Two-particle Problem Research Articles

Excitons in quantum disks and disk arrays

We present a theory which describes the influence of the Coulomb attraction on the optical spectra for quantum dots within the envelope function formalism. Starting from a generalized non-local Elliott formula absorption and luminescence spectra are traced back to two-particle wave functions and energies. They are solutions of the corresponding Schrodinger equation for interacting electron-hole pairs in confinement potentials determined by the spatial variation of the band edges of the considered microstructure. A complete numerical analysis of the two-particle problem for flat quantum dots, i.e. disks for which the size quantization in z-direction is much stronger than that in the xy-plane is presented

NONLINEAR QUANTUM MECHANICS WITH NONCLASSICAL GRAVITATIONAL SELF-INTERACTION III: RELATED TOPICS

Some problems are considered in the framework of general quantum mechanics with gravitational self-interaction constructed earlier. A number of them were analyzed for the stationary situation. Here, the problem of gauge invariance generated by translations which do not violate the 3 + 1 splitting is studied. The notions of position and momentum operators are extended to the general case. The uncertainty relations are obtained for the uncertainty of the Ricci tensor and for uncertainties of the position and momentum of a particle. The correspondence between the stationary and nonstationary cases is examined at the level of variational principles. At least, the one-particle and two-particle problems in the Newtonian–Schrödingerian limit are considered; the latter problem is compared with the standard two-particle quantum problem to demonstrate the advantage of our approach.

Exact calculations of the ground state properties of hydrogen chemisorbed on a transition metal with one electron or hole in the energy band for the Anderson-Newns model

We investigate the exact solution of the Anderson-Newns model for H chemisorption obtained in the limiting case when the metal d band contains one electron (or one hole) and the adsorption problem becomes a two-particle problem. The chemisorption energy and the excess adatom charge are calculated exactly for an arbitrary electron density of states Δ(ϵ) of the substrate metal. It is shown that for realistic functions Δ(ϵ) the critical region appears in the Anderson model parameter space where the metal-adatom system exhibits a local moment at zero temperature. Within this region and outside it the chemisorption energy coincides with two different analytic functions of V and ϵ a which go smoothly into one another at the region boundary. The charge susceptibility and the set of characteristic functions, describing the electron-electron correlations, are calculated, and the strong correlation, intermediate valence and weak correlation regions are identified. The adatom density of states ρ(ω) is obtained exactly in the region ω < ϵ F for the two-electron system and in the region ϵ > ϵ F for the two-hole system.

Light scattering from a spherical particle on a conducting plane: I Normal incidence

An efficient procedure is developed for calculating light scattering from a spherical particle on a conducting plane. The problem is converted into an equivalent two-particle scattering problem by the method of images. The scattering solution for this two-particle problem is then obtained by the multipole expansion method. Mirror symmetry is exploited to simplify the calculation. Formulas are developed for the differential-scattering cross section and for the extinction and the absorption cross sections. Several example problems are calculated, and the exact solutions are compared with the results from two approximation procedures.

The two-body plus potential problem between quantum field theory and relativistic quantum mechanics (two-fermion and fermion-boson cases)

Starting with a pair of integrodifferential Bethe-Salpeter equations for two fermions interacting mutually and with an external static potential, we obtain a pair of compatible and separable coupled Dirac equations, between which the unwanted relative time variable can be easily eliminated. We use two different adaptations of the method proposed by Sazdjian for the two-particle (without external potential) problem, leading to two different (although equivalent) systems of coupled Dirac equations. We examine the instantaneous approximation and we test our methods in the helium atom problem. The last section is devoted to the fermion-boson plus external potential problem.

The two-body plus potential problem between quantum field theory and relativistic quantum mechanics (spinless case)

Starting with a pair of integrodifferential Bethe-Salpeter equations for two spinless particles interacting mutually and with an external static potential, we obtain a pair of compatible and separable coupled Klein-Gordon equations, between which the unwanted relative time variable can be easily eliminated. The method we use is a generalization of that proposed by Sazdijan for the two-particle problem, and the resulting equations are a generalization of the well-known Droz-Vincent-Todorov-Komar equations of relativistic quantum mechanics. We examine the instantaneous approximation and we test our methods in a simple case (Bethe-Salpeter kernel given by a single scalar particle exchange graph).

Spin effects in highly relativistic systems.

Coupled integral equations are developed variationally and solved numerically for the two-particle bound-state problem in quantum electrodynamics. The ansatz incorporates explicit photon degrees of freedom, and the resulting equations, when perturbatively reduced, yield the correct perturbative ${\ensuremath{\alpha}}^{4}$ coefficients. The results are of interest as they may be applied to the large spin-dependent effects in the highly relativistic bound systems of quantum chromodynamics.

Gauge symmetry and supersymmetric two-particle problem

An analogy between the removal of nonphysical relative time (or relative energy) in the supersymmetric two-particle problem and the account of local gauge invariance in supersymmetric quantum field theory is discussed. A group of gauge transformations for the Bethe-Salpeter amplitudes is suggested, the invariants of which are the relativistic three-dimensional (quasipotential) wave functions in the Logunov-Tavkhelidze approach. Subsidiary conditions imposed on the Bethe-Salpeter amplitudes in the Todorov approach are shown to be equivalent to appropriate gauge fixing.

The effects of elastic stress on the kinetics of ostwald ripening: The two-particle problem

We show that the kinetics of the classical two-particle Ostwald ripening problem can be altered radically by elastic stress induced by misfit (coherency) strains. The growth rates of the secondphase particles in the presence of elastic stress are shown to differ not only in magnitude but also in sign from the classical zero-stress treatment. For example, the larger particle does not always grow at the expense of the smaller particle, but there are large ranges of the thermophysical parameters in which inverse ripening occurs;i.e., the smaller particle grows at the expense of the larger particle. Elastic stresses also may induce significant morphological changes in the particles, as well as far stronger spatial correlations between the coarsening particles than in the classical zero-stress limit. The analysis is cast in terms of an initial value problem, allowing the growth rates of the particles to be determined as a function of readily obtainable materials parameters. We find that the influence of misfit strains on particle growth rates and regimes of inverse coarsening is significantly greater than that predicted from elastic energy analyses. We expect many of the qualitative features of the ripening behavior discussed herein to be observed in the multibody problem found in nature.

Exact two-particle effective interaction and superconductivity in the two-level Hubbard model

A two-dimensional two-level Hubbard model with on-site repulsions only, is studied. By solving the two-particle problem, the effective interaction in the lower band is calculated exactly in the empty band limit. This interaction is found to consist of two terms: an on-site repulsion and a nearest-neighbor resonant superexchange. Within the Hartree-Fock approximation it is found that high-${T}_{c}$ s-wave superconductivity develops in a variety of situations. The validity of this model for the high-${T}_{c}$ Cu-O superconductors is discussed.

Positron annihilation fromFcenters of alkali halide crystals

Two-photon annihilation of a positron from an F center in an alkali halide crystal has been studied theoretically by accurately solving with the integral transform (generator coordinate method) the two-particle (electron-positron) problem for two different model crystal potentials (an effective Coulomb potential and the Krumhansl-Schwarz cavity potential) which are often used in the theoretical description of F centers. Calculations are presented of the stability and relevant expectation values of the positron-F-center system, the two-photon annihilation lifetime \ensuremath{\tau} and the angular correlation distribution N(theta) for sixteen different alkali halides. Comparison is made with available experimental data and previous, less accurate calculations. It is demonstrated that a proper description of electron-positron correlation in the wave function is necessary in order to obtain meaningful results. We have also established a range of effective charges, ${Z}^{\mathrm{*}}$, of the defect that can capture an electron-positron pair in a stable state. For the Krumhansl-Schwarz potential, the present highly accurate calculations show that all sixteen crystals have stable electron-positron states in the defect.

Remarks on the spectral theory for the multiparticle-type Schrödinger operator

Mourre's method is used to prove the limiting absorption principle for the multiparticle Schrödinger operator under the same assumptions on the pair potentials as in the two-particle problem. It is shown that at high energies this principle is valid under wider conditions than on the whole spectral axis. The scattering theory for a Schrödinger operator whose potential decays at infinity in an essentially anisotropic manner is constructed in analogy with the three-particle problem.

Memory-function approach to retardation effects in the relativistic two-body problem.

The memory-function approach is applied to the relativistic two-particle problem. We thus obtain integral equations similar to the Bethe-Salpeter equation, taking all retardation effects fully into account and eliminating the relative time variable. The formalism is studied for scalar Klein-Gordon fields as well as for spinorial Dirac fields. For both cases the memory function and the integral kernels are calculated for a given model explicitly. The scalar-scalar model of Cutkosky is treated in detail and numerical binding energies are obtained, very close to those of the original Bethe-Salpeter equation, thus showing the extreme importance of retardation effects. As an example for Dirac fields the memory function for a proton-antiproton pair interacting via exchange of a ..pi../sup 0/ is established.

An exactly soluble one-dimensional, two-particle problem

We examine a one-dimensional problem involving two interacting particles in an external field for which the Schrödinger equation can be solved analytically for the ground state. The Hartree approximation for this system also has a simple analytic solution of the form previously derived by Foldy. Comparison of exact and Hartree solutions leads to precise estimates of the effect of correlations on the binding energy and on the two-particle probability distribution.

Relativistic two-particle effects in exotic atoms

The relativistic two-particle problem is reduced to the motion of a single 'effective particle', for which a new Dirac equation is constructed. The method is used to calculate recoil corrections to the Uehling potential. The case of a nuclear form factor is also discussed.

The influence of finite masses on the threshold behaviour of scattering in a three charged particle system

Excitation cross sections in new channels remain finite above threshold because of the degeneracy of the levels in the two-particle Coulomb problem. Below the thresholds there are series of resonances distributed according to the geometrical progression law. In cases with one complex eigenvalue lambda , the S-matrix elements in the vicinity of the threshold continuously trace circles on a complex plane: one circle below the threshold, the other above. The properties of the circle are investigated. The logarithmic angular dependence is obtained for the elastic scattering amplitude on excited states at any threshold and for any mass.

Rigorous second-order cumulant expansion

In several occasions (typically in the quantum-mechanical cases) the random variables of a many-body stochastic process are q-number variables rather than c-numbers. The canonical ensemble part i t ion function, which can be thought of as the moment generating function for the interaction part of the hamil tonian H of the system, is usually computed through a cumulang cluster expansion (1). Suck an expansion however is always performed under the assumption that the interaction part of H might be considered small enough to be kept only up to the first perturbative order. The motivation is twofold. On one hand the interaction energy is indeed often muck smaller than the single-particle energy. On the other hand the interaction part of H is very frequently restricted to a linear combination of pair interaction terms, and the lat ter circumstance allows a very convenient and elegant procedure whereby the problem is reduced to a two-particle problem. Briefly, the argument goes as follows (2). Let

Cluster model in reaction theory

A recent work by Rosenberg on cluster states in reaction theory is reexamined and generalized to include energies above the threshold for breakup into four composite fragments. The problem of elastic scattering between two interacting composite fragments is reduced to an equivalent two-particle problem with an effective potential to be determined by extremum principles. For energies above the threshold for breakup into three or four composite fragments effective few-particle potentials are introduced and the problem is reduced to effective three- and four-particle problems. The equivalent three-particle equation contains effective two- and three-particle potentials. The effective potential in the equivalent four-particle equation has two-, three-, and four-body connected parts and a piece which has two independent two-body connected parts. In the equivalent three-particle problem we show how to include the effect of a weak three-body potential perturbatively. In the equivalent four-body problem an approximate simple calculational scheme is given when one neglects the four-particle potential the effect of which is presumably very small.NUCLEAR REACTIONS Scattering theory. Effective few-particle formulation of multiparticle problems. Approximation schemes.

Multidimensional two-particle problem with noncentral potential

Multidimensional two-particle problem with noncentral potential

Theoretical study of positron annihilation fromFcenters of alkali halide crystals

The present work is a theoretical study of the two-photon annihilation characteristics of a positron from an $F$ center in alkali halide crystals. It has been experimentally established that positrons can annihilate with $F$-center electrons in KCl. In the present study, the two-particle problem is solved using several types of trial wave functions and for two choices of the model crystal potentials. The model crystal potentials used are those which have been found useful in theoretical descriptions of $F$ centers. The stability of such systems, the two-photon annihilation lifetime $\ensuremath{\tau}$ and the angular distribution $N(\ensuremath{\theta})$ have been calculated in all cases considered; $\mathrm{Li}X$, $\mathrm{Na}X$, $\mathrm{K}X$, and $\mathrm{Rb}X$, with $X$ being fluoride, choride, bromide, and iodide ions. A theoretical assessment is given of the prospects of the $F$-center-electron trapping and annihilating with the positron. Internal trends between the various alkali halides are presented. The lifetimes and angular distribution curves are compared with the experimental data which are available for KCl and NaCl. These calculations lend strong support to the experimental evidence of positron annihilation from $F$-center electrons, notwithstanding the relative simplicity of the crystal potential employed.