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.
Although the fundamental principles of time-independent perturbation theory are well-established and widely applied, the specific treatment of degeneracy lifted to the second order remains largely unexplored. This work fills the gap by presenting a systematic derivation of time-independent perturbation theory that accounts for the added complexity introduced by non-trivial degeneracies. We provide a general procedure and corresponding formulae for calculating state and energy corrections to any order, with degeneracy lifted to the second order. Additionally, we apply these formulae to solve a prototype problem of the dressed quark system. This work not only advances the theoretical framework but also provides practical tools for tackling complex problems in quantum physics.
Calculations of the electronic energy of the Be-atom ground state are performed using a configuration-interaction technique and a nondegenerate formulation of diagrammatic many-body perturbation theory. Both methods are applied within the algebraic approximation defined by three finite-basis sets. One of the basis sets contains only
s functions, and it defines an algebraic problem without degeneracies where agreement between the two methods is excellent. The other two basis sets, although more complete, both contain
p functions, and they define algebraic problems involving a near degeneracy where agreement between the two methods is found to be less satisfactory. It is concluded that either a degenerate formulation of perturbation theory is required, or certain diagrams in the model perturbative scheme need to be summed to high order through denominator shifts in order to properly handle the nondynamic-correlation effect found in the Be atom.