A three-dimensional WKB approximation for the dirac equation

1964 ◽  
Vol 33 (6) ◽  
pp. 1667-1679 ◽  
Author(s):  
M. Rosen
Keyword(s):  
Dirac Equation ◽  
Three Dimensional ◽  
Wkb Approximation

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 Convergence of a three-dimensional quantum lattice Boltzmann scheme towards solutions of the Dirac equation

2011 ◽  
Vol 369 (1944) ◽  
pp. 2155-2163 ◽  
Author(s):  
Denis Lapitski ◽  
Paul J. Dellar
Keyword(s):  
Dirac Equation ◽  
Schrödinger Equation ◽  
Lattice Boltzmann ◽  
Schrodinger Equation ◽  
Fourier Transforms ◽  
Operator Splitting ◽  
Three Dimensional ◽  
Relativistic Limit ◽  
Quantum Lattice ◽  
Lattice Boltzmann Scheme

We investigate the convergence properties of a three-dimensional quantum lattice Boltzmann scheme for the Dirac equation. These schemes were constructed as discretizations of the Dirac equation based on operator splitting to separate the streaming along the three coordinate axes, but their output has previously only been compared against solutions of the Schrödinger equation. The Schrödinger equation arises as the non-relativistic limit of the Dirac equation, describing solutions that vary slowly compared with the Compton frequency. We demonstrate first-order convergence towards solutions of the Dirac equation obtained by an independent numerical method based on fast Fourier transforms and matrix exponentiation.

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.

A NUMERICAL IMPLEMENTATION OF THE DIRAC EQUATION ON A HYPERCUBE MULTICOMPUTER

1993 ◽  
Vol 04 (03) ◽  
pp. 459-492 ◽  
Author(s):  
J. C. WELLS ◽  
A. S. UMAR ◽  
V. E. OBERACKER ◽  
C. BOTTCHER ◽  
M. R. STRAYER ◽  
...  
Keyword(s):  
Dirac Equation ◽  
Three Dimensional ◽  
Heavy Ion ◽  
Relativistic Heavy Ion Collisions ◽  
Coordinate Space ◽  
Communication Overhead ◽  
Spline Collocation ◽  
Ion Collisions ◽  
Spatial Lattice ◽  
Computationally Intensive

We describe the numerical methods used to solve the time-dependent Dirac equation on a three-dimensional Cartesian lattice. Efficient algorithms are required for computationally intensive studies of nonperturbative electromagnetic lepton-pair production in relativistic heavy-ion collisions. Discretization is achieved through the lattice basis-spline collocation method, in which quantum-state vectors and coordinate-space operators are expressed in terms of basis-spline functions on a spatial lattice. For relativistic lepton fields on a lattice, the fermion-doubling problem is central in the formulation of the numerical method. All numerical procedures reduce to a series of matrix-vector operations which we perform on the Intel iPSC/860 hypercube, making full use of parallelism. We discuss solutions to the problems of limited node memory and node-to-node communication overhead inherent in using distributed-memory, multiple-instruction, multiple-data stream parallel computers.

scholarly journals DIRAC EQUATION FOR EMBEDDED FOUR-GEOMETRIES

2013 ◽  
Vol 22 (06) ◽  
pp. 1350024
Author(s):  
MACIEJ TRZETRZELEWSKI ◽  
NORDITA
Keyword(s):  
Dirac Equation ◽  
Three Dimensional ◽  
Square Root ◽  
Quantum Level ◽  
Singularity Formation ◽  
Frw Metric ◽  
Friedmann Robertson Walker

We apply the Dirac's square root idea to constraints for embedded four-geometries swept by a three-dimensional membrane. The resulting Dirac-like equation is then analyzed for general coordinates as well as for the case of a Friedmann–Robertson–Walker (FRW) metric for spatially closed geometries. The problem of the singularity formation at the quantum level is addressed.

scholarly journals Classical equations of an electron from the special form of the Dirac equation

2021 ◽  
Vol 10 (4) ◽  
Author(s):  
Miroslav Pardy ◽  
Keyword(s):  
Dirac Equation ◽  
Special Form ◽  
Wkb Approximation ◽  
System Of Equations ◽  
Equivalent System ◽  
Tensor Equation ◽  
Lorentz Equation

The equivalent system of equations corresponding to the Dirac equation is derived and the WKB approximation of this system is found. Similarly, the WKB approximation for the equivalent system of equation corresponding to the squared Dirac equation is found and it is proved that the Lorentz equation and the Bargmann-Michel-Telegdi iquations follow from the new Dirac-Pardy system. The new tensor equation with sigma matrix is derived for the verification by adequate laboratories.

scholarly journals A Novel Exactly Theoretical Solvable of Bound States of the Dirac-Kratzer-Fues Problem with Spin and Pseudo-Spin Symmetry

2016 ◽  
Vol 10 ◽  
pp. 8-22
Author(s):  
Abdelmadjid Maireche
Keyword(s):  
Dirac Equation ◽  
Bound States ◽  
Star Product ◽  
Three Dimensional ◽  
Spin Symmetry ◽  
Hamiltonian Operator ◽  
Bound State ◽  
Exact Bound ◽  
Quantum Numbers ◽  
Quantum Symmetries

New exact bound state solutions of the deformed radial upper and lower components of Dirac equation and corresponding Hermitian anisotropic Hamiltonian operator are studied for the modified Kratzer-Fues potential (m.k.f.) potential by using Bopp’s shift method instead to solving deformed Dirac equation with star product. The corrections of energy eigenvalues are obtained by applying standard perturbation theory for interactions in one-electron atoms. Moreover, the obtained corrections of energies are depended on two infinitesimal parameters (θ,χ), which induced by position-position noncommutativity, in addition to the discreet nonrelativistic atomic quantum numbers: (j=l±1/1,s=±1/2,landm) and we have also shown that, the usual relativistic states in ordinary three dimensional spaces are canceled and has been replaced by new degenerated 2(2l+1) sub-states in the extended quantum symmetries (NC: 3D-RS).

WKB approximation for the Dirac equation at Z > 137

1978 ◽  
Vol 80 (1-2) ◽  
pp. 68-72 ◽  
Author(s):  
V.S. Popov ◽  
V.L. Eletsky ◽  
V.D. Mur ◽  
D.N. Voskresensky
Keyword(s):  
Dirac Equation ◽  
Wkb Approximation

ON THE PHYSICAL PROPERTIES RELATED TO THE ALGEBRAIC STRUCTURE OF THE DIRAC EQUATION IN THREE-DIMENSIONAL SPACE–TIME

1998 ◽  
Vol 13 (09) ◽  
pp. 1523-1542
Author(s):  
C. A. LINHARES ◽  
JUAN A. MIGNACO
Keyword(s):  
Dirac Equation ◽  
Algebraic Structure ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Space Time ◽  
Four Dimensions ◽  
Dimensional Space Time ◽  
The One ◽  
Physical Consequences ◽  
Three Dimensional Space

We look for the physical consequences resulting from the SU(2) ⊗ SU(2) algebraic structure of the Dirac equation in three-dimensional space–time. We show how this is obtained from the general result we have proven relating the matrices of the Clifford–Dirac ring and the Lie algebra of unitary groups. It allows the introduction of a notion of chirality closely analogous to the one used in four dimensions. The irreducible representations for the Dirac matrices may be labelled with different chirality eigenvalues, and they are related through inversion of any single coordinate axis. We analyze the different discrete transformations for the space of solutions. Finally, we show that the spinor propagator is a direct sum of components with different chirality; the photon propagator receive separate contributions for both chiralities, and the result is that there is no generation of a topological mass at one-loop level. In the case of a charged particle in a constant "magnetic" field we have a good example where chirality plays a determinant role for the degeneracy of states.

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