scholarly journals Iterants, Majorana Fermions and the Majorana-Dirac Equation

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1373
Author(s):  
Louis H. Kauffman
Keyword(s):  
Dirac Equation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Clifford Algebras ◽  
Discrete Systems ◽  
Complex Numbers ◽  
Majorana Fermions ◽  
Matrix Algebras ◽  
Dirac Equations ◽  
Points Of View

This paper explains a method of constructing algebras, starting with the properties of discrimination in elementary discrete systems. We show how to use points of view about these systems to construct what we call iterant algebras and how these algebras naturally give rise to the complex numbers, Clifford algebras and matrix algebras. The paper discusses the structure of the Schrödinger equation, the Dirac equation and the Majorana Dirac equations, finding solutions via the nilpotent method initiated by Peter Rowlands.

scholarly journals Dispersion Lemmas Revisited

VLSI Design ◽  
1999 ◽  
Vol 9 (4) ◽  
pp. 365-375
Author(s):  
I. Gasser ◽  
P. A. Markowich ◽  
B. Perthame
Keyword(s):  
Dirac Equation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Semiclassical Limit ◽  
Kinetic Equations ◽  
Wigner Transform ◽  
Dirac Equations

We investigate regularizing dispersive effects for various classical equations, e.g., the Schrödinger and Dirac equations. After Wigner transform, these dispersive estimates are reduced to moment lemmas for kinetic equations. They yield new regularization results for the Schrödinger equation (valid up to the semiclassical limit) and the Dirac equation.

Elementare Herleitung der Dirac-Gleichung / Elementary Derivation of the Dirac Equation

1978 ◽  
Vol 33 (11) ◽  
pp. 1378-1379 ◽  
Author(s):  
Hans Sallhofer
Keyword(s):  
Dirac Equation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Classical Equation ◽  
Inhomogeneous Media ◽  
Elementary Derivation ◽  
Complete Analogy

As is well known, the Schrödinger equation can be derived by suitably arranging the refraction in the classical equation for light inhomogeneous media. In this paper it is shown that one may derive the Dirac equation in complete analogy by arranging for the refraction in the electrodynamics of inhomogeneous media.

Energy and momentum operator substitution method derived from Schrödinger equation for light and matter waves

2019 ◽  
Vol 33 (24) ◽  
pp. 1950285
Author(s):  
Saviour Worlanyo Akuamoah ◽  
Aly R. Seadawy ◽  
Dianchen Lu
Keyword(s):  
Wave Equation ◽  
Dirac Equation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Momentum Operator ◽  
Matter Wave ◽  
Relativistic Wave Equation ◽  
Relativistic Energy ◽  
Substitution Method ◽  
Energy And Momentum

In this paper, the energy and momentum operator substitution method derived from the Schrödinger equation is used to list all possible light and matter wave equations, among which the first light wave equation and relativistic approximation equation are proposed for the first time. We expect that we will have some practical application value. The negative sign pairing of energy and momentum operators are important characteristics of this paper. Then the Klein–Gordon equation and Dirac equation are introduced. The process of deriving relativistic energy–momentum relationship by undetermined coefficient method and establishing Dirac equation are mainly introduced. Dirac’s idea of treating negative energy in relativity into positrons is also discussed. Finally, the four-dimensional space-time representation of relativistic wave equation is introduced, which is usually the main representation of quantum electrodynamics and quantum field theory.

scholarly journals Dirac Equation in the Field of a Charged Cosmic String and in the Field of a Point Charge with Z > 137

1991 ◽  
Vol 44 (6) ◽  
pp. 585
Author(s):  
TJ Allen ◽  
LJ Tassie
Keyword(s):  
Dirac Equation ◽  
Schrödinger Equation ◽  
Internal Structure ◽  
Effective Potential ◽  
Cosmic String ◽  
Point Charge ◽  
Schrodinger Equation ◽  
Cylindrical Coordinates ◽  
Line Charge ◽  
Radial Dirac Equation

In both spherical and cylindrical coordinates, the radial Dirac equation can be written in the form of a Schrodinger equation with an effective potential. It is shown that the difficulties at r -+ 0 for the Dirac equation in the field of a point charge for Z > 137 are the same as those for the Schrodinger equation with a l/r2 potential. The effective potential is used to show that similar difficulties do not arise for the field of a line charge, so allowing the consideration of the motion of electrons in the field of a charged superconducting cosmic string without considering the internal structure of the string.

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.

Eigenvalues of energy in j (l + 1/2, l ‐ 1/2) electron state in the hydrogen atom derived from the relativistically modified Schrödinger equation

2021 ◽  
Vol 34 (2) ◽  
pp. 111-115
Author(s):  
Noboru Kohiyama
Keyword(s):  
Wave Equation ◽  
Hydrogen Atom ◽  
Dirac Equation ◽  
Schrödinger Equation ◽  
Radio Wave ◽  
Schrodinger Equation ◽  
Electron State ◽  
Microwave Emission

In the hydrogen atom, the eigenvalues of energy in j (l + 1/2, l ‐ 1/2) electron state cannot be correctly evaluated from the nonrelativistic Schrödinger equation. In order to express the relativistic properties of the wave equation for a particle with 1/2 spin, the Schrödinger equation is relativistically modified. The modified Schrödinger equation is solved for consistency with the eigenvalues of electron's energy derived from the Dirac equation. Based on the consistency of their eigenvalues, the different electron state is expressed. The microwave emission (e.g., 21 cm radio wave) by the hydrogen atom was thus predicted from this state.

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