Mapping degenerate perturbations in atoms onto an asymmetric rotor

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Abstract
Several problems in atomic physics, such as diamagnetism in high Rydberg states, or the electron-electron interaction in high doubly excited states, and analogous problems at high excitation in nuclear physics or in other branches of physics, display certain characteristic and common features. Among these are the existence of classes of sharply localized states formed through large superpositions of basis states that are degenerate in the absence of the interaction. The asymmetric rotor at high angular momentum J displays these same features: In particular, most of its eigenstates divide into two groups with different localizations (along the directions of minimum and maximum moments of inertia). A few states lie in a ‘‘separatrix’’ region in between, corresponding to localization along the axis with an intermediate moment of inertia. A 1:1 mapping is made from the atomic and nuclear problems into the asymmetric rotor, wherein the principal quantum number n and orbital angular momentum l of the former are put in correspondence, respectively, with J and M, the azimuthal projection of the rotor’s angular momentum. For each given problem, the ‘‘asymmetry parameter’’ of the corresponding rotor is identified, and eigenvalues and eigenfunctions are presented. The concept of a conjugate rotor is also introduced. The key, common feature that underlies localization is the vanishing as l→n of the off-diagonal coupling between the states in a degenerate manifold {nl} just as in the asymmetric rotor where the matrix elements of the step-up and step-down operators in the ‖JM〉 basis vanish when M→±J.

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