The asymmetric rotor as a model for localization

Abstract

Together with the pendulum, the symmetric and asymmetric rotors are standard models, well studied in both classical and quantum physics. In the limit of high angular momentum $J$, the quantum problem of an asymmetric rotor approaches the classical limit, which is characterized by one unstable and two stable motions. Rotations with respect to the axes with maximum and minimum moments of inertia are stable, while the rotations about the intermediate axis are unstable. Upon diagonalization of the rotor's Hamiltonian at fixed $J$, most eigenvectors are seen to be localized around the directions of maximum and minimum moment of inertia. The asymmetric rotor displays in this regard localization similar to that in many atomic and nuclear few-body systems, wherein external interactions mix degenerate states, particularly at high excitation. Indeed, there is a complete one-to-one correspondence between the rotor and these systems, with the principal ($n$) and angular momentum ($l$) quantum numbers mapping onto $J$, and its azimuthal projection $M$, respectively, for the rotor. Localization at a maximum or a saddle point of a potential is of considerable significance in a variety of problems.