scholarly journals Nonlocal Inverse Square Law in Quantum Dynamics

2018 ◽  
Author(s):  
Er'el Granot
Keyword(s):  
Wave Equation ◽  
Quantum Dynamics ◽  
Local Perturbation ◽  
Entire Space ◽  
Inverse Square Law ◽  
Local Areas ◽  
Spectrally Bounded ◽  
Universal Coupling ◽  
First Time ◽  
Local Value

Schrödinger dynamics is a nonlocal process. Not only does local perturbation affect instantaneously the entire space, but the effect decays slowly. When the wavefunction is spectrally bounded, the Schrödinger equation can be written as a universal set of ordinary differential equations, with universal coupling between them, which is related to Euler’s formula. Since every variable represents a different local value of the wave equation, the coupling represents the dynamics’ nonlocality. It is shown that the nonlocal coefficient is inversely proportional to the distance between the centers of these local areas. As far as we know, this is the first time that this inverse square law was formulated.

THE WAVE EQUATIONS FOR THE WEYL TENSOR/SPINOR AND DIMENSIONALLY DEPENDENT TENSOR IDENTITIES

1996 ◽  
Vol 05 (03) ◽  
pp. 217-225 ◽  
Author(s):  
FREDRIK ANDERSSON ◽  
S. BRIAN EDGAR
Keyword(s):  
Wave Equation ◽  
Large Class ◽  
Weyl Tensor ◽  
Wave Equations ◽  
Dimensional Case ◽  
Weyl Spinor ◽  
Four Dimensions ◽  
Identity Matching ◽  
Tensor Identity ◽  
First Time

By reconciling the wave equation for the Weyl tensor with the corresponding wave equation for the Weyl spinor, we establish a new tensor identity—involving the sum of terms each consisting of a product of the Weyl and Ricci tensors—valid in four (and only four) dimensions. This enables us to give, for the first time, the correct and simplest form of the wave equation for the Weyl tensor in four-dimensional nonvacuum spacetimes. The wave equation for the Weyl tensor in n(> 4) dimensional nonvacuum spaces is also presented for the first time; we show that there does not exist an analogous n-dimensional tensor identity matching the four-dimensional one, and so it follows that there does not exist an analogous simplification of the Weyl wave equation in the n-dimensional case. It is also shown how our new identity, and some other recently discovered identities, relate to a large class of dimensionally dependent identities found some time ago by Lovelock.

scholarly journals On the interpretation of the relativity wave equation for two electrons

1929 ◽  
Vol 124 (794) ◽  
pp. 422-425 ◽  
Keyword(s):  
Dynamical System ◽  
Wave Function ◽  
Wave Equation ◽  
Quantum Dynamics ◽  
Conjugate Function ◽  
Volume Element ◽  
Single Electron

In the general non-relativity quantum dynamics that has been developed by Dirac and others, the motion of a dynamical system is described by a wave function ψ (ξ 1 ... ξ n ; t ); and one interprets this wave function by postulating that |ψ (ξ 1 ' ξ 2 ' ... ξ n ' ; t ) | 2 d ξ 1 ' ... d ξ n ' is the probability that, at time t ξ r ' ≤ ξ r ≤ ξ r ' + d ξ r ' ( r = 1 ... n ). The relativity wave function for a single electron can be interpreted in a similar manner. If ψ( xyzt ) be the wave function, and φ the conjugate function, then φψ ( xyzt ) dx dy dz : is the probability that the electron will be found, at time t , in the volume element dx dy dz .

scholarly journals Explicit Stencil Computation Schemes Generated by Poisson's Formula for the 2D Wave Equation

2019 ◽  
Author(s):  
Naum Khutoryansky
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Integral Representation ◽  
Initial Conditions ◽  
Representation Formula ◽  
Time Step ◽  
Stencil Computation ◽  
Integral Representation Formula ◽  
Time Marching ◽  
First Time

An approach to building explicit time-marching stencil computation schemes for the transient 2D acoustic wave equation without using finite-difference approximations is proposed and implemented. It is based on using the integral representation formula (Poisson's formula) that provides the exact solution of the initial-value problem for the transient 2D scalar wave equation at any time point through the initial conditions. For the purpose of constructing a two-step time-marching algorithm, a modified integral representation formula involving three time levels is also employed. It is shown that integrals in the two representation formulas are exactly calculated if the initial conditions and the sought solution at each time level as functions of spatial coordinates are approximated by stencil interpolation polynomials in the neighborhood of any point in a 2D Cartesian grid. As a result, if a uniform time grid is chosen, the proposed time-marching algorithm consists of two numerical procedures: 1) the solution calculation at the first time-step through the initial conditions; 2) the solution calculation at the second and next time-steps using a generated two-step numerical scheme. Three particular explicit stencil schemes (with five, nine and 13 space points) are built using the proposed approach. Their stability regions are presented. The obtained stencil expressions are compared with the corresponding finite-difference schemes available in the literature. Their novelty features are discussed. Simulation results with new and conventional schemes are presented for two benchmark problems that have exact solutions. It is demonstrated that using the new first time-step calculation procedure instead of the conventional one can provide a significant improvement of accuracy even for later time steps.

scholarly journals Using the method of integral Fourier transform to solve the problem of nonstationary deformation of an elastic medium

2019 ◽  
Vol 109 ◽  
pp. 00080
Author(s):  
Volodymyr Sapehin
Keyword(s):  
Fourier Transform ◽  
Wave Equation ◽  
Internal Pressure ◽  
Elastic Medium ◽  
Plane Deformation ◽  
Integral Fourier Transform ◽  
Nonstationary Deformation ◽  
Variable Internal Pressure ◽  
First Time

For the first time, for solving the wave equation, the method of the integral Fourier transform for an elastic medium under the action of variable internal pressure was used. This equation contains singularities of the second kind, which were excluded from the calculation by the half division method. The main parameters of the process of non-stationary deformation affecting the magnitude of stresses and displacements for the case of plane deformation are established.

scholarly journals The collision between two electrons

1930 ◽  
Vol 126 (801) ◽  
pp. 259-267 ◽  
Keyword(s):  
Wave Equation ◽  
Classical Theory ◽  
Wave Functions ◽  
The Other ◽  
Exclusion Principle ◽  
Wave Mechanics ◽  
Inverse Square Law ◽  
Symmetry Properties ◽  
Α Particles

It is well-known that the problem of the collision between two particles interacting according to the inverse square law is exactly soluble on the wave mechanics, and that the solution yields the same scattering laws as the classical theory. If, however, the two particles are identical, e.g. , two electrons or two α-particles, this is not necessarily the case; for the wave functions used must be antisymmetrical or symmetrical in the co-ordinates of the two particles; and this may affect the scattering laws. In this paper we shall discuss the collision between two particles possessing spin, such as electrons, and also between two particles without spin, such as α-particles. Assuming an inverse square law force between the particles, and neglecting the actual spin forces, we shall deduce from the symmetry properties of the wave functions a scatter­ing law differing considerably from the classical. We shall also mention the various methods by which the effect could be observed, and give some experi­mental evidence in its favour. The application of the exclusion principle to collision problems has been discussed by the author in a previous paper. Suppose we wish to describe the motion of two particles interacting in any field of force. We obtain a solution w (r 1 r 2 ) of the wave equation, where r 1 refers to the position of the first particle, and r 2 to that of the second. If we did not use antisymmetrical wave functions, we should argue that the probability that the first particle should be at r 1 and the second at r 2 would be | w (r 1 r 2 )| 2 , and therefore the probability that one particle should be at r 1 and the other at r 2 would be | w (r 1 r 2 )| 2 + | w (r 2 r 1 )|

Higher modes of continental Rayleigh waves

1957 ◽  
Vol 47 (3) ◽  
pp. 187-204 ◽  
Author(s):  
Jack Oliver ◽  
Maurice Ewing
Keyword(s):  
Wave Equation ◽  
Particle Motion ◽  
High Frequency ◽  
Rayleigh Waves ◽  
Velocity Curve ◽  
Theoretical Studies ◽  
Higher Modes ◽  
Short Period ◽  
First Time ◽  
Rayleigh Mode

ABSTRACT A long dispersive train of waves corresponding to higher modes of the Rayleigh-wave equation (including Sezawa's M2 wave) for the continental crust-mantle system is positively identified, apparently for the first time. Observed particle motion is elliptical and retrograde, in agreement with theory. Although several theoretical studies have been published in which progressive elliptical particle motion was found, all of these involved values of the elastic constants unsuitable for the present problem. The beginnings of the short-period branches of the higher modes can account for the high-frequency longitudinal and vertical components of the continental surface-wave phase Lg. The large amplitudes and the peculiar appearance of Rg appear to depend on the broad flat minimum of the group velocity curve of the lowest or Rayleigh mode.

Distribution and frequent occurrence of diatom taxa (Bacillariophyta) inhabiting warmer oceans in Seogwipo coast of Jeju Island, southernmost Korea

Phytotaxa ◽  
2021 ◽  
Vol 517 (1) ◽  
pp. 1-67
Author(s):  
GYEONGJE JOH
Keyword(s):  
South Korea ◽  
Western Indian Ocean ◽  
Local Area ◽  
Jeju Island ◽  
Diatom Flora ◽  
The Pacific ◽  
Local Areas ◽  
The Pacific Ocean ◽  
The Caribbean ◽  
First Time

The epiphytic diatoms attached to seaweeds and benthic diatom assemblages, have been collected along the Seogwipo coast in Jeju Island, South Korea. In the last three-year survey, 80 diatom taxa inhabiting warmer oceans have been found, including those in the tropical and subtropical regions such as the Pacific Ocean, the Caribbean Sea, the western Indian Ocean, and other warmer regions. These diatom flora shows a worldwide pattern that concentrated around the Tropic of Cancer. This survey of local areas reveals the existence of many rare or retrieved taxa that were not well known until recently. Furthermore, some of them, including Aulacodiscus affinis, Leudugeria janischii, Hyalosynedra lanceolata, Trachyneis velata var. ornata, Lyrella concilians have been reported for the first time since the first entry or haven been reported only once. Of the diatom flora described here, 48 taxa are new to South Korea and previously unrecorded in the local area.

scholarly journals Charitable organisations, the Great Recession and the Age of Austerity: Longitudinal Evidence for England and Wales

2016 ◽  
Vol 46 (1) ◽  
pp. 1-30 ◽  
Author(s):  
DAVID CLIFFORD
Keyword(s):  
Great Recession ◽  
Public Spending ◽  
Real Income ◽  
Annual Income ◽  
England And Wales ◽  
The Great Recession ◽  
Local Areas ◽  
The Impact ◽  
First Time ◽  
Longitudinal Dataset

AbstractThere has been extensive concern about the effect of recession and of subsequent public spending austerity on the voluntary sector – but a lack of comprehensive sector-wide data to examine this empirically. We construct a unique longitudinal dataset, which follows through time the population of charitable organisations in England and Wales since 1999, and assess the impact of recession and austerity by placing organisations’ recent annual income within the context of longer-term trends. The results reveal the scale of the impact on charities’ incomes for the first time: since 2008 median real annual growth in income has been negative for six consecutive years, leading to sizeable cumulative real income decline over the period. Mid-sized charities, and those in more deprived local areas, have been most significantly affected, consistent with concerns about a ‘hollowing out’ of the charitable sector and about the uneven impact of austerity. However, there has also been considerable variation in the fortunes of charities working in different fields of activity. The analysis in this paper helps to widen our perspective on the implications of the Great Recession and of public spending austerity for social policy.

Sign in / Sign up

Export Citation Format

Share Document