Relations between propagators and Green functions of Hamiltonians which are SUSY partners have been obtained. New exact propagators for the family of multi-well, time-dependent and non-hermitian potentials have been calculated. Non-conservative SUSY transformation has been studied in the case of multichannel Schrodinger equation with different thresholds. Spectrum (bound/virtual states and resonances) of the non-conservative SUSY partner of zero potential has been founded. Exactly solvable model of the magnetic induced Feshbach resonance has been constructed. This model was tested in the case of Rb$^{85}$. Conservative SUSY transformations of the first and the second order have been studied in the case of multichannel Schrodinger equation with equal thresholds. Transformations which introduce non-trivial coupling between scattering channels have been founded. The first order SUSY transformation which preserves $S$-matrix eigen-phase shifts and modifies mixing parameter has been founded in the case of two channel scattering with partial waves of different parities. In the case of coinciding parities we have found the second order SUSY transformation which preserves $S$-matrix eigen-phase shifts and modifies mixing parameter. Phenomenological two channel $^3S_1-^3D_1$ neutron-proton potential has been obtained by using single channel, phase equivalent and coupling SUSY transformations applied to zero potential.

In this thesis I address two different topics in quantum theory. The first one is the long discussed Schrodinger's cat problem, and the issues related to having a macroscopic superposition state. I show that the quantum theory provides full explanation to the problem. In the second part, I discuss the time measurement related issues in quantum mechanics. Since there does not exist any time operator in quantum mechanics generally, time is not directly measurable. Therefore we should devise other methods to register time. We study different time-energy relations and will find that accurate clocks have high energy uncertainties. If we use accurate clocks in quantum systems to observe their time evolutions, their high energy uncertainties interfere with system's normal evolution and slows it down. I also provide a formal proof to a previously suggested limiting accuracy relation on the measurements of the time-of-arrival experiments.

This thesis consists of the four papers which consider different aspects of stochastic process modeling, error analysis, and minimization of computational cost.      In Paper I, we construct a Multipath Fading Channel (MFC) model for wireless channels with noise introduced through scatterers flipping on and off. By coarse graining the MFC model a Gaussian process channel model is developed. Complexity and accuracy comparisons of the models are conducted.      In Paper II, we generalize a multilevel Forward Euler Monte Carlo method introduced by Mike Giles for the approximation of expected values depending on solutions of Ito stochastic differential equations. Giles' work proposed and analyzed a Forward Euler Multilevel Monte Carlo (MLMC) method based on realizations on a hierarchy of uniform time discretizations and a coarse graining based control variates idea to reduce the computational cost required by a standard single level Forward Euler Monte Carlo method. This work is an extension of Giles' MLMC method from uniform to adaptive time grids. It has the same improvement in computational cost and is applicable to a larger set of problems.      In paper III, we consider the problem to estimate the mean of a random variable by a sequential stopping rule Monte Carlo method. The performance of a typical second moment based sequential stopping rule is shown to be unreliable both by numerical examples and by analytical arguments. Based on analysis and approximation of error bounds we construct a higher moment based stopping rule which performs more reliably.      In paper IV, Born-Oppenheimer dynamics is shown to provide an accurate approximation of time-independent Schrödinger observables for a molecular system with an electron spectral gap, in the limit of large ratio of nuclei and electron masses, without assuming that the nuclei are localized to vanishing domains. The derivation, based on a Hamiltonian system interpretation of the Schrödinger equation and stability of the corresponding hitting time Hamilton-Jacobi equation for non ergodic dynamics, bypasses the usual separation of nuclei and electron wave functions, includes caustic states and gives a different perspective on the Born-Oppenheimer approximation, Schrödinger Hamiltonian systems and numerical simulation in molecular dynamics modeling at constant energy. / <p>QC 20120508</p>

This study is about quantum physics and the role it can play in a religious perspective. I intend to examine how moderna physics may be related and understood as New age and Myths. My material consists of two books in the new age genre: "The Tao of Physics" by Fritjof Capra and "The Quantum Self" by Danah Zohar. My first two question to these books are which 1) religious and 2) scientific content they have. My other questions are too 3) investigate and demonstrate how these books are new age and 4) to exanine how the books can function as myths and adopt mythical features. And last 5) to compare important content and ideas between the books. My methods are a simple hermeneutic close reading with a qualitative inductive analysis grounded in previous research and theory. My method is also inspired by a contextual analysis of ideas and an exploratory study. In the investigation I start by showing what religious and scientific materials the books contain. My conslusion is that these two books thoroughly fulfill many criteria for the New age genre. They should however not be counted as Myths becuse they do not meet the requirenents I have defined in the theory section. Moreover, the books share many similarities, such as many common references and resistance to some ideas from the Western tradition of ideas. But there are also some differences, mainly in how they use quantum physics. Firstly, a difference is that Capra primarily make parallels between quantum physics and Eastern philosophical tradition which Zohar does not. Secondly, Capra uses quantum physics to approach the mesage, that is primarily derived from Buddhism, which is that the reality is dynamic and in the end that the world is a "non-existing thing". Zohar is opposed to this and she would call it an extreme solipsism. She argues that the reality consists of both a particle and wave structure wich she applices at the consciousness.

We introduce the de Broglie-Bohm causal interpreation of quantum mechanics and compare it to the standard interpretation of quantum mechanics, the Copenhagen interpretation. We examine the possibility of experimentally distinguishing between the two theories, as well as the potential for the causal interpretation to more easily bridge the gap between the physics of the quantum and classical worlds. We then use the causal interpretation to construct a deterministic model of the helium atom in which the two electrons move along trajectories through space and time about a stationary nucleus. The dynamics are governed by the non-relativistic Schrödinger equation and the spin vectors of both electrons are assumed to be constant along their respective trajectories. We examine the Bohmian trajectories associated with (approximations to) eigenstates of the helium Hamiltonian as well as the trajectories associated with some non-eigenstates. We also compute an approximation to the ground state energy of the helium atom using a representation of the helium wavefunction in terms of hydrogenic eigenfunctions which is motivated by a perturbation approach.

Abstract This thesis has two parts, in the first, the Caldeira-Leggett model is introduced; its derivation and general consequences are explored following a paper by Caldeira and Leggett[1]. An operator-formalism shortcut through some of the more mathematically cumbersome parts of the derivation of the model is also developed. The correlation of the force resulting from reservoir-interaction is examined in the high- and low-temperature limits, and the Langevin equation is shown to emerge in the classical limit.Abstract The second part introduces decoherence through a thought experiment that demonstrates the destructive effect of random phase shifts on interference terms, and then follows another paper by Caldeira and Leggett[2] in applying their model further to study the phenomenon of dissipative decoherence. The time-evolution of the interference terms in a superposition of Gaussian wave packets in a harmonic oscillator potential is studied when interacting with a heat bath, and they are shown to vanish at a rate much faster than the relaxation of the system.

We introduce the de Broglie-Bohm causal interpreation of quantum mechanics and compare it to the standard interpretation of quantum mechanics, the Copenhagen interpretation. We examine the possibility of experimentally distinguishing between the two theories, as well as the potential for the causal interpretation to more easily bridge the gap between the physics of the quantum and classical worlds. We then use the causal interpretation to construct a deterministic model of the helium atom in which the two electrons move along trajectories through space and time about a stationary nucleus. The dynamics are governed by the non-relativistic Schrödinger equation and the spin vectors of both electrons are assumed to be constant along their respective trajectories. We examine the Bohmian trajectories associated with (approximations to) eigenstates of the helium Hamiltonian as well as the trajectories associated with some non-eigenstates. We also compute an approximation to the ground state energy of the helium atom using a representation of the helium wavefunction in terms of hydrogenic eigenfunctions which is motivated by a perturbation approach.

We study the integral equation related to the three and higher dimensional superradiance problem. Collective radiation phenomena has attracted the attention of many physicists and chemists since the pioneering work of R. H. Dicke in 1954. We first consider the three-dimensional superradiance problem and find a differential operator that commutes with the integral operator related to the problem. We find all the eigenfunctions of the differential operator and obtain a complete set of eigensolutions for the three-dimensional superradiance problem. Generalization of the three-dimensional superradiance integral equation is provided. A commuting differential operator is found for this generalized problem. For the three dimensional superradiance problem, an alternative set of complete eigenfunctions is also provided. The kernel for the superradiance problem when restricted to one-dimension is the same as appeared in the works of Slepian, Landau and Pollak. The uniqueness of the differential operator commuting with that kernel is indicated. Finally, a concentration problem for the signals which are bandlimited in disjoint frequency-intervals is considered. The problem is to determine which bandlimited signals lose the smallest fraction of their energy when restricted in a given time interval. A numerical algorithm for solution and convergence theorems are given. Orthogonality properties of analytically extended eigenfunctions over L2(−∞,∞) are also proved. Numerical computations are carried out in support of the theory.

Contemporary materialism, which tries to explain the working principles of the mind and the universe, become less meaningful after the developments in the modern physics. The modern physics showed that the definition of matter, as it is used in defining materialism, is no longer valid. Chomsky states his position as &ldquo / Chomsky&#039 / s challenge to materialism&rdquo / by claiming that with the abolishment of the definition of the matter, there is no reason to defend materialism, which depends on that definition. Therefore, materialism becomes an empty doctrine thus cannot be used in explaining the mind. The developments in the modern physics creates the need for a new doctrine, which can explain the mind and at the same time be compatible with the modern physics and possible any future physics. This new doctrine, the methodological physicalism, aims to explain the mind by using the modern physics. Creating such a doctrine requires understanding of materialism and its form as well as understanding the problems of materialism and its forms. By identifying the defects in materialism and by using the modern physics as a standing point methodological physicalism can achieve a more successful understanding of the working of the mind. By using the modern physics, the methodological physicalism can explain why the currents models of the mind fail. Moreover, it can explain how certain models of the mind constructed, which employs the quantum mechanics while explaining the mind. The methodological physicalism will help understanding the mind where materialism fails to do so.

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