Two-dimensional Hydrogen-like Atom Research Articles

Spin Effects in the Emission of a Photon by a Two-Dimensional Hydrogen-Like Atom

It is shown that the result of calculation of the probability of single-photon emission by a two-dimensional hydrogen-like atom can depend on the choice of the spin operator. Specifically, if it is chosen to be the projection operator of the spin onto the plane of motion, then the result, as expected, agrees with the classical theory of emission of a three-dimensional atom, as was demonstrated by the author in one of his previous works. If, however, it is chosen as the projection operator onto the perpendicular to the plane of motion, the result does not agree with the classical theory. This puts in doubt the results presented in [S. H. Guo, X. L. Yang, F. T. Chan, Phys. Rev. A, 43, No. 3, 1197 (1991)].

Two-dimensional hydrogen-like atom in magnetic field in the presence of Rashba spin–orbit coupling

Abstract We investigate the spin–orbit (SO) Rashba coupling effect on bound states of an electron confined to the two-dimensional (2D) hydrogen-like atom (TDHA) in the presence of an external magnetic field (B), applied perpendicularly to the atomic plane. Energy levels and wavefunctions for arbitrary B and SO Rashba coupling constant have been obtained by the diagonalization of the Schrodinger Hamiltonian with the SO Rashba coupling term, using the basis of discrete Coulomb Sturmian functions. The solutions are used to investigate the influence of the SO Rashba interaction on phase coherence of spin states in terms of the von Neumann (vN) entropy and the spin polarization of the system. We shown that there exists unique correspondence between the vN spin entropy and the magnitude of spin-polarization vector. We also study the influence of SO interaction on magnetic properties of the planar H-like exciton using parameters appropriate for InSb semiconductor nanostructure.

Two-dimensional hydrogen-like atom in a constant magnetic field

The two-dimensional hydrogen-like atom in a constant magnetic field is considered. It is found that this is actually two separate problems. One for which the magnetic field causes an effective attraction between the nucleus and the electron and one for which it causes an effective repulsion. Each of the two problems has three separate cases depending on the sign of a shifted energy eigenvalue. For two of the six possibilities (shifted energy eigenvalue that is negative) it is shown that the first four solutions can be obtained exactly. For another two of the six possibilities (shifted energy eigenvalue that is positive) it is shown that the first eight solutions can be obtained exactly. For higher order states the energy eigenvalue is the root of a fifth or higher order polynomial, hence, the eigenvalue must be obtained numerically. Once the energy eigenvalue is known the solution to the radial wave equation is also known. Exact solutions for the radial wave equation, for the remaining two possibilities (shifted energy eigenvalue that is zero), are given to any desired order by means of a recursion relation.

Relativistic two-dimensional hydrogen-like atom in a weak magnetic field

A two-dimensional (2D) hydrogen-like atom with a relativistic Dirac electron, placed in a weak, static, uniform magnetic field perpendicular to the atomic plane, is considered. Closed forms of the first- and second-order Zeeman corrections to energy levels are calculated analytically, within the framework of the Rayleigh–Schrödinger perturbation theory, for an arbitrary electronic bound state. The second-order calculations are carried out with the use of the Sturmian expansion of the two-dimensional generalized radial Dirac–Coulomb Green function derived in the paper. It is found that, in contrast to the case of the three-dimensional atom (Stefańska, 2015), in two spatial dimensions atomic magnetizabilities (magnetic susceptibilities) are expressible in terms of elementary algebraic functions of a nuclear charge and electron quantum numbers. The problem considered here is related to the Coulomb impurity problem for graphene in a weak magnetic field.

Two-dimensional hydrogen-like atom in a uniform magnetic field: Large-order perturbation theory

The well-known hypervirial perturbation method (HPM), based on hypervirial relations and the Hellmann-Feynman theorem, is suitable for the calculation of perturbation corrections of large order for the non-relativistic, spinless, two-dimensional hydrogen-like atom in a uniform magnetic field. We show analytical results in terms of the quantum numbers and large-order corrections for particular states. This approach appears to be simpler and more efficient than the recently proposed one based on the Green’s functions for this particular model.

Second-order Stark effect and polarizability of a relativistic two-dimensional hydrogenlike atom in the ground state

The second-order Stark effect for a planar Dirac one-electron atom in the ground state is analyzed within the framework of the Rayleigh-Schr\"odinger perturbation theory, with the use of the Sturmian series expansion of the generalized Dirac-Coulomb Green function. A closed-form analytical expression for the static dipole polarizability of that system is found. The formula involves a generalized hypergeometric function ${}_{3}F_{2}$ with the unit argument. Numerical values of the polarizabilities for relativistic planar hydrogenic atoms with atomic numbers $1\leqslant Z\leqslant68$ are provided in a tabular form. A simple formula for the polarizability of a nonrelativistic two-dimensional hydrogenic atom, reported previously by several other authors, is recovered from our result in the nonrelativistic limit.

Two-dimensional hydrogen-like atom in a weak magnetic field

.We consider a non-relativistic two-dimensional (2D) hydrogen-like atom in a weak, static, uniform magnetic field perpendicular to the atomic plane. Within the framework of the Rayleigh-Schrödinger perturbation theory, using the Sturmian expansion of the generalized radial Coulomb Green function, we derive explicit analytical expressions for corrections to an arbitrary planar hydrogenic bound-state energy level, up to the fourth order in the strength of the perturbing magnetic field. In the case of the ground state, we correct an expression for the fourth-order correction to energy available in the literature.

Characteristics of a Two-Dimensional Hydrogen-like Atom

Using the customary and well-known representation of the radiation probability of a hydrogen-like atom in the three-dimensional case, a general expression for the probability of single-photon emission of a twodimensional atom has been obtained along with an expression for the particular case of the transition from the first excited state to the ground state, in the latter case in comparison with corresponding expressions for the three-dimensional atom and the one-dimensional atom. Arguments are presented in support of the claim that this method of calculation gives a value of the probability that is identical to the value given by exact methods of QED extended to the subspace {0, 1, 2}. Relativistic corrections ~ (Zα)4 to the usual Schrodinger value of the energy (~ (Zα)2) are also discussed.

Two-Dimensional Hydrogen-like Atom: Photon Emission and Relativistic Energy Corrections

Using the well-known solution Ψ S of the Schrodinger equation for an electron in the field of a nucleus (Ze) in polar coordinates, via which the spin-state-dependent Dirac spinor Ψ± obtained here is expressed, and by extending the QED methods to subspace {0; 1, 2}, we have calculated the probability of single-photon emission by a two-dimensional hydrogen-like atom with allowance made for the polarization and spin states. Relativistic energy corrections ∼(Zα)4 to the energy value have also been found. We show that the so-called contact interaction typical of a three-dimensional hydrogen-like atom also takes place in the twodimensional case, while the ordinary three-dimensional spin–orbit interaction is absent altogether.

Planar hydrogen-like atom in inhomogeneous magnetic fields: Exactly or quasi-exactly solvable models

We use a simple mathematical method to solve the problem of a two-dimensional hydrogen-like atom in the inhomogeneous magnetic fields B = (k/r)z and B = (k/r3)z. We construct a Hamiltonian that takes the same form as the Hamiltonian of a hydrogen-like atom in the homogeneous magnetic fields and obtain the energy spectrum by comparing the Hamiltonians. The results show that the whole spectrum of the atom in the magnetic field B = (k/r)z can be obtained, and the problem is exactly solvable in this case. We find analytic solutions of the Schrodinger equation for the atom in the magnetic field B = (k/r3)z for particular values of the magnetic strength k and thus present a quasi-exactly solvable model.

Relativistic corrections for a two-dimensional hydrogen-like atom in the presence of a constant magnetic field

In this paper, we study the relativistic corrections to energy eigenvalues of a 2D hydrogen atom in an external magnetic field. Using the power series expansion method, we have solved the nonrelativistic eigenvalue problem for arbitrary magnetic field strength. The first-order relativistic energy correction is calculated in the framework of the perturbation approach to the Dirac equation for the ground state and a few low-lying excited states.

Energy spectrum of a relativistic two-dimensional hydrogen-like atom in a constant magnetic field of arbitrary strength

We compute, via a variational mixed-base method, the energy spectrum of a two-dimensional relativistic atom in the presence of a constant magnetic field of arbitrary strength. The results are compared to those obtained in the non-relativistic and spinless case. We find that the relativistic spectrum does not present s states.

Planar Dirac electron in Coulomb and magnetic fields

The Dirac equation for an electron in two spatial dimensions in the Coulomb and homogeneous magnetic fields is discussed. This is connected to the problem of the two-dimensional hydrogenlike atom in the presence of an external magnetic field. For weak magnetic fields, the approximate energy values are obtained by a semiclassical method. In the case with strong magnetic fields, we present the exact recursion relations that determine the coefficients of the series expansion of wave functions, the possible energies, and the magnetic fields. It is found that analytic solutions are possible for a denumerably infinite set of magnetic field strengths. This system thus furnishes an example of the so-called quasiexactly solvable models. A distinctive feature in the Dirac case is that, depending on the strength of the Coulomb field, not all total angular momentum quantum numbers allow exact solutions with wave functions in reasonable polynomial forms. Solutions in the nonrelativistic limit with both attractive and repulsive Coulomb fields are briefly discussed by means of the method of factorization.

A (2+1)-dimensional fermion in the Coulomb and magnetic field backgrounds

In the problem of a two-dimensional hydrogen-like atom in a magnetic field background, we construct quasi-classical solutions and the energy spectrum of the Dirac equation in a strong Coulomb field and in a weak constant homogeneous magnetic field in 2+1 dimensions. We find some “exact” solutions of the Dirac and Pauli equations describing the “spinless” fermions in strong Coulomb fields and in homogeneous magnetic fields in 2+1 dimensions.

Analytic solution of a relativistic two-dimensional hydrogen-like atom in a constant magnetic field

We obtain exact solutions of the Klein-Gordon and Pauli-Schrödinger equations for a two-dimensional hydrogen-like atom in the presence of a constant magnetic field. Analytic solutions for the energy spectrum are obtained for particular values of the magnetic field strength. The results are compared to those obtained in the non-relativistic and spinless case. We obtain that the relativistic spectrum does not present s states.

Perturbation theory for the Stark effect in a two-dimensional hydrogenlike atom

We apply Rayleigh-Schrödinger perturbation theory to the Stark effect in a two-dimensional hydrogenlike atom and obtain large-order perturbation corrections to the energy by means of a recurrencerelation among moments of the wavefunction. The method is suitable for symbolic computation which enables one to calculate the perturbation corrections exactly. We explicity calculate the Stark shift for the ground state and the splitting of the first excited energy level thus showing, in the latter case, how to apply the method to degenerate states.

Theory of bound states associated withn-type inversion layers on silicon

Abstract : The ground state and two excited states of an electron bound to a charged impurity located at the interface between silicon and silicon dioxide have been investigated. The interface was taken to be parallel to a (001) plane and a static electric field p perpendicular to the surface was assumed. Calculations of the electron binding energies as functions of electric field were made for a bare charged impurity with screening by free carriers neglected. It was found that the binding energy increases with electric field and approaches that of the corresponding state of a two-dimensional hydrogen-like atom in the limit of infinite electric field. The case where the impurity is located within the oxide has also been investigated. (Author)