Classical-quantum correspondence and periodic orbits in doubly-excited states of the helium atom
This thesis studies double-excited states of the Helium atom. Extensive quantum mechanical calculations are carried out for collinear Helium, and the results are interpreted in terms of classical periodic orbits. The influence of unstable classical periodic orbits on quantum eigen-value spectra is examined via Gutzwiller's trace formula. The trace formula provides the only general approach presently known for investigating the properties of quantum systems in terms of completely classical quantities in the semiclassical limit, thereby establishing classical-quantum correspondence. Fluctuations in the quantum density of quasibound, doubly-excited states of the model Helium atom are interpreted in terms of short, unstable classical periodic orbits by use of Gutzwiller's trace formula in reverse. The quantum density of states is calculated by the stabilization method and its Fourier transform examined. The position of each peak in the Fourier transform corresponds quantitatively to the classical action of a corresponding periodic orbit. The absence of a peak corresponding to the symmetric stretching motion of the electron pair may be understood in terms of its relative instability compared to other, asymmetric stretching motions. The author has also studied the properties of quasibound, doubly-excited states of the colinear Helium by calculating their resonance energies and widths. The energies of the excited states in which both electrons are excited to the same level are well approximated by a double-Rydberg formula, which can be obtained by quantizing the asymmetric stretch type motion of the two electrons. Eigenfunctions for doubly-excited states are also examined. Many of the eigenfunctions are affect by short, classical periodic orbits, as had been seen in other systems. The wavefunctions of certain double-excited states show enhanced probability amplitude along these periodic orbits, and therefore clearly manifest the effect of scarring in a three-body system.
- Research Organization:
- Cornell Univ., Ithaca, NY (United States)
- OSTI ID:
- 6923572
- Resource Relation:
- Other Information: Thesis (Ph.D.)
- Country of Publication:
- United States
- Language:
- English
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