Quantum chaos of the hydrogen atom in a generalized van der Waals potential

Poisson-to-Wigner crossover transition in the nearest-neighbor statistics of random points on fractals

We show that the nearest-neighbor spacing distribution for a model that consists of random points uniformly distributed on a self-similar fractal is the Brody distribution of random matrix theory. In the usual context of Hamiltonian systems, the Brody parameter does not have a definite physical meaning, but in the model considered here, the Brody parameter is actually the fractal dimension. Exploiting this result, we introduce a new model for a crossover transition between Poisson and Wigner statistics: random points on a continuous family of self-similar curves with fractal dimensions between 1 and 2. The implications to quantum chaos are discussed, and a connection to conservative classical chaos is introduced.

Hydrogen atom in the presence of uniform magnetic and quadrupolar electric fields: integrability, bifurcations, and chaotic behavior

We investigate the classical dynamics of a hydrogen atom in the presence of uniform magnetic and quadrupolar electric fields. After some reductions, the system is described by a two degree of freedom Hamiltonian depending on two parameters. On the one hand, it depends on the z component of the canonical angular momentum P(phi), which is an integral because the system is axially symmetric; and on the other it also depends on a parameter representing the relative field strengths. We note that this Hamiltonian is closely related to the one describing the generalized van der Waals interaction. We report three cases of integrability. The structure and evolution of the phase space are explored intensively by means of Poincaré surfaces of section when the parameters vary. In this sense, we find several bifurcations that strongly change the phase space structure. The chaotic behavior of the system is studied and three order-chaos transitions are found when the system passes through the integrable cases. Finally, the ionization mechanics is studied.