published_or_final_version / Mechanical Engineering / Master / Master of Philosophy

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.

No description available.

No description available.

<p>Abstract: Considering the Schrödinger equation $\Delta_x u = i\partial{u}/\partial{t}$, we have a solution $u$ on the form $$u(x, t)= (2\pi)^{-n} \int_{\RR} {e^{i x\cdot \xi}e^{it|\xi|^2}\widehat{f}(\xi)}\, d \xi, x \in \RR, t \in \mathbf{R}$$ where $f$ belongs to the Sobolev space. It was shown by Sjögren and Sjölin, that assuming $\gamma : \mathbf{R}_+ \rightarrow \mathbf{R}_+ $ being a strictly increasing function, with $\gamma(0) = 0$ and $u$ and $f$ as above, there exists an $f \in H^{n/2} (\RR)$ such that $u$ is continuous in $\{ (x, t); t>0 \}$ and $$\limsup_{(y,t)\rightarrow (x,0),|y-x|<\gamma (t), t>0} |u(y,t)|= + \infty$$ for all $x \in \RR$. This theorem was proved by choosing $$\widehat{f}(\xi )=\widehat{f_a}(\xi )= | \xi | ^{-n} (\log | \xi |)^{-3/4} \sum_{j=1}^{\infty} \chi _j(\xi)e^{- i( x_{n_j} \cdot \xi + t_j | \xi | ^a)}, \, a=2,$$ where $\chi_j$ is the characteristic function of shells $S_j$ with the inner radius rapidly increasing with respect to $j$. The purpose of this essay is to explain the proof given by Sjögren and Sjölin, by first showing that the theorem is true for $\gamma (t)=t$, and to investigate the result when we use $$S^a f_a (x, t)= (2 \pi)^{-n}\int_{\RR} {e^{i x\cdot \xi}e^{it |\xi|^a}\widehat{f_a}(\xi)}\, d \xi$$ instead of $u$.</p>

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2012. / Unable to load abstract.

The theme of this thesis is combinatorics, complex analysis and algebraic geometry. The thesis consists of six articles divided into four parts. Part A: Spectral properties of the Schrödinger equation This part consists of Papers I-II, where we study a univariate Schrödinger equation with a complex polynomial potential. We prove that the set of polynomial potentials that admit solutions to the Schrödingerequation is connected, under certain boundary conditions. We also study a similar result for even polynomial potentials, where a similar result is obtained. Part B: Graph monomials and sums of squares In this part, consisting of Paper III, we study natural bases for the space of homogeneous, symmetric and translation-invariant polynomials in terms of multigraphs. We find all multigraphs with at most six edges that give rise to non-negative polynomials, and which of these that can be expressed as a sum of squares. Such polynomials appear naturally in connection to expressing certain non-negative polynomials as sums of squares. Part C: Eigenvalue asymptotics of banded Toeplitz matrices This part consists of Papers IV-V. We give a new and generalized proof of a theorem by P. Schmidt and F. Spitzer concerning asymptotics of eigenvalues of Toeplitz matrices. We also generalize the notion of eigenvalues to rectangular matrices, and partially prove the a multivariate analogue of the above. Part D: Stretched Schur polynomials This part consists of Paper VI, where we give a combinatorial proof that certain sequences of skew Schur polynomials satisfy linear recurrences with polynomial coefficients. / <p>At the time of doctoral defence the following papers were unpublished and had a status as follows: Paper 5: Manuscript; Paper 6: Manuscript</p>

Abstract: Considering the Schrödinger equation $\Delta_x u = i\partial{u}/\partial{t}$, we have a solution $u$ on the form $$u(x, t)= (2\pi)^{-n} \int_{\RR} {e^{i x\cdot \xi}e^{it|\xi|^2}\widehat{f}(\xi)}\, d \xi, x \in \RR, t \in \mathbf{R}$$ where $f$ belongs to the Sobolev space. It was shown by Sjögren and Sjölin, that assuming $\gamma : \mathbf{R}_+ \rightarrow \mathbf{R}_+ $ being a strictly increasing function, with $\gamma(0) = 0$ and $u$ and $f$ as above, there exists an $f \in H^{n/2} (\RR)$ such that $u$ is continuous in $\{ (x, t); t&gt;0 \}$ and $$\limsup_{(y,t)\rightarrow (x,0),|y-x|&lt;\gamma (t), t&gt;0} |u(y,t)|= + \infty$$ for all $x \in \RR$. This theorem was proved by choosing $$\widehat{f}(\xi )=\widehat{f_a}(\xi )= | \xi | ^{-n} (\log | \xi |)^{-3/4} \sum_{j=1}^{\infty} \chi _j(\xi)e^{- i( x_{n_j} \cdot \xi + t_j | \xi | ^a)}, \, a=2,$$ where $\chi_j$ is the characteristic function of shells $S_j$ with the inner radius rapidly increasing with respect to $j$. The purpose of this essay is to explain the proof given by Sjögren and Sjölin, by first showing that the theorem is true for $\gamma (t)=t$, and to investigate the result when we use $$S^a f_a (x, t)= (2 \pi)^{-n}\int_{\RR} {e^{i x\cdot \xi}e^{it |\xi|^a}\widehat{f_a}(\xi)}\, d \xi$$ instead of $u$.

published_or_final_version / Mechanical Engineering / Master / Master of Philosophy