Fractal Model of the Atom in the Hydrodynamic Approach of Scale Relativity Theory

Abstract

In this paper the fractal model of the atom, using the hydrodynamic approach of the scale relativity theory, is obtained. Thus, assuming that the electron motion around the nucleus takes place on fractal curves of fractal dimension DF (continuous but non-differentiable curves), it is shown that its dynamics, in the second order approximation of the equation of motion, is described in complex speed field by a generalized Navier-Stokes type equation with imaginary viscosity coefficient. Applying this model to study the atom, it resulted that the real part of the complex velocity field describes the electron averaged movement. The electron moves on stationary orbits according to a quantification condition and the imaginary part of the complex velocity describes the fractality through a fractal potential. In the DF=2 fractal dimension and for the D=h / 2m viscosity coefficient, the classical results of quantum mechanics for the hydrogen atom are obtained.