Hwang Cheuk Man. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 34-35). / Abstracts in English and Chinese. / Chapter 1 --- Introduction and Main Results --- p.5 / Chapter 2 --- Some qualitative results --- p.9 / Chapter 3 --- Existence result --- p.13 / Chapter 4 --- Asymtoptic behavior of the gound state solution --- p.21 / Bibliography --- p.34

published_or_final_version / Mechanical Engineering / Master / Master of Philosophy

by Kang Xiaosong. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 44-47). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.4 / Chapter 2 --- Preliminary Analysis --- p.11 / Chapter 3 --- Liapunov-Schmidt Reduction --- p.16 / Chapter 4 --- A Maximizing Procedure --- p.27 / Chapter 5 --- Proof of Theorem 1.1 --- p.30 / Chapter 6 --- Proof of Theorem 1.2 --- p.33 / Chapter 7 --- Concluding Remarks --- p.42

by Tso Man Kit. / Thesis submitted in: December 2001. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 58-61). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Preliminary analysis --- p.14 / Chapter 3 --- Liapunov-Schmidt reduction --- p.23 / Chapter 4 --- A minimizing procedure --- p.36 / Chapter 5 --- Proof of theorem 11 --- p.40 / Chapter 6 --- Proof of theorem 12 --- p.43 / Chapter 7 --- Proof of theorem 13 --- p.55 / Bibliography --- p.58

In this thesis I look at the theory of compact operators in a general Hilbert space, as well as the inverse of the Hamiltonian operator in the specific case of L2[a,b]. I show that this inverse is a compact, positive, and bounded linear operator. Also the eigenfunctions of this operator form a basis for the space of continuous functions as a subspace of L2[a,b]. A numerical method is proposed to solve for these eigenfunctions when the Hamiltonian is considered as an operator on Rn. The paper finishes with a discussion of examples of Schrödinger equations and the solutions.

In principle Quantum Mechanics allows the creation of macroscopic mass superposition states - so called "Schrödinger Cat States". This has not been confirmed experimentally largely due to the difficulty of isolating such states from environmental decoherence. It is of interest to create massive superpositions both in order to test Quantum Mechanics and to shed light on the elusive 'measurement problem'. This thesis presents the theoretical analysis of, and the initial experimental steps towards, an ambitious proposal to test the superposition principle of Quantum Mechanics at the 10^-12 kg-scale, approximately nine orders of magnitude more massive than any superposition observed to date. The experimental principle is that a small mirror mounted on a micro-mechanical oscillator (cantilever) forms one end of a high-finesse cavity in one arm of a Michelson interferometer and is coupled to a single photon by radiation pressure. The photon, in a superposition of each arm, and the cantilever evolve into a superposition involving two distinct locations of the cantilever. By observing the interference of the photon only, one can study the creation and decoherence of the combined state. Firstly, a detailed analysis of the experimental requirements is given based on (1) the need for sufficient momentum transfer from the photon to displace the micro-mirror/cantilever to a distinguishable degree, (2) the need to isolate the cantilever to avoid significant environmental decoherence, and (3) the need to have sufficient interferometric stability to perform the measurement. An iterative analysis was performed to optimise these to a set that is feasible with current technology. This demands: (1) cavity mirrors with a reflectivity of R ≥ 0.9999998 at visible wavelength, (2) a system temperature of ≤ 3mK, (3) a cantilever mechanical quality Q ≥ 10⁵ , (4) a vacuum with gas particle density of 1012/m3, (5) a relative position stability of the cavity mirrors of ≤ 10^-13 m/min, and (6) optical mirror switching to 50% for ≤ lμs. Whilst extremely demanding, all of these goals appear to be within reach of current technology. Secondly, initial experimental results are described: (1) the fabrication of a 10μm radius dielectric mirror designed for peak reflectivity R > 0.99997 and the attaching of this to an AFM-type cantilever of mechanical quality Q > 4 x 10⁴ ; (2) the alignment of a cavity of length 2.5cm involving this micro-mirror/cantilever at one end and the demonstration of a finesse of F > 1000 using two independent measurement techniques. The diffraction losses for the cavity are calculated numerically to be < 10^-6 . Other mechanisms limiting the finesse are investigated and the dominant one is determined to be accoustic noise which can be alleviated by placing the cavity into a vacuum. In addition, results demonstrating ultra-fast optical switching of high reflectivity mirrors are shown.

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The Schrödinger equation ... is considered. The solution of this equation is reduced to the problem of finding the eigenvectors of an infinite matrix. The infinite matrix is truncated to a finite matrix. The approximation due to the truncation is carefully studied. The band structure of the eigenvalues is shown. The eigenvectors of the multiwells potential are presented. The solutions of Schrödinger equation are calculated. The results are very sensitive to the value of the parameter y. Localized solutions, in the case that the energy is slightly greater than the maximum value of the potential, are presented. Wigner and Weyl functions, corresponding to the solutions of Schrödinger equation, are also studied. It is also shown that they are very sensitive to the value of the parameter y. / Garyounis University and Libyan Cultural Affairs

published_or_final_version / abstract / toc / Mechanical Engineering / Master / Master of Philosophy