Showing papers on "Configuration space published in 1976"
TL;DR: In this article, a fast and accurate method of solving the Vlasov equation numerically in configuration space is described. But the method is very accurate and efficient, and it does not handle nonperiodic spatial boundary conditions.
802 citations
TL;DR: The R-matrix concept of atomic processes was first introduced by Wigner and Eisenbud as mentioned in this paper with the fundamental idea that configuration space describes the scattered particle, and the target is divided into two regions.
Abstract: Publisher Summary This chapter discusses the R-matrix concept of atomic processes. The R-matrix method was first introduced by Wigner and Eisenbud with a fundamental idea that configuration space describes the scattered particle, and the target is divided into two regions. In the internal region (r
498 citations
TL;DR: In this paper, asymptotic forms of the wavefunction and Faddeev components in configuration space are shown to determine uniquely the solutions of the Schrodinger or Faddeiv differential equations for 2 → (2, 3) and 3 → ( 2, 3), processes.
114 citations
TL;DR: In this paper, the possibility of replacing the nonrelativistic, three-dimensional configuration space by a so-called fuzzy configuration space, having an isomorphic Borel structure, is discussed.
Abstract: Starting from the idea that physical measurements may have residual imprecisions, the possibility of replacing the nonrelativistic, three‐dimensional configuration space by a so‐called fuzzy configuration space, having an isomorphic Borel structure, is discussed. A quantization procedure with respect to such a space is developed, and the invariance of nonrelativistic quantum mechanics under such Borel isomorphisms is exploited to prove the equivalence of this quantization procedure to the usual quantization procedure on a fuzzy‐free configuration space. Further, any Galilean invariant dynamics is shown to be insensitive to such imprecisions in the measurements of position and momentum.
61 citations
TL;DR: In this paper, the renormalization group method is reformulated for application to quantum mechanical many-particle systems situated on regular lattices in two or three-dimensional configuration space.
Abstract: The renormalization group method is reformulated for application to quantum mechanical many-particle systems situated on regular lattices in two or three-dimensional configuration space. The method is applied to the spin 1/2 XY model on the square lattice. The free energy curve is in excellent agreement with that obtained from high temperature series expansion. No critical temperature is found.
26 citations
TL;DR: In this paper, it was shown that the set of nth-order perturbation theory graphs for the N-particle in-state wave function can be combined and reduced to an equivalent set of factorized graphs in which the particle lines are numbered according to their ordering in momentum space at time t = -infinity.
Abstract: The one-dimensional many-body problem described by the nonrelativistic vertical-bar phi vertical-bar/sup 4/ theory of a complex scalar boson field (also known as the delta-function model) is studied. By induction, it is shown that the set of nth-order perturbation theory graphs for the N-particle in-state wave function can be combined and reduced to an equivalent set of factorized graphs in which the particle lines are numbered according to their ordering in momentum space at time t = -infinity. By carrying out loop integrations, each factorized graph is reduced to one of a finite number of skeleton graphs multiplied by a dressing function. The dressing functions are related to the multiparticle phase shifts which appear in Bethe's form of the N-body wave function. The skeleton graphs are shown to be associated with the ordering of particles in configuration space which characterizes Bethe's hypothesis. The analogy of a classical system of billiard balls is found to be helpful in interpreting the form of Bethe's hypothesis and the physical significance of the skeleton graphs. The use of factorized graphs in the scattering theory of statistical mechanics is demonstrated by a graphical calculation of the second virial coefficient. (AIP)
18 citations
TL;DR: The concept of nodal phase correlation (NPC) was introduced into many-particle quantum mechanics by as mentioned in this paper, which is a particular quantum correlation among particles, which is important for current-carrying states of isolated systems.
Abstract: The new concept of ''nodal phase correlation'' (NPC) is introduced into many-particle quantum mechanics. NPC is a particular quantum correlation among particles, which is important for current-carrying states of isolated systems. Mathematically it is characterized by a definite relation in configuration space between the phase and the modulus of the many-particle wave function. The significance of NPC is illustrated by showing the crucial role it plays in magnetic flux quantization and in certain macroscopic interference effects in superconducting rings. To this end the theory of these phenomena is partly rederived. The method id based on the BCS theory. Previous work is critically examined, in particular the single-particle pairing theory, which is considered to be a standard approach to magnetic flux quantization. This theory does not give rise to NPC. It is shown that the corresponding many particle wave function is in fact unrealistic and not consistent with the BSC theory of superconductivity. (AIP(
17 citations
TL;DR: Magnetic dipole states in 208 Pb were calculated in random phase approximation using a configuration space which includes also 2 h ω excitations in this article, and they found very collective 1 + -resonances around 19 MeV and 25 MeV.
16 citations
TL;DR: In this article, a mathematical model for the dynamics of a string of fuel bundles in axial flow is described, where each bundle is composed of fuel elements fixed between end plates, and the bundles are held together by a central support tube running through the assembly.
16 citations
TL;DR: In this paper, the radial equation for symmetric nonlocal potentials is investigated, using a configuration space approach and restricting the analysis to real k. The Fredholm determinants associated with the integral equations for the physical, regular, and Jost solutions are central to the development.
Abstract: Nonlocality is characteristic of potentials describing processes in which degrees of freedom are eliminated, and provides for a description of a much wider variety of phenomena than that encountered with short range local potentials. In this paper, properties of the radial equation for symmetric nonlocal potentials are investigated, using a configuration space approach and restricting the analysis to real k. Emphasis is placed on identifying those constraints associated with a local potential which are relaxed in going from a local to a nonlocal potential. The Fredholm determinants associated with the integral equations for the physical, regular, and Jost solutions are central to the development. Unlike the case for a short range local potential, for a nonlocal potential these Fredholm determinants can vanish for real knot-equal0. It is shown that when D/sup plus-or-minus/(k) =0 all other Fredholm determinants are zero as well. The properties of the solutions at the zeroes of the Fredholm determinants are discussed in this context, and the concepts ''spurious state'' and ''continuum bound state'' are clarified. The behavior of solutions is illustrated with examples. (AIP)
14 citations
TL;DR: In this paper, a relation between the size of the model space used in a specific calculation and the relevant properties of the effective residual interaction is established, which can be used to construct simple although realistic effective forces.
Abstract: Starting from a momentum space analysis of the two-body matrix elements, a relation has been established between the size of the model space actually used in a specific calculation and the relevant properties of the effective residual interaction. It turns out that the two-body transition density acts like a filter function on the Fourier transform of the force; it exhibits a distinct structure which clearly reflects the size and the detailed properties of the configuration space actually used. From an investigation of this filter function an equivalence criterion for different effective residual two-body interactions has been established both for closed and open shell nuclei. This result can be used to construct simple although realistic effective forces. As an example, a model for a separable residual interaction is proposed in which the corresponding parameters are being clearly related to the nuclear radius (i.e., the mass number), to the quantum numbers (i.e., the angular momentum) of the state under consideration and to the size of the configuration space used. For a number of examples this force has been applied successfully for the description of low energy properties of both closed and open shell nuclei. (AIP)
TL;DR: In this article, the most probable laser mode amplitudes are calculated on the basis of the stationary momentum space solution of the Fokker-Planck equation for a continuum mode laser oscillator.
01 Aug 1976
TL;DR: In this article, the coordinate-space asymptotic behavior of the wave functions for a system of three charged particles is described in all directions of configuration space, and boundary-value problems uniquely determining the wave function are formulated for the modified Faddeev differential equations.
Abstract: The coordinate-space asymptotic behavior of the wave functions for a system of three charged particles is described in all directions of configuration space. Boundary-value problems uniquely determining the wave functions are formulated for the modified Faddeev differential equations.
01 Jan 1976
TL;DR: The dynamical systems discussed in these lectures are systems with a finite number of degrees of freedom as discussed by the authors, and they can be obtained from a Lagrangian L(f(t), f(t, t), t).
Abstract: The dynamical systems discussed in these lectures are systems with a finite number of degrees of freedom. We shall consider both classical and quantum mechanical systems whose dynamical equations can be obtained from a Lagrangian L(f(t), f(t), t). The paths f map a time interval π = [ta, tb] into the configuration space M of the system. For nearly all physical systems M is a Riemannian manifold often multiply connected. A classical path q : T → M is a solution of the Euler Lagrange equation satisfying some boundary conditions which are chosen here to be q(ta) = a, q(tb) = b unless otherwise specified.
TL;DR: In this article, Dirac's charge quantization condition is derived by means of a canonical quantization procedure of an enlarged classical phase space in which the interaction constant is a dynamical variable.
Abstract: Dirac’s charge quantization condition is derived by means of a canonical quantization procedure of an enlarged classical phase space in which the interaction constant is a dynamical variable. The charge quantization condition follows by imposing a superselection rule. The method avoids string singularities and does not depend on spherical symmetry. The charge quantization condition is due solely to the topology of the enlarged classical configuration space.
TL;DR: The group structure associated with lattice systems without constraints is extended to systems with constraints and some of its physical consequences are investigated in this article, in particular low and high-temperature expansions, equilibrium equations and duality transformations.
Abstract: The group structure associated with lattice systems without constraints is extended to systems with constraints and some of its physical consequences are investigated. In particular low- and high-temperature expansions, equilibrium equations and duality transformations are analyzed within this framework. It is shown that a large class of systems with constraints whose configuration space has a group structure are “equivalent” to systems without constraints. These structures are illustrated with several examples.
TL;DR: In this article, the authors define a canonical system as a canonical manifold M plus a canonical vectorfield on M, and derive a unique kinematical interpretation from a set of Kinematical Axioms satisfied by the algebra of differentiable functions on M. If the phase space interpretation is adopted they are shown to describe the motion of masspoints in some configuration space under the influence of and interacting by arbitrary vector and scalar potentials.
Abstract: We define a canonical system as a canonical manifoldM plus a canonical vectorfield onM. For such systems a unique kinematical interpretation is deduced from a set of Kinematical Axioms satisfied by the algebra of differentiable functions onM. This algebra is required to contain a subalgebra which is maximal commutative under the Poisson bracket.M is shown to be diffeomorphic to the cotangent bundle over its quotient manifold, which is defined by the given subalgebra. Canonical systems satisfying these axioms are then classified. If the “phase space interpretation” is adopted they are shown to describe the motion of masspoints in some configuration space under the influence of and interacting by arbitrary vector and scalar potentials.
TL;DR: In this article, the authors consider a system consisting of one kind of atoms, each of which has only a set of p-levels with three-fold degeneracy and regard the p-state as a three dimensional vector, because the angular relations of any two p-states both in the configuration space and in the Hilbert space coincide each other.
Abstract: Why is a certain crystalline structure preferred to others in each element substance? This question seems to be a still difficult problem within the framework of the present solid state physics, in spite of the success of the band theory. One should, however, find out the origin which stabilizes the structure of the aggregate of atoms in the properties of individual constituent atoms. The aim of this paper is to report on an idea of our attempt to construct a primitive understanding of this problem on the basis of an idealized model Let us consider a system consisting of one kind of atoms each of which has only a set of p-levels with three-fold degeneracy. It is allowed to regard the p-state as a three dimensional vector, because the angular relations of any two p-states both in the configuration space and in the Hilbert space coincide each other: If the p-state with the angular wave function V3/4i-(J"·e) /lrl is denoted by the ket ]e), then
TL;DR: An addition theorem for the regular and irregular Coulomb functions by means of the symmetry properties of the Coulomb problem in analogy to that for the spherical Bessel functions is derived in this article.
Abstract: An addition theorem is derived for the regular and irregular Coulomb functions by means of the symmetry properties of the Coulomb problem in analogy to that for the spherical Bessel functions. The coefficients which enter in the addition theorem are closely related to 9j symbols of complex angular moment. For the computation of these coefficients a complete set of recurrence relations is given. In addition, some useful relations are presented for the Coulomb function in configuration space as well as in momentum space.
TL;DR: In this paper, the elastic scattering from an assembly of nonoverlapping potentials is treated in a $T$-matrix formalism in configuration space and the relation to the equations in $N$-body theory is briefly discussed.
Abstract: Quantum-mechanical elastic scattering from an assembly of nonoverlapping potentials is treated in a $T$-matrix formalism in configuration space. For potentials satisfying a certain fairly weak separation condition, the total $T$ matrix for the assembly of potentials is expressed in terms of the $T$ matrices for the individual potentials. The relation to the equations in $N$-body theory is briefly discussed.
TL;DR: In this paper, momentum space techniques were used to sum the complete class of outer rainbow graphs contributing to the IPI two-point function in λφ2ϕ2 theory.
TL;DR: In this article, the authors constructed the solution to a linearized Dyson equation for the IPI two-point function in λφ2ϕ2, where ϕ denotes a massless field.
Abstract: We construct the solution to a linearized Dyson equation for the IPI two-point function in λφ2ϕ2, where ϕ denotes a massless field. Special attention is given to the ambiguities arising in a conventional configuration space treatment. It is shown that proper account has to be taken of these ambiguities if one is to avoid serious discrepancies with regard to the anomalous dimension of ϕ. The connection between our nonperturbative result and perturbation theory as well as the cut-off dependence of the solution are also discussed.
01 Jun 1976
TL;DR: In this article, the configuration space representation of the operator-valued function Gα(z)=(z−K−Vα)−1 where K is the total kinetic energy of the three-particle system and Vα is the interaction of the pairΒγ.
Abstract: Analytic expressions for Green's functions of three-particle subresolvents (including one spectator particle) are given, i.e. the configuration space representation of the operator-valued functionGα(z)=(z−K−Vα)−1 whereK is the total kinetic energy of the three-particle system andVα is the interaction of the pairΒγ. A sum of separable potentials is well suited to exhibit the structure of the corresponding Riemann surface such that from these results also some conclusions may be drawn about the full Green's operator of the three-particle system.
TL;DR: In this paper, the authors presented a direct derivation of the three-body equations using a method developed in the nonrelativistic context, which applied the two-body scattering boundary condition at zero range to each of the pairs in the 3-body configuration space wave function.
Abstract: Publisher Summary This chapter analyzes zero range covariant three-body equations The fixed past-uncertain future program for a particulate quantum mechanics maintains covariance by using only free particle wave functions To make a dynamical theory out of this kinematic phenomenology, the first step is to obtain three particle dynamical equations containing only two particle observables as input Rather than simply taking the zero range limit in Brayshaw's theory, this chapter also presents a direct derivation of the three-body equations using a method developed in the nonrelativistic context The method described in the chapter applies the two-body scattering boundary condition at zero range to each of the pairs in the three-body configuration space wave function Essentially, the same derivation is given by Brayshaw applying the boundary condition at finite range, but it is felt that his more abstract treatment tends to conceal the essential simplicity of the approach As he gives the general result for a finite number of angular momentum states and also proves the on-shell unitarity of the resulting three-body amplitude, this chapter focuses only to the case of three spinless particles with zero total and relative angular momentum