Homeomorphism connecting the hydrogenic system and the free-particle system

The topological equivalence between the hydrogenic system and the free-particle system is established by a homeomorphic embedding of R₃ into R₄. 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 D(0,(-1/2ZE)1/2), $E>0$, in the case of the hydrogen atom.