A 2D spinless version of Dirac's equation written in a noninertial frame of reference

Abstract

ABSTRACT:In this article, we present a quantized classical-like wave equation. It is obtained by considering the equivalence between the Hamilton-Jacobi eq. (which belongs to the usual manifold R 3 R of General Relativity), with our two spinor versions of the Dirac eq. (which belongs to the associated complex manifold C C). In which the electron is considered as a particle-like entity, instead of the usual wave-like interpretation of standard Quantum Mechanics. We also consider the transformation properties between the two manifolds through the corresponding groups O(3,1) and SU(2), but we assume a flat Minkowsky space as the background space. We made an illustrative application to the rather standard problem of the hydrogen and helium atoms. We propose a variational extension of the approach, through a Hylleraas-like computational method, to take into account a many electron problem too futurely.

A 2D Formulation for the Helium Atom Using a Four-Spinor Dirac-like Equation and the Discussion of an Approximate Ground State Solution

“…Now, the homogeneous system of equations in the coefficients 00 k a corresponding to the determinant Eq. (15) has a kernel generated by the eigenvalue 1  , whose base is given by the two linearly independent column vectors   12 (20)…”

“…In a previous work [1] we have shown that, by taken into account the classical conservation theorems before the Dirac matrices are found, it is possible to use 2x2 matrices instead the usual 4x4 matrices in the solution of the hydrogen atom, which yields a considerable reduction in the complexity of the problem. In this work, we try to get the same reduction in the dimensions of the Dirac matrices for the problem of the helium atom, which allows us to use 4x4 matrices instead of the 16x16 matrices of the Breit theory of the helium [2,3], which in spite of this is our starting point.…”

Quantization and Stable Attractors in a DissipativeOrbital Motion

“…From this non-linear momentum variation equation for a non-conservative system, which is analogous to (2) for a conservative system, the electron trajectories resulting in the stable attractor mentioned above will be obtained. Thus, Equation (11) is the linear differential equation associated with the non-linear momentum equation, (14).…”

“…We can now perform the reduction of both the equations of motion, (11) and (14). Starting with the former, we consider a variation in path with a constant angular momentum 3 0 L   , so that .…”

“…We follow an alternative approach in this work, in which we show that it is possible to obtain the exact energy of the bound states of the hydrogen atom by searching for stable orbital attractors in a non-conservative Hamilton-Jacobi dynamics [12][13][14]. Thus, the quantiza-tion problem is solved by selecting, from all of the possible electron paths in which energy is dissipated, those that tend to stable closed paths in which bound states of motion are reached in the limit as time approaches infinity, that is, to stable orbital attractors.…”