Relativistic recursion relations for transition matrix elements

摘要

We review some recent results on recursion relations which help evaluatingarbitrary non-diagonal, radial hydrogenic matrix elements of $r^\lambda$ and of$\beta r^\lambda$ ($\beta$ a Dirac matrix) derived in the context of Diracrelativistic quantum mechanics. Similar recursion relations were derived someyears ago by Blanchard in the non relativistic limit. Our approach is based ona generalization of the second hypervirial method previously employed in thenon-relativistic Schr\"odinger case. An extension of the relations to the caseof two potentials in the so-called unshifted case, but using an arbitraryradial function instead of a power one, is also given. Several importantresults are obtained as special instances of our recurrence relations, such asa generalization to the relativistic case of the Pasternack-Sternheimer rule.Our results are useful in any atomic or molecular calculation which take intoaccount relativistic corrections.