Quantum Mechanical Rigid Rotator with an Arbitrary Deformation. I

Abstract

A rigorous definition is given of rigidity" in the framework of quantum mechanics. It is argued that corresponding to the measurement of the shape of the bodyat least within Gedanken experimentsthere must exist a set of operators associated with this shape. The rigidity of the body is defined such that the shape is in principle measurable without any quantum mechanical uncertainties. For a quadratically deformed rigid body, the commutation relations among the shape operators and angular momentum form a Lie algebra, which is shown to be the semi-direct product of the O(3) to a five-dimensional Abelian group; O(3) T5. The irreducible representations of this dynamical group are explicitly constructed by employing an elementary algebraic method. It is found that the representation can be specified naturally in terms of Bohr and Mottelson's deformation parameters and as ( 2, 3cos 3 ) and that every member of all the possible rotation bands of the rigid rotator is contained in our single irreducible representation once and only once. Several remarks are added concerning the relation of our dynamical group to the more general rotation-vibration group SL(3R).