Solutions In Configuration Space Research Articles

Semiclassical theory and the Koopman-van Hove equation * *Contribution to the Special Collection ‘Koopman Methods in Classical and Quantum-Classical Mechanics’.

The phase space Koopman-van Hove (KvH) equation can be derived from the asymptotic semiclassical analysis of partial differential equations. Semiclassical theory yields the Hamilton–Jacobi equation for the complex phase factor and the transport equation for the amplitude. These two equations can be combined to form a nonlinear semiclassical version of the KvH equation in configuration space. There is a natural injection of configuration space solutions into phase space and a natural projection of phase space solutions onto configuration space. Hence, every solution of the configuration space KvH equation satisfies both the semiclassical phase space KvH equation and the Hamilton–Jacobi constraint. For configuration space solutions, this constraint resolves the paradox that there are two different conserved densities in phase space. For integrable systems, the KvH spectrum is the Cartesian product of a classical and a semiclassical spectrum. If the classical spectrum is eliminated, then, with the correct choice of Jeffreys–Wentzel–Kramers–Brillouin (JWKB) matching conditions, the semiclassical spectrum satisfies the Einstein–Brillouin–Keller quantization conditions which include the correction due to the Maslov index. However, semiclassical analysis uses different choices for boundary conditions, continuity requirements, and the domain of definition. For example, use of the complex JWKB method allows for the treatment of tunneling through the complexification of phase space. Finally, although KvH wavefunctions include the possibility of interference effects, interference is not observable when all observables are approximated as local operators on phase space. Observing interference effects requires consideration of nonlocal operations, e.g. through higher orders in the asymptotic theory.

Searching Multiple Approximate Solutions in Configuration Space to Guide Sampling-Based Motion Planning

High-dimensional configuration space is usually searched using sampling-based motion planning methods. The well-known issue of sampling-based planners is the narrow passage problem caused by small regions of the configuration space that are difficult to cover by random samples. Practically, the presence of narrow passages decreases the probability of finding a solution, and to cope with it, the number of random samples has to be significantly increased, which also increases the planning time. By dilating the free space, e.g., by scaling-down or thinning the robot (or obstacles), narrow passages become wider, which allows us to compute an approximate solution. Then, the configuration space can be sampled densely around the approximate solution to find the solution of the original problem. However, this process may fail if the final solution is too far from the approximate one. In this paper, we propose a method to find multiple approximate solutions in the configuration space to increase the chance of finding the final solution. The approximate solutions are computed by repeated search of the configuration space while avoiding, if possible, the already discovered solutions. This enables us to search for distinct solutions leading through different parts of the configuration space. The number of approximate solutions is automatically determined based on their similarity. All approximate solutions are then used to guide the sampling of the configuration space. The performance of the proposed approach is verified in scenarios with multiple narrow passages and the benefits of the method are demonstrated by comparing the results with the state-of-the-art planners.

Poisson–Vlasov in a strong magnetic field: A stochastic solution approach

Stochastic solutions are obtained for the Maxwell–Vlasov equation in the approximation where magnetic field fluctuations are neglected and the electrostatic potential is used to compute the electric field. This is a reasonable approximation for plasmas in a strong external magnetic field. Both Fourier and configuration space solutions are constructed.

Vortex solutions in the noncommutative torus

Vortex configurations in the two-dimensional torus are considered in noncommutative space. We analyze the BPS equations of the Abelian Higgs model. Numerical solutions are constructed for the self-dual and anti-self dual cases by extending an algorithm originally developed for ordinary commutative space. We work within the Fock space approach to noncommutative theories and the Moyal-Weyl connection is used in the final stage to express the solutions in configuration space.

Multiple solutions in configuration space in a chiral nuclear model

A saturatinf chiral lagrangian allows for two different types of nuclear structures: strongly bound shell like nuclei and weakly bound “normal” nuclei. These two distinct spectra exist due to a bifurcation in configuration space, that occurs when the nuclear repulsion (parametrized by the vector coupling constant) is increased to its nuclear matter value.

Configuration space Faddeev continuum calculations. I.n−dscattering length

Formulation of the zero-energy Faddeev-type scattering equations in configuration space is discussed. Numerical solutions are obtained using spline techniques. Scattering lengths are calculated for the $s$-wave $\mathrm{NN}$ interaction models of Malfliet and Tjon; comparison with previously published results and Kohn variational estimates are made. Faddeev amplitudes and Schr\"odinger wave functions are plotted. Significant features of the configuration space solutions are identified.NUCLEAR REACTIONS $^{2}\mathrm{H}(n, n)^{2}\mathrm{H}$, $E=0$ MeV; $n\ensuremath{-}d$ scattering lengths; configuration space Faddeev calculation.

Mode properties of optical fibers with lossy components by the propagating beam method

The theory of optical fibers has been concerned mainly with propagation in lossless waveguides. Although real fibers are composed of core materials with extremely small absorption coefficients, they may have claddings and jackets with substantially higher losses. These components can in turn selectively attenuate certain higher-order guided modes. The propagating beam method utilizes configuration space solutions to a scalar wave equation to generate the mode propagation constants and eigenfunctions for a general class of weakly guiding fibers. This method has now been generalized to treat fibers with absorbing components such as claddings and jackets. It is now possible to compute with this generalized method both the mode attenuation coefficients and mode eigenfunctions for such fibers. Results are given for planar and circularly symmetric waveguides with lossy claddings and jackets.

Light-cone quantization of dual models and structure functions

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

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.

Bethe-Salpeter Equation for Triplet Amplitude in Intermediate-Vector-Boson Theory of Weak Interactions

The Bethe-Salpeter equation for the triplet amplitude in the intermediate-vector-boson theory of weak interactions is shown to have a unique solution in configuration space; the solution has an essential singularity at the light cone, and does not have a Fourier transform. If the neutrino mass is zero, there exists a prescription that regularizes the amplitude to zero on the mass shell.

Inequivalent Hamiltonian Formalisms in Electrodynamics

In treating the electric dipole approximation of a charged particle undergoing self-interaction, it is shown how, in the cases of a charged free particle and a charged oscillator, the same equations of motion may be derived from two inequivalent Hamiltonian formalisms. The first formalism is the Dirac formalism of Schiller and Schwartz, which must be satisfied by all configuration space solutions generated by the total Hamiltonian. It is pointed out that in order to ensure that the configuration space solutions are consistent with the latter formalism, one must recognize that the constraints impose modifications on the commutation rules satisfied by averaged field variables. While the first formalism is general, the second formalism can be constructed only after one has solutions expressed in terms of a Fourier decomposition of the radiation field. The second formalism is an interaction picture formalism where the radiation field part of the total Hamiltonian is the generator of time translations. The latter formalism is used to give a new derivation of Kramers' integral for the zero-point level shift of an oscillator. We also discuss some asymptotic boundary conditions and comment on a recent paper by Sokolov and Lysov.

Electric Dipole Approximation and the Canonical Formalism in Electrodynamics

We use the electric dipole approximation to study the problem of finding commuting solutions of coupled equations of motion. We point out that for a charged particle in an external radiation field, the solutions of the coupled equations cannot be considered independent in the sense of commuting with one another if the homogeneous solutions are assumed to have the commutation properties of uncoupled variables. We explicitly treat the case of a charged free particle and a charged harmonic oscillator in an external radiation field. We indicate that for a retarded (advanced) self-field, the free particle fits into a canonical formalism while the oscillator does not. For a stationary self-field, both the free particle and the oscillator fit into a canonical formalism. We show that the Fourier transforms of the configuration space solutions (based on ${e}^{i\mathrm{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathrm{x}}$ and ${e}^{i\ensuremath{\omega}t}$) do not exist. In the latter connection, we point out that earlier treatments of the oscillator by Sokolov and Tumanov and Norton and Watson contain misleading results as a consequence of their using Fourier transforms.