Showing papers on "Configuration space published in 1973"

Formulation of the Binary-Encounter Approximation in Configuration Space and Its Application to Ionization by Light Ions

Abstract: The binary-encounter approximation (BEA) is transformed from momentum space to configuration space. In this frame the impact-parameter representation allows one to calculate a variety of quantities pertinent to the general problem of ionization. Among these are cross sections for proton ionization of hydrogen and helium; in the latter case, croas sections for ejection of both electrons are also given. A number of tables and formulas are given, enabling one to correct the simple hydrogenlike-model predictions of the BEA for effects which arise in multielectron atoms. Multiple-ionization probabilities (K + L shell) are calculated and compared to experimental results and to the predictions of the semiclassical approximation.

An Infinite Particle System with Zero Range Interactions

Abstract: Existence and uniqueness is proved for Spitzer's zero range interaction process. It is then shown that a certain class of measures on the configuration space is invariant for this process.

Related integral theorem. II. A method for obtaining quadratic constants of the motion for conservative dynamical systems admitting symmetries

Abstract: By use of Lie derivatives, symmetry mappings for conservative dynamical systems are formulated in terms of continuous groups of infinitesimal transformations within the configuration space. Such symmetry transformations, called trajectory collineations, may be interpreted as point mappings which drag along coordinates and geometric objects as they map trajectories into trajectories. It is shown that if a conservative dynamical system admits a trajectory collineation, then in general a new quadratic (in the velocity) constant of the motion will result from the deformation of a given quadratic constant of the motion under such a symmetry mapping. The theory is applied to obtain the group of symmetry transformations and concomitant constants of the motion associated with the deformations of the energy integral for the Kepler problem and for the isotropic simple harmonic oscillator. The Runge‐Lenz vector of the Kepler problem and the symmetric tensor constant of the motion for the three‐dimensional oscillator...

General relativity, partial differential equations, and dynamical systems

Abstract: In this paper we study two aspects of the Einstein equations of evolution for an empty spacetime. In the first part (§§1-3) we give a simple direct proof that the differentiability of the Cauchy data is maintained for short time. In the second part (§§4-5) we sketch how, on a suitable configuration space. Einstein's equations can be considered as forced geodesics modified by terms which reflect a moving coordinate system equipped with its own system of clocks. Both of these topics will be presented in more detail in paper [15], [12] by Fischer and Marsden.

Symmetry mappings of constrained dynamical systems and an associated related integral theorem

Abstract: By use of Lie derivatives symmetry mappings of constrained conservative dynamical systems are formulated in terms of continuous groups of infinitesimal transformations within the configuration space. Such symmetries are called ``natural trajectory collineations'' in that the total energy has the same fixed value along each trajectory of the natural family, this value being preserved by the symmetry. It is found that these natural trajectory collineations must be conformal motions subject to an additional restriction dependent upon the potential. The corresponding groups of natural trajectory collineations are obtained for a flat configuration space with potential energy functions with rotational invariance about a point. A specialization of the theory to an indefinite Riemannian space‐time shows that homothetic transformations are necessary and sufficient to map a natural family of time (space)‐like geodesics into itself. A related integral theorem for constrained dynamical systems admitting linear or quadratic constants of the motion is obtained and illustrated. This theorem shows that in general a new constant of the motion will be obtained by deformation of an existing constant of the motion under a natural trajectory collineation.

Thermodynamic duality for classical systems of arbitrary spin

Abstract: Given a classical spins system, namely, a set of spin sites of maximum spins inv-dimensional space along with a Hamiltonian defined on the possible spin configurations, a general method is described for constructing a large class of dual lattices of the same spin. The method utilizes the commutative group structure with which the configuration space is endowed.

Uncertainty epidemics among interacting particles

Abstract: It is shown that a small uncertainty in the data describing a system ofN particles in a configuration space of more than one dimension is amplified in time due to their interaction, and that after a timeti, called the total ignorance time, only statistical predictions about the system are possible. Expressions forti are derived and it is argued that in a universe constructed according to a hierarchy of levels a deterministic description is impossible.

On relativistic quantum field theory in configuration space

Abstract: A relativistic quantum field theory in configuration space is considered and the consequences discussed. (UK)

Invariant Manifolds in Celestial Mechanics

Abstract: Several examples of invariant manifold problems in celestial mechanics are discussed including the determination of the topological character of the Kepler energy manifold and the derivation of a priori bounds for the artificial satellite problem.

Light-cone quantization of dual models and structure functions

Abstract: The aim of this work is to study some of the light-cone aspects of the dual-resonance models and to show the agreement with the usual results of the light-cone analysis based on other models. To accomplish this program we quantize on the light-cone the Lagrangian formulation of the DRM. Our quantization procedure is based on the derivation of the completeness relation for thec-number solutions of the wave equation. An interesting result of this analysis is that thec-number solutions in configuration space form a complete set atx+=const only in the space of the functionsf(x) such that \( \mathop {\lim }\limits_{x^ - \to \pm \infty } \)f(x)=0. From the quantization of the field we obtain the current commutators and the explicit expressions for the bilocal operators. Using such expressions we calculate the structure functions of the deep inelastic scattering up to the second order in the interaction term which reproduces the dual amplitude. We find thatF2(ξ) has the correct Regge behaviour forξ→0.

A Soluble Self-Consistent Nuclear Model

Abstract: The self-consistent, independent-particle Hartree equations for N fermions interacting with a separable (in configuration space) two-body potential are solved exactly in one and three dimensions. The exclusion principle is accounted for by allowing only one particle per orbital state. The possibility of only a finite number of particles being bound self-consistently, for given two-body force parameters, is studied and the effect of introducing the observed (in real nuclei) approximate constancy with particle number of particle density and common well depth is also considered.

A new form of the eigenfunctions of the Heisenberg Hamiltonian in one dimension

Abstract: The exact solution of the eigenvalue problem of a symmetric Heisenberg Hamiltonian in one dimension is investigated in a new form, in a spin-wave representation. This avoids the specific ordering of lattice points which characterizes the one-dimensional solution in configuration space, and may indicate a way to an extension to several dimensions. The familiar phase factors of the configuration space solution are determined independently of the dynamics of the equations, from the condition that there can be only one spin 1/2 reversal on a lattice point, and from a zero-momentum condition resulting from the symmetry of the Hamiltonian. The new form of the solution reveals a number of new relationships, and a new method is presented to verify that these wavefunctions satisfy the wave equation.

On Relativistic Quantum Field Theory in Configuration Space

Abstract: Let us consider a relativistic quantum field theory. Let ak* be the creation operator of a free particle of momentum k and q~t-)(x) be the corresponding standard creation part of the configuration space field at time equal to zero. Here the word 'standard' means that qSt-)(x) is obtained from ak* by means of a Fourier transformation with the relativistic weighing factor (e.g. kol). A non-standard creation configuration field q~'(-)(x) will be a different linear combination of the ak*'S. It has been stated that while ak* ]0) is an eigenstate of momentum the state ~b(-)(x)10) is not an eigenstate of position (cf. e.g. Lurif, 1968). If right, this ill-configuration behaviour of the field (which does not happen in the non-relativistic case) would obscure the physical interpretation of the theory. In order to have a better configuration field (if it exists), one should try a field q~'c-)(X) which would be a different linear combination of the ak*'S, i.e. the creation part of a non-standard configuration field; cf. Wightman & Schweber (1955) who have shown that the change ~bc-)(x) ~ q~'(-)(X) corresponds to the replacement, in the one-particle subspace, of an incorrect position operator Xep by the correct one Xop. It would then be expected that