Recoil effects in the coordinate space Dirac equation
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Abstract:
Recoil corrections are incorporated into the coordinate space Dirac equation. Numerical calculations with and without the recoil corrections are performed.Received 5 August 1986DOI:https://doi.org/10.1103/PhysRevC.35.369©1987 American Physical SocietyKeywords:
Recoil
Two-body Dirac equations
The Dirac and Klein-Gordon equations provide a full relativistic description for particles with spin ½ and 0, respectively. A calculation now shows how to extend this description to particles, such as nuclei, with spin greater than ½.
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In this chapter, we investigate the Dirac equation, which is named after P. A. M. Dirac, who is one of the fathers of quantum field theory. The Dirac equation is a relativistic quantum mechanical wave equation for spin-1/2 particles (e.g. electrons), which was derived by Dirac in 1928. The difficulties in finding a consistent single-particle theory from the Klein–Gordon equation led Dirac to search for an equation that
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In the model of topological particles we have four types of topologically stable dual Dirac monopoles with soft core and finite mass. We discuss the steps how to get a Dirac equation for these particles. We show for the free and the interacting case that we arrive at the Dirac equation in the limit, where the soft solitons approach singular dual Dirac monopoles.
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The Dirac equation is a relativistic wave equation and was the first equation to capture spin in relativistic quantum mechanics. Here, the Dirac equation will be derived and solved for a particle in a spherical box potential. Comparisons to the non-relativistic Schrodinger equation as well as to a relativistically corrected Schrodinger equation will be made. Applications of the Dirac equation, such as the Bogoliubov model, are examined and discussed and it is validated that the Dirac equation can provide knowledge about elementary particles.
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Relativistic wave equations
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We consider the behavior of the particles at ultra relativistic energies, for both the Klein-Gordon and Dirac equations. We observe that the usual description is valid for energies such that we are outside the particle's Compton wavelength. For higher energies however, both the Klein-Gordon and Dirac equations get modified and this leads to some new effects for the particles, including the appearance of anti particles with a slightly different energy.
Klein–Gordon equation
Two-body Dirac equations
Compton wavelength
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We demonstrate that if one adheres to a method akin to Dirac's method of arriving at the Dirac equation -- then, the Dirac equation is not the only equation that one can generate but that there is a whole new twenty four equations that Dirac left out. Off these new equations -- interesting is that; some of them violate C, P, T, CT, CP, PT and CPT-Symmetry. If these equations are acceptable on the basis of them flowing from the widely -- if not universally accepted Dirac prescription, then, the great riddle of why the preponderance of matter over antimatter might find a solution.are acceptable on the basis of them flowing from the widely -- if not universally accepted Dirac prescription, then, the great riddle of why the preponderance of matter over antimatter might find a solution.
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The author has previously derived an energy-momentum relationship applicable in a hydrogen atom. Since this relationship is taken as a departure point, there is a similarity with the Dirac’s relativistic wave equation, but an equation more profound than the Dirac equation is derived. When determining the coefficients and β of the Dirac equation, Dirac assumed that the equation satisfies the Klein-Gordon equation. The Klein-Gordon equation is an equation which quantizes Einstein's energy-momentum relationship. This paper derives an equation similar to the Klein-Gordon equation by quantizing the relationship between energy and momentum of the electron in a hydrogen atom. By looking to the Dirac equation, it is predicted that there is a relativistic wave equation which satisfies that equation, and its coefficients are determined. With the Dirac equation it is necessary to insert a term for potential energy into the equation when describing the state of the electron in a hydrogen atom. However, in this paper, a potential energy term is not introduced into the relativistic wave equation. Instead, potential energy is incorporated into the equation by changing the coefficient of the Dirac equation.
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Klein–Gordon equation
Hydrogen atom
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Applications of the Dirac equation with an anomalous magnetic moment are considered for description of characteristics of electrons, muons and quarks. The Dirac equation with four-dimensional scalar and vector potentials is reduced to a form suitable for a numerical integration. When a certain type of the potential is chosen, solutions can approximate quark states inside hadrons. In view of complicated behaviour of quarks in a confinement domain some generalizations are considered such as the Dirac-Gursey-Lee equation, the Dirac equation in a five-dimensional Minkowski space, the Dirac equation in a quantum phase space. Extended symmetries for the Dirac equation and its generalizations are considered, which can be used for investigation of properties of solutions of these equations and subsequent applications in particle physics.
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