Transition Probabilities of the Hydrogen Atom from Noncompact Dynamical Groups

Abstract

In order to describe electromagnetic dipole transitions of the H atom within the framework of dynamical groups, an explicit irreducible representation of the Lie algebra $O\left(4.2\right)\mathrm{}$ has been found on the space of bound-state wave functions. This representation remains irreducible when restricting to the subalgebra $O(4,1)\mathrm{}$. The transformation properties of the dipole operator in $O(4,2)\mathrm{}$ have been specified. The description becomes particularly simple by the introduction of a one-parameter family of representations of $O(4,2)\mathrm{}$. Finally, position representations of the generators of $O(4,2)\mathrm{}$ have been given.