum plasmas are investigated by solving the corresponding equations. It is found that applying a uniform magnetic field on the hydrogen atom embedded in a plasma leads to change in the profile of the total interaction potential. Thus, confinement effects of magnetic field on hydrogen atom embedded in Debye and quantum plasmas modeled by a MGECSC potential lead to shift bound state energies. This effect would be important to isolate the plasma from the external environment in the experimental applications of plasma physics.

Bound states for the quantum dipole moment in two dimensions

We calculate accurate eigenvalues and eigenfunctions of the Schrödinger equation for a two-dimensional quantum dipole. This model proved useful for the study of elastic effects of a single edge dislocation. We show that the Rayleigh–Ritz variational method with a basis set of Slater-type functions is considerably more efficient than the same approach with the basis set of point-spectrum eigenfunctions of the two-dimensional hydrogen atom used in earlier calculations.

Localization of Dirac-like excitations in graphene in the presence of smooth inhomogeneous magnetic fields

The present paper discusses magnetic confinement of the Dirac excitations in graphene in the presence of inhomogeneous magnetic fields. In the first case a magnetic field directed along the z axis whose magnitude is proportional to 1/r is chosen. In the next case we choose a more realistic magnetic field which does not blow up at the origin and gradually fades away from the origin. The magnetic fields chosen do not have any finite/infinite discontinuity for finite values of the radial coordinate. The novelty of the two magnetic fields is related to the equations which are used to find the excited spectra of the excitations. It turns out that the bound state solutions of the two-dimensional hydrogen atom problem are related to the spectra of graphene excitations in the presence of the 1/r (inverse-radial) magnetic field. For the other magnetic field profile one can use the knowledge of the bound state spectrum of a two-dimensional cutoff Coulomb potential to dictate the excitation spectra of graphene. The spectrum of the graphene excitations in the presence of the inverse-radial magnetic field can be exactly solved while the other case cannot be. In the later case we give the localized solutions of the zero-energy states in graphene.

Coulomb states in nanostructures, accidental degeneracy, and the Laplace-Runge-Lenz operator

The energy-level structure of spatially indirect Coulomb complexes is investigated. Two limiting cases that correspond to the two-dimensional hydrogen atom and planar isotropic harmonic oscillator are characterized by accidental degeneracy are considered. Numerical calculations are used to trace the evolution of the energy-level structure between these two limiting cases.