O(4,2) operator replacements for highly excited atomic states.

Abstract

It is shown that for highly excited atomic states, quantum-mechanical operators involving position and momentum can be approximated by O(4,2) generators. The resulting Schr\"odinger equation is solved exactly in the case of hydrogenlike systems, for both eigenvalues and eigenvectors. The precision of the approximations involved in these replacements can thus be tested. It is verified that the new solutions converge to the hydrogenlike solutions as n/Z goes to infinity. n is the principal quantum number and Z the charge of the nucleus in atomic units. The O(4) dynamics symmetry of the hydrogenlike atoms is preserved and the new constant of the motion identified. The resulting Schr\"odinger equation for two electron atoms is then presented and formulated in terms of scaled doubly excited states basis vectors previously introduced with the o(4) algebra. This equation is equivalent to a sparse infinite system of linear equations whose nonzero coefficients have very simple analytical expressions.