| Summary: | 博士 === 國立成功大學 === 航空太空工程學系碩博士班 === 101 === The geometrical phase is noticed after the development of conventional quantum mechanics gradually matured. It describes the single-value property of the wavefunction phase accumulation when varying adiabatically around a closed circuit. However, due to the lack of the concept of a dynamic trajectory in the probability-interpreted quantum mechanics, it is difficult to verify the applicability of the geometric phase. Complex mechanics, as a bridge between the macroscopic classical mechanics and the microscopic quantum mechanics, provides a nonlinear dynamic trajectory interpretation that cannot be revealed in conventional quantum mechanics.
The nonlinear system analysis tools used in classical mechanics now can be applied to the quantum world via complex nonlinear dynamic representation. Hence, the quantum world can be displayed in terms of a dynamic trajectory interpretation instead of using probability interpretation. Complex mechanics provides the complex Schrödinger equation with a new description by considering the complex space extension and a quantum potential within this complex space.
The dissertation begins with the introduction of the nonlinear dynamic trajectory on the basis of complex mechanics, than applies the index theorem in classical nonlinear systems to the quantum system. This is done by presenting the quantum index theorem and also is able to find the corresponding physical meanings for new defined quantum indices. It is found that the geometrical phase of the wavefunction gives an important geometrical meaning in the quantum index theorem. It shows that the closed loop dynamic trajectory in complex space satisfies the single-value property of the wavefunction and classically exhibits the geometrical phase which originally could not be explained properly in conventional quantum mechanics. The quantum geometrical phase interpreted by the dynamic trajectory has the coherent appearance of the geometrical phase proposed by probability-interpreted quantum mechanics. This coherence relation is discussed via the study of the electronic dynamics in a uniform magnetic field and its related Aharonov-Bohm effect. The simulation result of the study is compatible with the experimental result, which provides strong evidence that the trajectory interpretation in complex space is indeed a correct framework of quantum mechanics.
Finally, the author actually applies the quantum geometrical phase to the field of engineering via a superconducting device which has macroscopic quantum behavior. Josephson junction is a basic device used in superconducting engineering. The characteristics of the junction are described by a fictitious particle’s tunneling behavior on the wavefunction phase difference plane in the macroscopic scale. Complex mechanics connects quantum mechanics to classical mechanics, providing the dynamic trajectory interpretation such that it can discuss and clarify those problems which are not represented fully in quantum mechanics.
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