Alternative interpretation of the 1D-box solution and the Bargmann theorem

Abstract

Primitive mapping of 2D fractal spaces yields a formulation of the Schro¨ dinger equation and endows its solutions and the respective 3D objects with specific geometric images. In particular, it is shown that the simplest 1D-box solution comprising no parameters of particles motion can be interpreted as a 2D inhomogeneous string oscillating on a real-imaginary fractal surface or as a 3D static spindle with a harmonically distributed mass spectrum. The description of an inertially moving similar object is obtained using a Bargmann-type theorem applied to the Bohm equations, and, as their exact solution, a fractal function containing explicit kinematic terms.