Geometry and Topology of Escape I: Epistrophes

We consider a dynamical system given by an area-preserving map on a two-dimensional phase plane and consider a one-dimensional line of initial conditions within this plane. We record the number of iterates it takes a trajectory to escape from a bounded region of the plane as a function along the line of initial conditions, forming an ``escape-time plot''. For a chaotic system, this plot is in general not a smooth function, but rather has many singularities at which the escape time is infinite; these singularities form a complicated fractal set. In this article we prove the existence of regular repeated sequences, called ``epistrophes'', which occur at all levels of resolution within the escape-time plot. (The word ``epistrophe'' comes from rhetoric and means ``a repeated ending following a variable beginning''.) The epistrophes give the escape-time plot a certain self-similarity, called ``epistrophic'' self-similarity, which need not imply either strict or asymptotic self-similarity.Comment: 15 pages, 9 figures, to appear in Chaos, first of two paper

Resonances and Recurrences in the Absorption Spectrum of an Atom in an Electric Field

We use closed-orbit theory to study the absorption spectrum of an atom in an electric field. In previous work we examined absorption spectra above the zero-field ionization threshold. Only one closed orbit exists there, and it is unstable. Now we examine the situation below threshold. Here, the orbit parallel to the electric field is stable and, as the energy decreases, many other closed orbits bifurcate out of it. These closed orbits have simple patterns, and the associated recurrences are most clear if the absorption spectrum is measured using a scaled-variables method. The relation between the semiclassical Einstein-Brillouin-Keller-Marcus (EBKM) theory and periodic-orbit or closed-orbit theory is examined: they are complementary methods in the same sense that energy and time are complementary variables in quantum mechanics. Our numerical calculations show that sinusoidal fluctuations contributed by the closed orbits combine into peaks, and these peaks are in the locations predicted by EBKM theory

Closed-Orbit Theory of Oscillations in Atomic Photoabsorption Cross Sections in a Strong Electric Field. II. Derivation of Formulas

A formula for photoabsorption cross sections of hydrogen and alkali-metal atoms in a static electric field is derived, based on the closed-orbit theory previously used to study hydrogen in a magnetic field. Electric fields are simpler than magnetic fields, because the classical motion is regular and closed orbits can be enumerated. In alkali metals the core modifies the relevant dipole matrix elements, and it produces additional phase shifts

Geometry and Topology of Escape II: Homotopic Lobe Dynamics

We continue our study of the fractal structure of escape-time plots for chaotic maps. In the preceding paper, we showed that the escape-time plot contains regular sequences of successive escape segments, called epistrophes, which converge geometrically upon each endpoint of every escape segment. In the present paper, we use topological techniques to: (1) show that there exists a minimal required set of escape segments within the escape-time plot; (2) develop an algorithm which computes this minimal set; (3) show that the minimal set eventually displays a recursive structure governed by an ``Epistrophe Start Rule'': a new epistrophe is spawned Delta = D+1 iterates after the segment to which it converges, where D is the minimum delay time of the complex.Comment: 13 pages, 8 figures, to appear in Chaos, second of two paper

Semiclassical Picture of Collision-induced Λ-doublet Transitions in Diatomic Molecules

We investigate collision-induced Λ-doublet transitions in a system similar to NO+Ar, based on a semiclassical model in which nuclear motion is treated classically and electronic motion quantum mechanically. We present a picture of this process by monitoring〈Λ〉, the expectation value of the projection of electronic orbital-angular momentum onto the molecular NO axis, over the duration of the collision. In a typical collision, the interaction with Ar would cause the electronic orbital-angular momentum to precess about the rotating NO–Ar vector. However, since this angular momentum is locked tightly to the diatomic axis, it is restricted to oscillation along this axis. This oscillation leads to transitions between Λ-doublet states. In addition to providing this physical picture of the collision process, we calculate an alignment effect of 1.2 for a hypothetical three-vector correlation experiment, neglecting spin

Quantum Manifestations of Bifurcations of Closed Orbits in the Photoabsorption Spectra of Atoms in Electric Fields

The methods developed in the preceding paper (paper II) are used to construct a wave function near a bifurcation of classical orbits of an atomic electron in an electric field. A formula for the recurrence strength near the bifurcation is derived and compared with experimental measurements

Energy‐Moment Methods in Quantum Mechanics

Three quantum‐mechanical computational techniques based on energy moments, μk = ∫ dqψ*(q)Hkψ(q)μk=∫dqψ*(q)Hkψ(q), and semimoments, νk(q′) = [Hkψ(q)]q = q′νk(q′)=[Hkψ(q)]q=q′, are formulated. The μ method, which employs the μk, is connected to the method of moments in probability theory, to the variational method, and to eigenvalue spectroscopy. The ν and λ methods, which employ semimoments, are related to local energy methods using one and several configuration points, respectively. An Nth‐order calculation, requiring 2N moments or semimoments, yields N approximate eigenvalues and eigenfunctions. In accordance with a conjectured convergence criterion, exact eigenstates are approached in the limit N→∞. From quantities obtained in a moments calculation, a lower bound on the ground‐state eigenvalue can also be determined using a refinement of Weinstein's criterion. A computational method for generating moments and semimoments is given and the μ method is applied to the linear harmonic oscillator.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70024/2/JCPSA6-47-8-2784-1.pd

Semiclassical Model of Λ-doublet States in Diatomic Molecules

An intuitive picture of Λ-doubling in diatomic molecules is presented using a semiclassical theory. A common view of Λ-doubling as arising from electrons “lagging” behind the rotating internuclear axis is shown to be misleading; rather, the eigenfunctions are symmetric about the molecular axes and can be expressed as a superposition of pure nonrotating orbitals and travelling waves. These results are shown to be consistent with a full quantum treatment. We also examine, for the first time, time-dependent states, by monitoring expectation values of electronic- and nuclear-angular momenta. For low rotation frequency, the expectation value of the electronic-angular momentum locks onto the rotating internuclear axis, while for high rotation frequency it locks onto the space-fixed total-angular momentum axis. At intermediate frequencies is a complicated behavior

Closed-Orbit Theory of Oscillations in Atomic Photoabsorption Cross Sections in a Strongelectric Field. I. Comparison Between Theory and Experiments on Hydrogen and Sodium Above Threshold

Using a simple analytic formula from closed-orbit theory, we calculate photoabsorption cross sections of hydrogen and sodium in a strong electric field. The theoretical spectra show good agreement with experimental results. A scaled variable measurement is also suggested