Dieter Wintgen

The semiclassical helium atom

Recent progress in the semiclassical description of two-electron atoms is reported herein. It is shown that the classical dynamics for the helium atom is of mixed phase space structure, i.e., regular and chaotic motion coexists. Semiclassically, both types of motion require separate treatment. Stability islands are quantized via a torus-quantization-type procedure, whereas a periodic-orbit cycle expansion approach accounts for the states associated with hyperbolic electron pair motion. The results are compared with highly accurate ab initio quantum calculations, most of which are reported here for the first time. The results are discussed with an emphasis on previous interpretations of doubly excited electron states

Analysis of classical motion on the Wannier ridge

The authors study the classical electronic motion of a two-electron atom on the Wannier ridge. Stability properties of certain periodic orbits are calculated. The analysis shows that the helium atom is not ergodic. Semiclassical implications of the classical analysis are discussed.

Novel dynamical symmetries of asymmetrically doubly excited two-electron atoms

The authors study a new class of long-lived resonances of doubly excited two-electron atoms. The states are characterized by a highly polarized inner electron located near the axis between the nucleus and a dynamically localized outer electron. Classical mechanics studies prove the stability of the corresponding classical motion and allow for EBK quantization to obtain semiclassical energies. The resonance states are treated further within the framework of a single-channel adiabatic approximation, where the inner electron is prescribed by a polarized molecular type wavefunction. The adiabatic energies as well as the semiclassical results are in good agreement with resonance energies obtained by highly accurate solutions of the full three-body Schrodinger equation. There is a one-to-one correspondence between approximate quantum numbers derived from the semiclassical and from the adiabatic approach, both of which explain the nodal structures of the ab initio quantum wavefunctions reflecting the approximate dynamical symmetries of the problem.

The nodal structure of doubly-excited resonant states of helium

The authors examine the nodal structure of accurate helium wavefunctions calculated by direct diagonalization of the full six-dimensional problem. It is shown that for fixed interelectronic distance R (or hyperspherical radius R) the symmetric doubly-excited resonant states have well defined lambda , mu nodal structure indicating a near separability in prolate spheroidal coordinates. For fixed lambda , however, a clear mixing of R, mu nodes is demonstrated. This corresponds to a breakdown of the adiabatic approximation and can be understood in terms of the classical two-electron motion.

Quantization of chaotic systems.

Starting from the semiclassical dynamical zeta function for chaotic Hamiltonian systems we use a combination of the cycle expansion method and a functional equation to obtain highly excited semiclassical eigenvalues. The power of this method is demonstrated for the anisotropic Kepler problem, a strongly chaotic system with good symbolic dynamics. An application of the transfer matrix approach of Bogomolny is presented leading to a significant reduction of the classical input and to comparable accuracy for the calculated eigenvalues.

Scars in wavefunctions of the diamagnetic Kepler problem

The localization of eigenfunctions around classical periodic orbits is studied numerically for the H-atom in a strong magnetic field by calculating their Husimi distribution in phase space. In contrast to the configuration space representation, the phase space distributions are simply structured: about 90% of eigenstates may be unambigously related to fixed points and invariant manifolds of periodic orbits, indicating that scars are the rule rather than the exception. In order to measure the influence of one particular orbit, we calculate the integrals of the energetically lowest 500 Husimi distributions along the orbit. Their incoherent superposition defines the scar strength distribution for the particular periodic orbit which is analyzed by Fourier transformation. The Husimi distribution at (q, p) in phase space may be represented as a scalar product of the wavefunction with a coherent state of the unperturbed system, i.e., a radial Gaussian wave packet located at (q, p) in the (regularized) Coulomb system. This simplifies the actual calculation of the Husimi distribution and allows to treat their incoherent superposition within Gutzwillers theory extended to matrix elements of an operator A, if we choose A to be the projector on a coherent state.

Calculations of planetary atom states

The authors prove the existence of a new type of resonance formation in highly doubly excited atoms and ions. The resonant states represent the quantum analogue of a stable planetary atom configuration discovered recently in classical mechanics. The energies and total decay widths of the resonances are calculated quantum mechanically by solving the full three-body Coulomb problem without any approximation. The resonance energies and the structure of the associated wavefunctions coincide with expectations derived from simple semiclassical models. The authors discuss excitation schemes which should allow an experimental observation of these strongly correlated electron states.

Calculation of the Liapunov exponent for periodic orbits of the magnetised hydrogen atom

The author presents a classical analysis of the energy and field strength dependence of the stability coefficients (Liapunov exponents) for certain periodic orbits of a hydrogen atom in a uniform magnetic field.

Efficient quantisation scheme for the anisotropic Kepler problem

We have developed an efficient quantisation scheme for highly excited states of the anisotropic Kepler problem. This applies to the level spectrum of a donor impurity in semiconductors. Results for silicon and germanium are presented.

Classical and semiclassical zeta functions in terms of transition probabilities

Abstract We propose a semiclassical quantization scheme for bound hyperbolic systems based on the properties of a single ergodic trajectory. The dynamics of the system is approximated by transition probabilities between cells of a partition of the phase-space. We construct a transfer matrix of the corresponding Markov graph which approaches the classical Frobenius-Perron (transfer) operator in the limit of infinitesimal tesselations of the phase-space. A semiclassical zeta function may be obtained as the determinant of an appropriately weighted transfer operator and leads to a product over the closed paths of the graph in close analogy to the Gutzwiller-Voros zeta function which is a product over periodic orbits.