Few-particle Systems Research Articles

Poincaré-invariant Lee model.

A relativistic generalization of the Lee model is developed within the framework of light-front dynamics. The model describes three particles {ital V}, {ital N}, and {theta} interacting through the virtual process {ital V}{r reversible}{ital N}+{theta}, with no antiparticles present. The model consists of ten generators which satisfy the commutation relations of the Poincare group. The Lorentz invariance of the {ital S}-matrix elements that arise is demonstrated. The dressed {ital V}-particle state is given, and the integral equations that arise in the {theta}{ital V}-{theta}{theta}{ital N} sector are derived. The interacting spin operator for the model is analyzed in some detail. The integral equations for the {theta}{ital V}-{theta}{theta}{ital N} sector are transformed to a new picture and a partial-wave analysis of the transformed equations is carried out. It is shown that the {theta}{ital V} elastic scattering amplitudes can be obtained from the solution of uncoupled, one-dimensional integral equations. The model has many of the features of few-particle systems involving pions and nucleons and moreover provides insight into dealing with the interacting angular momentum operators of light-front dynamics.

The properties of off-shell coulomb radial wavefunctions

Approximations to exact wave functions for the scattering of few-particle systems often involve components corresponding to the interaction of two of the particles “off the energy-shell”. Several examples arising in the collision of ions and photons with atoms are given. An expansion in partial waves leads to an off-shell radial wave function. The defining differential equation is solved here numerically with particular emphasis on the behaviour arising from two-body potentials of long-range Coulomb form. The transition to shell of the radial wave functions, Jost functions and solutions andT-matrix elements is discussed for both short-range and Coulomb potentials. It is shown that the approximation of a Coulomb potential by a shorter-range form involves little error when sufficiently far off the energy shell.

Semiclassical calculations of (Z+e−+e+) energy spectra

A modified Einstein–Keller–Maslov quantization, based on a quasi-one-particle model has been applied in calculating resonance energy levels for a heavy charge Z plus electron and positron systems. The method starts with the strict correlation plane configuration and then allows for the vibrational modes excitation. Calculations have been carried out for [H;e+] and [He+;e+] transient systems, and series of autoionizing levels are obtained. Possible improvements of the method as well as further extensions to other few-particle systems are considered, too.

On exact analytical solutions for the few-particle Schrodinger equation. IV. The asymptotic form and normalisability of the wavefunction

For pt.III see ibid., vol.16, nol.18, p.4237-53 (1983). Formal series solutions for the Schrodinger equations of few-particle systems contain infinitely many associated with the normalisability, i.e. square-integrability of the wavefunction. The energy is one of these parameters. A method for determining the parameters by examining the asymptotic behaviour of the series has been developed. No integration, matrix inversions or trial and error procedures are involved. The method is directly applicable to the 1S states of two-electron atoms. If the wavefunction is expressed as a multipole expansion in spherical polar coordinates the radial functions have asymptotic properties which give rise to relations between the parameters. Values assigned to the parameters not specified by these relations (one per multipole) ensure that the wavefunction is normalisable.

On exact analytical solutions for the few-particle Schrödinger equation. II. The ground state of helium

Analytical methods for solving the Schrödinger equation directly can be applied to few-particle systems. This is illustrated by deriving a solution to the first-order perturbation equation for the ground state of helium. This solution is the in form of a partial wave expansion in spherical polar coordinates with Legendre polynomials as the angular functions. The radial functions include polynomials and exponential integral functions. Arbitrary parameters in the formal solution, associated with the square-integrability of the wavefunction, are identified. Their values are deter­mined by a least-squares method. The same arbitrary parameters occur in formal solutions of the higher-order perturbation equations. It is evident that a similar treatment can be applied to these equations, and to the eigenvalue problem.

Phenomenological Study of Three-Body Force Based on the Energy Levels of the Alpha-Particle

The contributions of the three· body force to the energies of the alpha-particle are estimated both in the ground state and in the first three T = 0 negative parity (j = 0', 2 -, l' ) excited states. These contributions just correspond to the deviations of the theoretical values of the energies of the alpha-particle, which are obtained with the realistic two-body force only, from the experimental ones not only in the ground state but also in the excited states. From this fact, it is shown that the three-body force is indispensable to reproducing the energies of nuclei with realistic nuclear interactions and that the three-body force may be determined in a phenomenological way. There have been no experimental phenomena which constitute a direct evidence of the nuclear three-body force. In the present paper, it is pointed out that there exists a fact taking the place of such phenomena, and it is shown that rough features of the three-body force such as its strength and type can be deduced. Recently A TMS method 1) has been proposed to solve few-particle systems under realistic two-body interactions. This method is of powerful one. In fact, the method gives the very close upper and lower bound energies of the ground states for three- and four-alpha-particle systems, where the alpha-particles in­ teract through phenomenological inter-particle potential with a strong repulsive part. 2 ) This means that ATMS method can make it possible to solve almost completely these alpha-particle systems. Using this method, wave functions and energy values have been evaluated for the triton and the alpha-particle with several realistic two-body forces. 1),3) We pay attention to the deviations of the theoretical energy values obtained with the realistic two-body force from the experimental ones in the first three T =0 negative parity (j=0-, 2-, r) excited states of the alpha-particle as well as in the ground state. The theoretical value of the ground state energy of the alpha-particle, which is obtained by the use of ATMS method, deviates to a large extent from the experimental one in any case of the realistic two-body forces, and this deviation is considerably large compared with the differences among several theoretical values calculated with different types of the two-body force.

Two-particle resonance poles and the energy spectra of three- and four-particle systems

The position of the two-particle resonance poles is shown to be the dominant factor in a number of interesting phenomena appearing in the energy spectra of bound three- and four-particle systems with separable two-particle interactions. The validity of the bound-state pole dominance assumption in few-particle systems is shown to depend on the location of the two-particle resonance poles. These results generalize the ground state collapse obtained earlier for the Tabakin rank-one separable interactions in the triton. However, our results also indicate that the collapse does not necessarily occur for more general nonlocal interactions.

Effective Hamiltonian for few-particle systems in polar semiconductors

The interaction between carriers in semiconductors is considered, including the coupling with LO-phonon field. The effective Hamiltonian, derived by the variational method, is valid for any values of the electron-phonon coupling constant. Applying it to the problem of exciton in Cu 2O, CuBr and CuCl, we obtain the energies which agree well with the experimental spectra. Owing to its simple form this Hamiltonian may be useful for few-particle systems.

Erratum: ’’Statistical analysis of classical few-particle systems’’

First Page

Dense-fluid shear viscosity via nonequilibrium molecular dynamics

A novel fluid-transport calculation by computer simulation, via nonequilibrium molecular dynamics, of laboratory methods of transport measurement is described. Shear viscosity of soft-sphere (${r}^{\ensuremath{-}12}$ potential) and Lennard-Jones particles (${r}^{\ensuremath{-}12}\ensuremath{-}{r}^{\ensuremath{-}6}$ potential) has been obtained from molecular dynamic modeling of Couette flow. Soft-sphere deviations from Enskog theory are similar to those found for hard spheres by Alder, Gass, and Wainwright, using time-correlations of equilibrium molecular dynamic system fluctuations. For the Lennard-Jones shear viscosity near the triple-point region, there is agreement between the equilibrium calculation of Levesque, Verlet, and Kurkijarvi and the nonequilibrium results using 108 atoms in a cube. However, systems two and three cubes wide give lower results, which, when extrapolated with inverse width, yield close agreement with the experimental argon shear viscosity. Comparison of the Lennard-Jones shear viscosity with experimental argon data along the saturated vapor-pressure line of argon confirms our successful simulation of macroscopic viscous flow with few-particle nonequilibrium molecular dynamic systems. A new result of the nonequilibrium molecular dynamics is the characterization of nonequilibrium distribution functions, which might provide the basis for a perturbation theory of transport. Since momentum transport is primarily accomplished by the repulsive potential core for high temperatures, the Lennard-Jones shear viscosity must behave like the soft-sphere system for high temperatures [viscosity divided by ${(\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e})}^{\frac{2}{3}}$ is a function of density divided by ${(\mathrm{temperature})}^{\mathrm{\textonequarter{}}}$]. In fact, the calculated excess shear viscosity (that part above the zero-density temperature dependence) has been successfully correlated in terms of the 12th-power scaling variables for temperatures as low as the critical value (along the freezing line). The utilization of soft-sphere scaling variables yields relatively simple functions for describing both the excess shear viscosity and the thermal-conductivity behavior throughout the fluid phase. The introduction of these scaling variables also clearly reveals two features: (i) weak temperature dependence, and (ii) the sign of the temperature derivative at constant density (negative for shear viscosity and positive for thermal conductivity). While both of these features have been experimentally observed in simple fluid experimental data, their cause has not been previously traced to the dominance of the core potential. Thus, the soft-sphere scaling variables should be useful for correlating experimental data.