Hamiltonian bifurcation theory of closed orbits in the diamagnetic Kepler problem

Abstract

Classically chaotic systems possess a proliferation of periodic orbits. This phenomenon was observed in a quantum system through measurements of the absorption spectrum of a hydrogen atom in a magnetic field. This paper gives a theoretical interpretation of the bifurcations of periodic or closed orbits of electrons in atoms in magnetic fields. We ask how new periodic orbits can be created out of existing ones or "out of nowhere" as the energy changes. Hamiltonian bifurcation theory provides the answer: it asserts the existence of just five typical types of bifurcation in conservative systems with two degrees of freedom. We show an example of each type. Every case we have examined falls into one of the patterns described by the theory.

Semiclassical trace formulas for pitchfork bifurcation sequences

“…Such a cascade can form a geometric progression reminiscent of the Feigenbaum scenario [26] (although there the bifurcations are generically period doubling), and the new periodic orbits born at the bifurcations may exhibit self-similarity properties [18,19]. Bifurcation cascades are frequently found in physical systems with discrete symmetries and mixed classical dynamics [16,17], so that a semiclassical approach to those situations would seem to have been required long ago. Here we develop a uniform approximation of codimension two for the contribution of a pair of pitchfork bifurcations to the semiclassical density of states and test it numerically by comparison with exact quantum-mechanical calculations.…”

“…Explicit semiclassical trace formulae have been given for various systems with continous symmetries [3,5,6,7], for symmetry breaking through the destruction of rational tori [8,9,10,11,12], and for isolated bifurcations [13,14,15]. However more complicated bifurcation scenarios which usually occur in realistic physical systems and, in particular, bifurcation cascades [16,17,18,19] still constitute one of the most serious problems of the semiclassical theory.…”

Closed orbits and their bifurcations in the crossed-field hydrogen atom

“…Therefore, any closed orbit is either itself periodic or it is one half of a periodic orbit. Due to the close link between closed orbits and periodic orbits, closed-orbit bifurcations can be described in the framework of periodic-orbit bifurcation theory developed by Mayer [8,22]. In particular, in a magnetic field closed orbits possess repetitions, so that arbitrary m-tupling bifurcations are possible.…”

“…in the hydrogen atom in a pure magnetic field. For these orbits a complete classification is available [3,4,5,6,7,8]. It will now be recapitulated briefly.…”

“…For the hydrogen atom in a magnetic field, the systematics of closed orbits and their bifurcations has been known for a long time [3,4,5,6,7,8]. For the hydrogen atom in crossed electric and magnetic fields, the classical mechanics is much more complicated because three non-separable degrees of freedom have to be dealt with.…”