Algebra of the Noncompact GroupO(3, 2)and the Hydrogen-Atom Radial Functions

The bound-state hydrogen-atom radial functions are used to construct a basis for a unitary representation of the algebra $O(3, 2)$. With the use of the $O\left(3\right)\mathrm{}$ subalgebra of this "radial algebra" matrix elements of $r^s$ between states of different energy are shown to be completely expressible in terms of the matrix elements of $r^{-2\mathrm{}}$. Matrix elements corresponding to the radial matrix elements of the dipole operator are shown to be proportional to the overlap of the radial functions. The continuum radial functions of constant energy are shown to provide a basis for a unitary representation of $O(2, 1)$.