Classical spin in a potential field

We consider spatially two dimensional Madelung fluid whose irrotational motion reduces into the Schr\"odinger equation for a single free particle. In this respect, we regard the former as a direct generalization of the latter, allowing a rotational quantum flow. We then ask for the most likely wave function possessing a given average energy by maximizing the Shannon information entropy over the quantum probability density. We show that there exists a class of solutions in which the wave function is self-trapped, rotationally symmetric, spatially localized with finite support, and spinning around its center, yet stationary. The stationarity comes from the balance between the attractive quantum force field of a trapping quantum potential generated by quantum probability density and the repulsive centrifugal force of a rotating velocity vector field. We further show that there is a limiting case where the wave function is non-spinning and yet still stationary. This special state turns out to be the lowest stationary state of the ordinary Schr\"odinger equation for a particle in a cylindrical tube classical potential.

A solution of the Schrodinger equation for the ground state of a particle in a potential field is analyzed. Since the wavefunctions of the ground state are nodeless, potentials of various kinds can be unambiguously determined. It turns out that the ground state corresponds to zero energy for a wide class of model potentials. Moreover, the zero level can be a single one at the boundary of the continuous spectrum. Crater-like potentials monotonically dependent on coordinates in one-, two-, and three-dimensional cases are studied. Instanton-type potentials with two local minima are of interest in the one-dimensional case. For the Coulomb potential, the energy of the ground state is stable with respect to both long- and short-range screening of this potential. Two-soliton solutions of the nonlinear Schrodinger equation are found. It is demonstrated that the proposed version of the inverse scattering transform is efficient in the analysis of solutions of differential equations.

Whether it is classical (cf. Chaps. 2 and 3) or quantum context (cf. Chaps. 4 and 5) so far we have talked of the systems which are free from additional restrictions (called constraints) on the canonical variables of the system. For this reason only we have freely used the Poisson brackets in the classical context and the commutators in the quantum context as and when their discussion is required. In the presence of constraints, however, some modifications at the foundation level itself are necessary. In this chapter, we aim at a brief discussion of these modifications in the context of both classical and quantal ideas.

This paper is divided into two sections. In the first section we describe the use depth first search trees and breadth first search trees in understanding self avoiding walks and restricted random walks. While we derive our results for Euclidean lattices the method is totally general and can be used for other lattices as well. In the second section derive an expression for the number of laces in the lace expansion of a memory four restricted random walk.

Fracture scaling parameters are an important input for modelling of naturally fractured reservoirs, but are very difficult to derive from subsurface data. Extensive areas of exposure in the northern Kurdistan Region of Iraq provide useful outcrop analogues for nearby producing and potential hydrocarbon fields. A variety of data acquisition methods are used to analyse fracture systems in carbonates of the Upper Cretaceous Aqra–Bekhme Formation across a wide range of scales. When plotted on length–intensity graphs, the collated data lie below an upper envelope that follows a power-law distribution over five orders of magnitude between 0.1 and 3000 m, and which defines the maximum likely intensity of background fracturing across the region. Contouring the length–intensity data shows the distribution of intensities below the upper envelope, and allows modal and minimum likely intensities to be estimated. Likely causes for the observed variation in fracture intensities include the domainal nature of deformation, the proximity to high strain zones including faults, second-order effects such as ladder fractures, and variations in the thickness of mechanical layering. Thematic collection: This article is part of the Naturally Fractured Reservoirs collection available at: https://www.lyellcollection.org/cc/naturally-fractured-reservoirs

Nonlinear d-dimensional vector σ models, such as O(N), SU(N), and CPN, are considered in the limit of an infinite number of components N. It is shown that the equation for the two-point correlation function in these models is similar to the Schrodinger equation for a quantum particle moving in a δ-function potential well (−T)δ(x), where T is the temperature. This equation adequately describes the systems under study both above and below the Curie point. Within this approach, the critical behavior of the SU(N)-invariant Ginzburg-Landau model in an external uniform magnetic field is determined in the vicinity of the upper critical magnetic field. The critical indices in this case are the same as in the spherical model in a random magnetic field. An exact equation describing the Hc2(T) curve of continuous phase transitions is derived, which allows one to determine the asymptotes of this curve in strong and weak fields. The relation between the one-particle Schrodinger equation and critical phenomena is analyzed, and applications of this method to various models in solid state physics and statistical mechanics are discussed.

We use the variational method within the Hamiltonian formalism of quantum field theory to derive relativistic two-, (four)-, and (six)-body wave equations for scalar particles interacting via a massive or massless mediating scalar field (the scalar Yukawa model). Fock-space variational trial states [$2+(4)+(6)$] are used to derive the relativistic two-body system. The equations are shown to have the Schr\odinger nonrelativistic limit, with Coulombic interparticle potentials in the case of a massless mediating field and Yukawa interparticle potentials in the case of a massive mediating field. The results show that the inclusion of virtual pairs has a large effect for the binding energy of the system at strong coupling. In the case of the discovery of a scalar Higgs particle in upcoming experiments, we may apply the present results to real, physical systems.

A new graph theoretical method is introduced to analyze a data set of objects described by a dissimilarity matrix d ij . This method is based on the generation of a series of random walks in the data set. We define a random walk in a data set by moving in each time step from one object to another one at random. In order that the random walk depends on the pattern of the data, a restriction depending on the previous moves is imposed during its generation, so that the random walk is attracted by clusters formed of similar objects. We define a hierarchical set of graphs consisting of all connections of a series of random walks at a certain time step to detect the structure of the data and to derive an similarity measure between the objects. In an example an application of the method is shown.

In this paper, which is largely review, I will discuss dissipative behavior in mechanical systems which preserve energy. The paper will encompass both the classical and quantum domains. In the classical context I will consider almost Poisson systems; systems which which have a Poisson braket which fails the Jacobi identity. This class of systems includes nonholonomic mechanical systems: systems with nonintegrable constraints such as rolling constraints. Such systems may either preserve or fail to preserve a natural measure. I will discuss also pure Hamiltonian systems such as the Toda lattice which can exhibit dissipative behavior in certain contexts as well as infinite-dimensional systems exhibiting radiative damping. In the quantum context I will discuss systems of quantum oscillators coupled to a heat bath, which also exhibit natural dissipative behavior.

The parallel mean free path of cosmic ray particles in partially turbulent electromagnetic fields is a key input param- eter for cosmic ray transport. Here the parallel mean free paths of cosmic ray protons, electrons and positrons are calculated for two particular turbulence models: slab-like dynamical and random sweeping turbulence. After outlining the general quasilinear formalism for deriving the pitch-angle Fokker-Planck coecient in weak turbulence from the particle's equation of motion, the rigidity dependence and the absolute value of the mean free path for these specific turbulence models are calculated. Approximations for the mean free path for realistic Kolmogorov-type turbulence power spectra which include the steepening at high wavenumbers due to turbulence dispersion and/or dissipation are given.

We present exact solutions of the Schrodinger equation for a charged particle constrained to move along a toroidal surface in the presence of uniform electric and magnetic fields.

In this paper, time-independent Schrödinger equation for a charged particle, in the presence of electric potential and vector potential, has been solved using He's Homotopy Perturbation Method (HPM).HPM is one of the newest analytical methods to solve linear and nonlinear differential equations.In contrast to the traditional perturbation methods, the Homotopy method does not require a small parameter in the equation.In this method, according to the homotopy technique, a Homotopy with an embedding parameter δ ∈ [0, 1] is constructed, and the embedding parameter is considered as a small parameter.Using cylindrical coordinates, it has been found that the z-equation of the charged particle is a one-dimensional harmonic oscillator and the r equation is actually a two-dimensional harmonic oscillator.The obtained results show the evidence of simplicity, usefulness, and effectiveness of the HPM for obtaining approximate analytical solutions for the time-independent Schrödinger equation for a charged particle in parallel electric and magnetic fields.

A wave equation for a noninteracting particle with zero mass and arbitrary spin $s$ is given in this paper. The Hamiltonian is proportional to the inner product of the momentum and spin operators so that the wave function has $2s+1$ components. As an auxiliary condition, solutions with spins not parallel or anti-parallel to the momentum are discarded. With this condition the theory is Lorentz-covariant. The energy, momentum, and angular momentum are defined in terms of expected values of the usual type of displacement operator. The specialization $s=\frac{1}{2}$ is the two-component neutrino theory and $s=1$ gives Maxwell's equations for the photon.

We construct exact solutions of the Schrodinger and Pauli equations for charged particles in the external field of the Redmond generalized configuration. We calculate the Green's functions of scalar and spinning particles in this field. Using them, we calculate the equations for the complex quasienergy of a bound particle (bound by a short-range potential) two different ways. In the example of an external constant electric field, we discuss the applicability domain of the obtained equations and the differences between their solutions.

Static step-step correlations for restricted and self-avoiding random walks (SARW) on quadratic and simple cubic lattices are studied with the help of Monte Carlo simulation technique. For the SARW ofN steps our results, for largeN tend to the theoretically predicted values. An analysis of the correlations for SARW in terms of the individual restrictions indicate that its value between steps at a distancer have a dominant contribution coming from the restriction prohibiting polygon closures of sides2r (r≧2). Our results also show that the contribution of successive restrictions to the mean-square end-to-end distance of a SARW decay with the same exponent as that of the correlations for the SARW.