Flow equations for the Hénon–Heiles Hamiltonian

Abstract

The Hénon–Heiles Hamiltonian was introduced in 1964 [M. Hénon, C. Heiles, Astron. J. 69 (1964) 73] as a mathematical model to describe the chaotic motion of stars in a galaxy. By canonically transforming the classical Hamiltonian to a Birkhoff–Gustavson normal form, Delos and Swimm obtained a discrete quantum mechanical energy spectrum. The aim of the present work is to first quantize the classical Hamiltonian and to then diagonalize it using different variants of flow equations, a method of continuous unitary transformations introduced by Wegner in 1994 [Ann. Physik (Leipzig) 3 (1994) 77]. The results of the diagonalization via flow equations are comparable to those obtained by the classical transformation. In the case of commensurate frequencies the transformation turns out to be less lengthy. In addition, the dynamics of the quantum mechanical system are analyzed on the basis of the transformed observables.