A thorough mathematic analysis was conducted to the negative energy solutions to the Dirac equation in a complex plane due to the fact that the solutions are vectors of complex number. An opposite vector with negative energy state was identified when a vector with positive energy state was defined in a complex plane. We concluded that any opposite vectors in a complex plane must have opposite energy state because their vector heads move always in opposite direction. We configured an electron as a small ball with dimetric vectors instead of a structureless mass point. Based on the new electron configuration, we explained the particle wave duality as the feature of electron’s spinvector in motion, explained the reason for the electrons with half spin quantum number. And we concluded that de Broglie’s standing waves theory would be more rational than the interpretation by current quantum theory.
Considers the Dirac equation with external electromagnetic fields and gives a careful formulation of the Stuckelberg-Feynman interpretation. This electron-positron theory may be applied whenever electron and positron states can be separated, which is possible for a wide class of time-independent potentials vanishing at infinity. One then may introduce a system of observables with a position operator X(t) which is free of Zitterbewegung and yields the usual relativistic connection between momentum and velocity. The Stuckelberg-Feynman interpretation is usually associated with a scattering theory (propagator formalism), which differs from the one of a one-particle interpretation (formalism imitating non-relativistic techniques). For the above-mentioned potentials there is a simple relationship between these physically different formalisms. There are, however, more general potentials for which the Stuckelberg-Feynman interpretation may still be applied asymptotically. With the help of Klein's paradox the authors show that the Dirac equation may then have unphysical solutions not conserving the total charge.
We underscore the need of relativistic quantum mechanics in dealing with particles. Various notations used in relativity are defined and explained. We derive the Klein–Gordon equation for free particle as well as for particle in electromagnetic field. We obtain a solution and interpret it. We derive the Dirac equation for free particle as well as for particle in electromagnetic field. Plane wave solution to the Dirac equation is obtained. We show how spin of electron, idea of antiparticle , phenomena of pair production and pair annihilation follow from the Dirac equation.
Abstract The Dirac equation requires a treatment of the step potential that differs fundamentally from the traditional treatment, because the Dirac plane waves, besides momentum and spin, are characterized by a quantum number with the physical meaning of sign of charge. Since the Hermitean operator corresponding to this quantum number does not commute with the step potential, the time displacement parameter used in the ansatz of the stationary state does not have the physical meaning of energy. Therefore there are no paradoxal values of the ‘energy’. The new solution of the Dirac equation with a step potential is obtained. This solution, again, allows for phenomena of the Klein paradox type, but in addition it contains a positron amplitude localized at the threshold point of the step potential.
The paper analyzes time propagation of Dirac observables - using Heisenberg representation - in the light of various pseudodifferential operator algebras (cf. [Co3], [Co15], [Co16]). Our theory gives (i) a mechanical angular momentum (the spin), but also (ii) another real 3-vector, travelling with the particle, with magnetic properties (its motion guided by the magnetic field around it, but not in the proper relativistic way). The above is discussed for potentials vanishing at infinity. But we also look at a Dirac particle in the field of a plane polarized X-ray wave, trying to analyze the Compton effect. The propagations of energy and momentum (in the X-ray's direction) are coupled - they allow a joint asymptotic expansion with terms representing change of energy by
nhν and momentum by
nhν/c, with n=0,1,2, ..., valid for large frequencies, i.e., large momentum coordinates. Possible interpretation: A collision of the electron-positron with 0, 1, 2, ... n Photons of energy
hν, affecting a change of energy-momentum. Note, this does not require QFT.