Quantum chaos ofSU3observables

Abstract

Hamiltonians built of SU 3 generators are presented which allow for qualitatively different classical limits. The generators of the group SU 3 give eight independent observables. In the classical limit two invariant functions of these observables, the so-called Casimir functions c 2 and c 3 , take fixed values in the ranges ½ c 2 (2/3) and -(2/9) c 3 (2/9) (an SU 2 system, in contrast, has just one such Casimir function, the squared total angular momentum). Generically, this leads to a six-dimensional phase space. However, for two special (`degenerate') pairs of values c 2 = (2/3),c 3 = -(2/9) and c 2 = (2/3),c 3 = (2/9) the classical dynamics is only four-dimensional. One and the same Hamilton function of the generators may be integrable in the four-dimensional phase space but generate chaos in the six-dimensional case. We give two examples of Hamiltonians for which that alternative does arise; one of these is closely related to the Lipkin Hamiltonian familiar from nuclear shell models. The transition from integrable to chaotic classical behaviour is accompanied by the usual quantum transition from level clustering to level repulsion.