Current Search: Department of Physics (x) » Beetle, Christopher (x)
View All Items
- Title
- General relativistic quasi-local angular momentum continuity and the stability of strongly elliptic eigenvalue problems.
- Creator
- Wilder, Shawn M., Beetle, Christopher, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Physics
- Abstract/Description
-
In general relativity, angular momentum of the gravitational field in some volume bounded by an axially symmetric sphere is well-defined as a boundary integral. The definition relies on the symmetry generating vector field, a Killing field, of the boundary. When no such symmetry exists, one defines angular momentum using an approximate Killing field. Contained in the literature are various approximations that capture certain properties of metric preserving vector fields. We explore the...
Show moreIn general relativity, angular momentum of the gravitational field in some volume bounded by an axially symmetric sphere is well-defined as a boundary integral. The definition relies on the symmetry generating vector field, a Killing field, of the boundary. When no such symmetry exists, one defines angular momentum using an approximate Killing field. Contained in the literature are various approximations that capture certain properties of metric preserving vector fields. We explore the continuity of an angular momentum definition that employs an approximate Killing field that is an eigenvector of a particular second-order differential operator. We find that the eigenvector varies continuously in Hilbert space under smooth perturbations of a smooth boundary geometry. Furthermore, we find that not only is the approximate Killing field continuous but that the eigenvalue problem which defines it is stable in the sense that all of its eigenvalues and eigenvectors are continuous in Hilbert space. We conclude that the stability follows because the eigenvalue problem is strongly elliptic. Additionally, we provide a practical introduction to the mathematical theory of strongly elliptic operators and generalize the above stability results for a large class of such operators.
Show less - Date Issued
- 2014
- PURL
- http://purl.flvc.org/fau/fd/FA00004235
- Subject Headings
- Boundary element methods, Boundary value problems, Differential equations, Elliptic -- Numerical solutions, Differential equations, Partial -- Numerical solutions, Eigenvalues, Spectral theory (Mathematics)
- Format
- Document (PDF)
- Title
- Improving In Vivo Two Photon Microscopy Without Adaptive Optics.
- Creator
- Estrada, Gerardo, Beetle, Christopher, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Physics
- Abstract/Description
-
Two photon microscopy is one of the fastest growing methods of in-vivo imaging of the brain. It has the capability of imaging structures on the scale of 1μm. At this scale the wavelength of the imaging field (usually near infra-red), is comparable to the size of the structures being imaged, which makes the use of ray optics invalid. A better understanding is needed to predict the result of introducing different media into the light path. We use Wolf's integral, which is capable of fulfilling...
Show moreTwo photon microscopy is one of the fastest growing methods of in-vivo imaging of the brain. It has the capability of imaging structures on the scale of 1μm. At this scale the wavelength of the imaging field (usually near infra-red), is comparable to the size of the structures being imaged, which makes the use of ray optics invalid. A better understanding is needed to predict the result of introducing different media into the light path. We use Wolf's integral, which is capable of fulfilling these needs without the shortcomings of ray optics. We predict the effects of aberrating media introduced into the light path like glass cover-slips and then correct the aberration using the same method. We also create a method to predict aberrations when the interfaces of the media in the light-path are not aligned with the propagation direction of the wavefront.
Show less - Date Issued
- 2015
- PURL
- http://purl.flvc.org/fau/fd/FA00004495
- Subject Headings
- Cellular signal transmission -- Measurement, Image analysis, Imaging systems in medicine, Membranes (Biology) -- Imaging, Neurons -- Imaging, Optics, Adaptive
- Format
- Document (PDF)
- Title
- ON THE PARAXIAL APPROXIMATION IN QUANTUM OPTICS.
- Creator
- Jongewaard, Maria Fernanda de Boer, Beetle, Christopher, Florida Atlantic University, Department of Physics, Charles E. Schmidt College of Science
- Abstract/Description
-
We examine how best to associate quantum states of a single particle to modes of a narrowly collimated beam of classical radiation modeled in the paraxial approximation, both for scalar particles and for photons. Our analysis stresses the importance of the relationship between the inner product used to define orthogonal modes of the paraxial beam, on the one hand, and the inner product underlying the statistical interpretation of the quantum theory, on the other. While several candidates for...
Show moreWe examine how best to associate quantum states of a single particle to modes of a narrowly collimated beam of classical radiation modeled in the paraxial approximation, both for scalar particles and for photons. Our analysis stresses the importance of the relationship between the inner product used to define orthogonal modes of the paraxial beam, on the one hand, and the inner product underlying the statistical interpretation of the quantum theory, on the other. While several candidates for such an association have been proposed in the literature, we argue that one of them is uniquely well suited to the task. Specifically, the mapping from beam modes to ”henochromatic” fields on spacetime is unique within a large class of similar mappings in that it is unitary in a mathematically precise sense. We also show that the single-particle quantum states associated to the orthogonal modes of a classical beam in the henochromatic approach are not only orthogonal, but also complete in the quantum Hilbert space.
Show less - Date Issued
- 2023
- PURL
- http://purl.flvc.org/fau/fd/FA00014212
- Subject Headings
- Quantum optics, Hilbert space, Quantum theory
- Format
- Document (PDF)