Knowledge Bank

Following a suggestion of Gell-Mann, Ne'eman and $Dothan1$, a number of $theorists2$ this last summer independently discovered that the bound state wave functions of the Kepler problem belong to a single (infinite dimensional) representation of the de Sitter group, $O(4,1)$---a non compact group whose underlying Lie algebra is the same as that of the 5 dimensional real orthogonal group $O(5)$. As a consequence, any operator which interconverts hydrogenic bound state functions may be expressed as a function of the 10 generators of this Lie algebra. Using these observations it is a simple matter to develop our recent $O(4)$ method of determining bound state molecular $orbitals2$ into a purely group theoretical approach---an approach in which every operation is expressible in terms of the operations of the group $O(4,1)$, and every operator and wave function can be classified according to the irreducible representations of the group. After a brief discussion of the Lie algebra and its representations we will show how it is used, and will in fact illustrate its use in a simple molecular calculation. The viewpoint implied in the method will be discussed and its advantages pointed out.