Homeomorphism connecting the hydrogenic system and the free-particle system

Abstract

The topological equivalence between the hydrogenic system and the free-particle system is established by a homeomorphic embedding of ${\mathrm{R}}_{3}$ into ${\mathrm{R}}_{4}$. The oscillator representation of the generators of the dynamical group SO(4,2) is realized for both systems in a unified way. It is shown that the free-particle wave functions form the basis of the unitary irreducible representation D(0,0) of the homogeneous Lorentz group instead of $\mathrm{D}{(0,(\ensuremath{-}\frac{\frac{1}{2}Z}{E})}^{\frac{1}{2}})$, $E>0$, in the case of the hydrogen atom.