On a bifurcation in the quantum system for an approximation to the Hénon-Heiles system

Abstract

It is known, in classical mechanics, that the Hénon-Heiles system exhibits bifurcation phenomena in its periodic solutions depending on a parameter. These phenomena are usually analyzed by applying the Birkhoff-Gustavson's normal form method; the Hénon-Heiles Hamiltonian is expanded into a series in normal form, and then truncated at a finite order. It is also known that the bifurcation taking place in the truncated system approximates the one taking place in the original Hénon-Heiles system in case of sufficiently small energy. In this regard, the aim of this article is to find what will happen when the truncated Hénon-Heiles system is quantized. Quantum analogue to the classical bifurcation phenomena are observed as follows: Degeneracy of energy eigenvalues proves to occur at a certain parameter value. At that value the quantum system happens to admit the SO(2)×D α symmetry, D α being the dihedral group. However, at a generic parameter value the symmetry reduces to SO(2)×D 2. Changes in eigenstates depending on the parameter are observed; a symmetry breaking of zero angular momentum state takes place when the parameter passes that particular value.