scholarly journals SOLUTION OF THE DIRAC EQUATION FOR POTENTIAL INTERACTION

2003 ◽  
Vol 18 (27) ◽  
pp. 4955-4973 ◽  
Author(s):  
A. D. ALHAIDARI
Keyword(s):  
Dirac Equation ◽  
Three Dimensional ◽  
Nonrelativistic Limit ◽  
Potential Interaction ◽  
Spherically Symmetric ◽  
Local Interactions ◽  
Shape Invariant ◽  
Exactly Solvable ◽  
Relativistic Extension

An effective approach for solving the three-dimensional Dirac equation for spherically symmetric local interactions, which we have introduced recently, is reviewed and consolidated. The merit of the approach is in producing Schrödinger-like equations for the spinor components that could simply be solved by correspondence with well-known exactly solvable nonrelativistic problems. Taking the nonrelativistic limit will reproduce the nonrelativistic problem. The approach has been used successfully in establishing the relativistic extension of all classes of shape-invariant potentials as well as other exactly solvable nonrelativistic problems. These include the Coulomb, Oscillator, Scarf, Pöschl–Teller, Woods–Saxon, etc.

scholarly journals DIRAC OSCILLATORS AND QUASI-EXACTLY SOLVABLE OPERATORS

2005 ◽  
Vol 20 (25) ◽  
pp. 1875-1885 ◽  
Author(s):  
Y. BRIHAYE ◽  
A. NININAHAZWE
Keyword(s):  
Dirac Equation ◽  
Dirac Operator ◽  
Spherically Symmetric ◽  
Dirac Oscillator ◽  
Generic Polynomial ◽  
Exactly Solvable ◽  
Spectral Equation ◽  
Radial Variable ◽  
Radial Dirac Equation

The Dirac equation is formulated in the background of three types of physically relevant potentials: scalar, vector and "Dirac-oscillator" potentials. Assuming these potentials to be spherically-symmetric and with generic polynomial forms in the radial variable, we construct the corresponding radial Dirac equation. Cases where this linear spectral equation is exactly solvable or quasi-exactly solvable are worked out in details. When available, relations between the radial Dirac operator and some super-algebra are pointed out.

scholarly journals EXACT SOLUTIONS OF DIRAC AND SCHRÖDINGER EQUATIONS FOR A LARGE CLASS OF POWER-LAW POTENTIALS AT ZERO ENERGY

2002 ◽  
Vol 17 (30) ◽  
pp. 4551-4566 ◽  
Author(s):  
A. D. ALHAIDARI
Keyword(s):  
Exact Solutions ◽  
Power Law ◽  
Bound States ◽  
Hypergeometric Functions ◽  
Rest Mass ◽  
Three Dimensional ◽  
Zero Energy ◽  
Confluent Hypergeometric Functions ◽  
Exactly Solvable ◽  
Relativistic Extension

We obtain exact solutions of Dirac equation for radial power-law relativistic potentials at rest mass energies. It turns out that these are the relativistic extension of a subclass of exact solutions of Schrödinger equation at zero energy carrying representations of SO(2,1) Lie algebra. The latter are obtained by point canonical transformations of the exactly solvable problem of the three dimensional oscillator. The wave function solutions are written in terms of the confluent hypergeometric functions and almost always square integrable. For most cases these solutions support bound states at zero energy. Some exceptional unbounded states are normalizable for nonzero angular momentum. Using a generalized definition, degeneracy of the nonrelativistic states is demonstrated and the associated degenerate observable is defined.

1995 ASME Gas Turbine Award Paper: Development and Application of a Multistage Navier–Stokes Solver: Part I—Multistage Modeling Using Bodyforces and Deterministic Stresses

1998 ◽  
Vol 120 (2) ◽  
pp. 205-214 ◽  
Author(s):  
C. M. Rhie ◽  
A. J. Gleixner ◽  
D. A. Spear ◽  
C. J. Fischberg ◽  
R. M. Zacharias
Keyword(s):  
Stokes Equations ◽  
Pressure Ratio ◽  
Three Dimensional ◽  
Design Procedure ◽  
Potential Interaction ◽  
Theoretical Development ◽  
Navier Stokes ◽  
Radial Profiles ◽  
Compressor Performance ◽  
High Level

A multistage compressor performance analysis method based on the three-dimensional Reynolds-averaged Navier-Stokes equations is presented in this paper. This method is an average passage approach where deterministic stresses are used to ensure continuous physical properties across interface planes. The average unsteady effects due to neighboring blades and/or vanes are approximated using deterministic stresses along with the application of bodyforces. Bodyforces are used to account for the “potential” interaction between closely coupled (staged) rows. Deterministic stresses account for the “average” wake blockage and mixing effects both axially and radially. The attempt here is to implement an approximate technique for incorporating periodic unsteady flow physics that provides for a robust multistage design procedure incorporating reasonable computational efficiency. The present paper gives the theoretical development of the stress/bodyforce models incorporated in the code, and demonstrates the usefulness of these models in practical compressor applications. Compressor performance prediction capability is then established through a rigorous code/model validation effort using the power of networked workstations. The numerical results are compared with experimental data in terms of one-dimensional performance parameters such as total pressure ratio and circumferentially averaged radial profiles deemed critical to compressor design. This methodology allows the designer to design from hub to tip with a high level of confidence in the procedure.

scholarly journals Construction of potential functions associated with a given energy spectrum — An inverse problem

2020 ◽  
Vol 35 (20) ◽  
pp. 2050104
Author(s):  
A. D. Alhaidari
Keyword(s):  
Energy Spectrum ◽  
Three Dimensional ◽  
Logarithmic Potential ◽  
Potential Functions ◽  
Physical Parameters ◽  
Linear Potential ◽  
Hahn Polynomial ◽  
Exactly Solvable ◽  
The One ◽  
Solvable Problems

Using a formulation of quantum mechanics based on orthogonal polynomials in the energy and physical parameters, we present a method that gives the class of potential functions for exactly solvable problems corresponding to a given energy spectrum. In this work, we study the class of problems associated with the continuous dual Hahn polynomial. These include the one-dimensional logarithmic potential and the three-dimensional Coulomb plus linear potential.

scholarly journals Non-Commutative Integration of the Dirac Equation in Homogeneous Spaces

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1867
Author(s):  
Alexander Breev ◽  
Alexander Shapovalov
Keyword(s):  
Dirac Equation ◽  
Exact Solutions ◽  
Homogeneous Space ◽  
Integration Method ◽  
Homogeneous Spaces ◽  
Separation Of Variables ◽  
General Structure ◽  
Three Dimensional ◽  
De Sitter ◽  
Elementary Functions

We develop a non-commutative integration method for the Dirac equation in homogeneous spaces. The Dirac equation with an invariant metric is shown to be equivalent to a system of equations on a Lie group of transformations of a homogeneous space. This allows us to effectively apply the non-commutative integration method of linear partial differential equations on Lie groups. This method differs from the well-known method of separation of variables and to some extent can often supplement it. The general structure of the method developed is illustrated with an example of a homogeneous space which does not admit separation of variables in the Dirac equation. However, the basis of exact solutions to the Dirac equation is constructed explicitly by the non-commutative integration method. In addition, we construct a complete set of new exact solutions to the Dirac equation in the three-dimensional de Sitter space-time AdS3 using the method developed. The solutions obtained are found in terms of elementary functions, which is characteristic of the non-commutative integration method.

scholarly journals Local interactions and their group-level consequences in flocking jackdaws

2019 ◽  
Vol 286 (1906) ◽  
pp. 20190865 ◽  
Author(s):  
Hangjian Ling ◽  
Guillam E. Mclvor ◽  
Kasper van der Vaart ◽  
Richard T. Vaughan ◽  
Alex Thornton ◽  
...  
Keyword(s):  
Group Structure ◽  
Three Dimensional ◽  
Movement Direction ◽  
Gravitational Potential Energy ◽  
Wingbeat Frequency ◽  
Local Interactions ◽  
Interaction Forces ◽  
Short Range Repulsion ◽  
Direction Of Movement ◽  
Repulsive Forces

As one of nature's most striking examples of collective behaviour, bird flocks have attracted extensive research. However, we still lack an understanding of the attractive and repulsive forces that govern interactions between individuals within flocks and how these forces influence neighbours' relative positions and ultimately determine the shape of flocks. We address these issues by analysing the three-dimensional movements of wild jackdaws ( Corvus monedula ) in flocks containing 2–338 individuals. We quantify the social interaction forces in large, airborne flocks and find that these forces are highly anisotropic. The long-range attraction in the direction perpendicular to the movement direction is stronger than that along it, and the short-range repulsion is generated mainly by turning rather than changing speed. We explain this phenomenon by considering wingbeat frequency and the change in kinetic and gravitational potential energy during flight, and find that changing the direction of movement is less energetically costly than adjusting speed for birds. Furthermore, our data show that collision avoidance by turning can alter local neighbour distributions and ultimately change the group shape. Our results illustrate the macroscopic consequences of anisotropic interaction forces in bird flocks, and help to draw links between group structure, local interactions and the biophysics of animal locomotion.

Three-Dimensional Time-Resolved Flow Field in the First and Last Turbine Stage of a Heavy Duty Gas Turbine: Part II — Interpretation of Blade Pressure Fluctuations

2000 ◽  
Author(s):  
Martin von Hoyningen-Huene ◽  
Wolfram Frank ◽  
Alexander R. Jung
Keyword(s):  
Flow Field ◽  
Gas Turbine ◽  
Gas Turbines ◽  
Pressure Fluctuations ◽  
Three Dimensional ◽  
Potential Interaction ◽  
Heavy Duty ◽  
Turbine Stage ◽  
Count Ratio ◽  
Heavy Duty Gas Turbine

Unsteady stator-rotor interaction in gas turbines has been investigated experimentally and numerically for some years now. Most investigations determine the pressure fluctuations in the flow field as well as on the blades. So far, little attention has been paid to a detailed analysis of the blade pressure fluctuations. For further progress in turbine design, however, it is mandatory to better understand the underlying mechanisms. Therefore, computed space–time maps of static pressure are presented on both the stator vanes and the rotor blades for two test cases, viz the first and the last turbine stage of a modern heavy duty gas turbine. These pressure fluctuation charts are used to explain the interaction of potential interaction, wake-blade interaction, deterministic pressure fluctuations, and acoustic waveswith the instantaneous surface pressure on vanes and blades. Part I of this two-part paper refers to the same computations, focusing on the unsteady secondary now field in these stages. The investigations have been performed with the flow solver ITSM3D which allows for efficient simulations that simulate the real blade count ratio. Accounting for the true blade count ratio is essential to obtain the correct frequencies and amplitudes of the fluctuations.

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