An evolution of the relativistic electron spin's equation in the Heisenberg representation

I. M. Ternov

Toyoki Koga

Loris D’Alessi

In this work, the action of the relativistic electron is derived from the hydrodynamic formulation of the Dirac equation. In particular, in the hydrodynamic scenario, the four-velocity of the electron is regarded as an Eulerian field and the electron spin interacts with the corresponding vorticity field. In the second part of the work, it is shown how the second order Dirac equation can be derived from a principle of Minimum Fisher Information.

J. Jahanpanah, Ali Vahedi, H . Khosrojerdi

The relativistic behavior of Hydrogen-like atoms (HLAs) is investigated in Heisenberg picture for the first time. The relativistic vibrational Hamiltonian (RVH) is first defined as a power series of harmonic oscillator Hamiltonian by using the relativistic energy eigenvalue . By applying the first-order RVH (proportional to ) to Heisenberg equation, a pair of coupled equations is turned out for the motion of electron position and its relativistic linear momentum. A simple comparison of the first-order relativistic and nonrelativistic equations reveals this reality that the natural (fundamental) frequency of electron oscillation (like entropy) is slowly raised by increasing the atomic number. The second-order RVH (proportional to ) have then been implemented to determine an exact expression for the electron relativistic frequency in the different atomic energy levels. In general, the physical role of RVH is fundamental because it not only specifies the temporal relativistic variations of position, velocity, and linear momentum of oscillating electron, but also identifies the corresponding relativistic potential, kinetic, and mechanical energies. The results will finally be testified by demonstrating the energy conservation.

Ralph F. Guertin. Physics Department

A. O. Barut, Anthony J. Bracken

For a free Dirac electron, the Heisenberg equations define an internal dynamical system in the rest frame, isomorphic to a finite three-dimensional oscillator with a compact SO(5) phase space, such that the spin of the electron is the orbital angular momentum of the internal dynamics (Barut and Bracken 1980, 1981a). In the present work, the change in this internal dynamics due to an external magnetic field is studied. In order that the internal motion can be distinguished from the centre of mass motion, the solutions of the corresponding Hamilton and Heisenberg equations for the relativistic classical motion and the relativistic quantum mechanical spinless motion are also presented. The solutions for the electron exhibit the effect of the spin terms both in the internal motion and external motion, and we are able to identify the properties of the Zitterbewegung in the external field.