Large order perturbation theory in the context of atomic and molecular physics—interdisciplinary aspects

Abstract

Abstract The paper is introduced by a description of the role of divergent series in mathematics and their particular use in quantum mechanics. It is emphasized that there is a formal similarity between divergent series in nonrelativistic quantum mechanics and those series occurring in quantum field theory. It is indicated that studies of divergent series in nonrelativistic quantum mechanics can be helpful in the verification of suminability techniques in quantum field theory. The general problems of the application of perturbation theory to the study of the hydrogen atom in external fields are discussed. It is shown how the problem of the continuous spectrum can be bypassed. Further it is shown that algebraic techniques are very useful for a transparent formulation of perturbation theory. The theory is applied to the study of the hydrogen atom in a magnetic field and in an electric field and also to the study of the hydrogen molecule ion. The results for the hydrogen atom in a magnetic field and the hydrogen molecule ion are discussed in detail and the large order behavior of perturbation terms is compared with analytic formulas. Finally the questions of the summability for the series corresponding to the quartic anharmonic oscillator and the hydrogen atom in a magnetic field are discussed. It is seen that large order perturbation theory can be built into a continued fraction representation of a divergent perturbation series. Numerical results are presented for the two perturbation series mentioned above.