Integral equations for three-particle systems with coulomb interaction

Modified Faddeev equations are formulated that may be used for studying the wave functions below and also above the threshold.     

Ground-state energy of strongly correlated electronic systems

We calculate the ground-state energy of strongly correlated electrons by using a twodimensional (CuO₂)_N system as an example. It constitutes the most important structural element of the high-T _c superconducting materials. The strong correlations are treated by projection technique. For that purpose a new form of the free energy is derived, which allows for applying that method. The ground-state energy for the half-filled band case is calculated by using a Padé approximation which agrees with series expansions up to order (t/( _p – _d ))⁶. Heret is thep-d hopping matrix element and _p , _d are the orbital energies of the O(2p _x(y)) and electrons.     

Adiabatic similarity and dependence of the energy of molecular systems on the masses of particles

The dependence of the energy of three-particle molecules on their masses is examined. It is shown that such molecules with the same values of the ratio of the reduced masses for motion in a “fast” and “slow” Jacobi coordinates have the property of adiabatic similarity: In the adiabatic approximation, their energies are proportional to the reduced masses. This allows information on the energy of molecules symmetric in the masses of particles to be extended to asymmetric molecules adiabatically similar to the symmetric molecules. For molecules with arbitrary masses of the particles, an analytic expression for the adiabatic energy and a formula approximating the exact energy are constructed using the principle of adiabatic similarity. Along with the adiabatic energy, which is the lower bound of the exact energy, a simple procedure is considered for determining the upper bound of the energy of asymmetric molecules from the energy of their symmetric counterparts. Based on these results, values of the lower and upper energy bounds are calculated and an approximation of the exact energy is obtained for 43 three-particle molecular systems.     

Screening effect of the coulomb interaction on the energy levels of atomic and molecular systems

The energies of atomic and molecular systems with a screened Coulomb interaction are expressed in terms of properties of the system with a conventional Coulomb interaction to the third order in the screening constant. The energy levels of an H atom, an H^? ion, and nonadiabatic systems, namely, the positronium ion e ⁺ e ^? e ^? and the positronium molecule e ⁺ e ⁺ e ^? e ^?, are calculated for screened Coulomb interactions of charged particles. It is demonstrated that, as the screening constant of the Coulomb interaction increases, the stability of the hydrogen atom, the H^? ion, and the positronium molecule e ⁺ e ⁺ e ^? e ^? decreases, whereas the stability of the positronium ion e ⁺ e ^? e ^? increases after a slight decrease.     

The upper and lower bounds of energy for nuclear and coulomb few-body systems

The upper and lower bounds of energy are found for three-, four-, and five-particle nuclear and Coulomb systems in the framework variational method with the trial functions of exponential and Gaussian types. The two-sided estimates of energy not only allow one to fix the limits for the exact value of energy but also provide an additional opportunity for extrapolation of the variational estimates to the exact value of energy. This allows one to reduce the volume of calculations by shortening the number of trial functions without the loss of accuracy.     

Charging energy and stability of charged metallic particles

The stability of a charged metallic particle is investigated within the classical theory. The minimal number of atoms for which a charged spherical particle is stable against the spheroidal deformation, which relaxes the energy cost due to the Coulomb repulsion between extra charges, is derived. It is suggested that, for negatively charged particles, an extra electron virtually bound to particle is emitted before the fragmentation due to the Coulomb repulsion between extra charges will occur.     

Ground-state energy of quantum liquids

The kinetic (K ₄ ⁰ (n) and K ₃ ⁰ (n)) and potential (V ₄ ⁰ (n) and V ₃ ⁰ (n)) energies of ⁴He and ³He atoms have been found from the law of corresponding states and the experimental data on the dependence of the ground-state energies E ₄ ⁰ (n) and E ₃ ⁰ (n) on the density of the isotopes ⁴He and ³He. In the approximation of structureless quantum liquid, the potential energies are equal, V ₄ ⁰ V ₃ ⁰ (n) = (n), and the kinetic energies are inversely proportional to the atomic mass, $K_4^0 (n) = \frac{3} {4}K_3^0 (n)$ . The potential energy given by the expression V ⁰ = 4E ₄ ⁰ ? 3E ₃ ⁰ to a high accuracy is linear in the density n, which is associated with nearly an absence of short-range order in liquid helium. The kinetic energy of liquid ⁴He is given by the expression K ₄ ⁰ = 3(E ₃ ⁰ ? E ₄ ⁰ ), which agrees with the experimental data on neutron scattering in liquid ⁴He. The quantities K ₄ ⁰ (n) and K ₃ ⁰ (n) determine the scale of all thermodynamic characteristics in the temperature range where the effects of the particle statistics can be neglected.     

Symmetry and boundness of four-particle coulomb systems

The problem of boundness of a ⁺ b ⁺ c ⁻ d ⁻ four-particle Coulomb systems (quadrions) is studied versus the masses of the particles involved. Inequalities that make it possible to deduce that, if some reference quadrions form a bound state, the same is true for a large number of quadrions formed by particles having various masses were derived. A compendium of calculations for energies of reference systems that possess various symmetries [positronium molecules (e ⁺ e ⁺ e ⁻ e ⁻) and quadrions of the a ⁺ b ⁺ b ⁻ b ⁻, a ⁺ b ⁺ a ⁻ b ⁻, and a ⁺ a ⁺ b ⁻ c ⁻ types] is given, and groups of bound asymmetric quadrions corresponding to them are determined. An inequality for kinetic energies of particles that makes it possible to find out, by using asymmetric reference systems, whether specific quadrions are bound is obtained. It is shown that the boundness of many quadrions is ensured by the boundness of respective three-particle systems. The entire body of the present results permits proving that, of the total number of 406 quadrions containing electrons, muons, pions, kaons, protons, deuterons, and tritons and their antiparticles, 227 quadrions are bound.     

On the Kohn variational principles for three-particle systems

The Kohn-type variational principles are formulated, which can be used for calculating the (2 → 3), (3 → 2), and (3 → 3) scattering amplitudes if the two-particle wave functions are known.     

On the masses of elementary particles

We make an attempt to describe the spectrum of masses of elementary particles, as it comes out empirically in six distinct scales. We argue for some rather well defined mass scales, like the electron mass; we elaborate on the assumption that there is a minimum mass associated to any electric charge. Another natural mass scale is Λ = Λ _QCD coming arbitrarily at quantizing a classically conformal SU(3) _c theory. Indeed, some scales of masses will cover also masses of composite particles or mass differences. We extend some plausible arguments for other scales, as binding or self-energy effects of the microscopic forces, plus some speculative uses, here and there, of gravitation. We also consider briefly exotics like supersymmetry and extra dimensions in relation to the mass scale problem, including some mathematical arguments (e.g. triality), which might throw light on the three-generation problem. We also address briefly the issues of dark matter and dark energy.     

Ground-state energy of a classical artificial molecule

We study the ground-state energy of a classical artificial molecule formed by two-dimensional clusters (artificial atoms) of N/2 charged particles separated by a distance d. For the small molecules of N = 2 and 4, we obtain analytical expressions for this energy. For the larger ones, we calculate the ground-state energy using molecular dynamics simulation for N up to 128. From our numerical results, we are able to find out a function to approximate the ground-state energy of the molecules covering the range from atoms to molecules for any inter-atom distance d and for particle number from N = 8 to 128 within a difference less than one percent from the MD data.     

Ground-state energy of the electron liquid in ultrathin wires

The ground-state energy and the density correlation function of the electron liquid in a thin one-dimensional wire are computed. The calculation is based on an approximate mapping of the problem with a realistic Coulomb interaction law onto exactly solvable models of mathematical physics. This approach becomes asymptotically exact in the limit of a small wire radius but remains numerically accurate even for modestly thin wires.     

Charge symmetry breaking interactions and coulomb energy differences

A detailed study of the possibility that the long standing discrepancies between theory and experiment in the values of Coulomb displacement energies have their origin in the charge asymmetry of the nuclear force is presented. A review of the present status of the theory of Coulomb displacement energies of mirror states is given. We have constructed a phenomenological charge asymmetric potential that can remove the Coulomb energy discrepancies for several mirror nuclei yet having only minimal effect on the ¹S₀ nucleon-nucleon scattering length. The form of this potential is however not compatible with the contributions due to charge asymmetric meson exchange processes that have been considered in the literature. We assume that these meson exchange effects can be reasonably described, up to strength factors, by a sum of one-pion, one-scalar-meson (σ) and one-vector-meson exchange potentials. The need for some additional type of charge asymmetric effect, e.g. three-body forces, is pointed out.