New perturbation treatment of scattering and bound-state problem

David Atkinson

George L. Johnston, Garrison Sposito

The Stark effect of the plane rotator is a pedagogically appealing problem for undergraduate courses on quantum mechanics and, accordingly, has appeared in a number of textbooks as an application of bound-state perturbation theory. Despite this fact and the apparent simplicity of the problem, all of the published accounts of it are erroneous in that they employ nondegenerate perturbation theory and present incorrect expressions for the perturbed eigenvalues and eigenstates. In this paper the problem is given a new treatment, based upon a simple analysis of symmetry features and a careful application of degenerate perturbation theory. The major shortcomings of previous discussions are pointed out and correct expressions are derived, to second order, for the perturbed state vectors and energy levels. The lifting of the twofold degeneracy of the mth unperturbed excitation energy in the 2mth order of perturbation theory is demonstrated. Finally, a comparison of the correct perturbation-theoretic results with the exact solution of the problem shows the two to be in identical correspondence.

Horacio E. Camblong, CARLOS R. ORDÓÑEZ

A path-integral approach for \ensuremath{\delta}-function potentials is presented. Particular attention is paid to the two-dimensional case, which illustrates the realization of a quantum anomaly for a scale-invariant problem in quantum mechanics. Our treatment is based on an infinite summation of perturbation theory that captures the nonperturbative nature of the \ensuremath{\delta}-function bound state. The well-known singular character of the two-dimensional \ensuremath{\delta}-function potential is dealt with by considering the renormalized path integral resulting from a variety of schemes: dimensional, momentum-cutoff, and real-space regularization. Moreover, compatibility of the bound-state and scattering sectors is shown.