Dissertation
Semiclassical Propagation of Coherent States and Wave Packets: Hidden Saddles and the Phase of the Complex Prefactor
Washington State University
Doctor of Philosophy (PhD), Washington State University
01/2021
DOI:
https://doi.org/10.7273/000001874
Handle:
https://hdl.handle.net/2376/120698
Abstract
The saddle point method is an important method used in semiclassical approximations involving coherent states and wave packets. A saddle point defines a complex trajectory that satisfies a two-point boundary value problem. The search of saddle points is very challenging when applied to multidimensional systems. Previous advances include finding a set of real boundary conditions by reducing the dimension of the search in real phase space. Each real boundary condition being in the convergence zone of a Newton-Raphson search will lead to a saddle point, and the real trajectories defined by these real boundary conditions are called reference trajectories. The saddle found in this way are called exposed saddles. This thesis broadens the search from exposed saddles to hidden saddles through parameter variations, where hidden saddles are defined as those that can not be found from real boundary conditions directly. The hidden saddles contribute over regions where a wave function decays, in a manner close to tunneling processes. The thesis also discusses a solution to a discrepancy in the phase calculation of the prefactor in the saddle point method. Since the prefactor is the square root of the inverse of a dynamical quantity associated with a complex trajectory, the regular prescription takes the accumulated phase of the dynamical quantity and divides it by half as the phase of the prefactor. This prescription sometimes leads to sign errors due to the fact that the zeros of the dynamical quantity defined on the complex time plane become branch points after the square root operation. Using a complex time path circumventing the zeros in a consistent way can solve this problem. However, for the case of exposed saddles, a more convenient way is to associate the phase to the Maslov index of the real, reference trajectory. The phase of the prefactor associated with a hidden saddle point can be linked to the phase associated with an exposed saddle point by varying the parameters of the trajectories.
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Details
- Title
- Semiclassical Propagation of Coherent States and Wave Packets: Hidden Saddles and the Phase of the Complex Prefactor
- Creators
- HUICHAO WANG
- Contributors
- Steven Tomsovic (Advisor)Philip Marston (Committee Member)Mark Kuzyk (Committee Member)
- Awarding Institution
- Washington State University
- Academic Unit
- Physics and Astronomy, Department of
- Theses and Dissertations
- Doctor of Philosophy (PhD), Washington State University
- Publisher
- Washington State University
- Number of pages
- 118
- Identifiers
- 99900606650801842
- Language
- English
- Resource Type
- Dissertation