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Quantum level structures at a Fermi resonance with angular momentum: classical periodic orbits, catastrophe maps and
1The University of Oxford, Physical and Theoretical Chemistry Department, South Parks Road, Oxford, UK.
Physical Chemistry Chemical Physics : PCCP
|September 29, 2005
Summary
The Xiao-Kellman catastrophe map now includes finite vibrational angular momentum. This reveals how classical orbits influence quantum eigenvalues, showing dislocations due to quantum monodromy.
Area of Science:
- Quantum mechanics
- Classical mechanics
- Chemical physics
Background:
- The Xiao-Kellman catastrophe map classifies periodic orbits in the 2:1 Fermi resonance Hamiltonian.
- Previous studies focused on systems without vibrational angular momentum.
Purpose of the Study:
- Extend the Xiao-Kellman map to include finite vibrational angular momentum.
- Investigate the impact of classical periodic orbit structure on quantum mechanical eigenvalues.
- Demonstrate quantum monodromy in this extended system.
Main Methods:
- Extension of the Xiao-Kellman catastrophe map.
- Analysis of classical periodic orbits.
- Examination of quantum mechanical eigenvalues in angular momentum and energy space.
Main Results:
- The extended map successfully classifies periodic orbits for systems with finite vibrational angular momentum.
- Classical orbit structure significantly influences quantum eigenvalue organization across the map's regions.
- Quantum eigenvalue lattices exhibit dislocations attributed to quantum monodromy.
Conclusions:
- Quantum monodromy is a topological effect influencing quantum eigenvalue structure.
- Analogies exist between this system and quantum monodromy in quasi-linear molecules and LiCN/LiNC isomerization.
- The extended map provides a framework for understanding quantum-classical correspondence in resonant systems.