As the search for life in our universe grows, it is important to not only locate planets outside of our solar system, but also to work towards the ability to understand and characterize their nature. Many current research endeavors focus on the discovery of exoplanets throughout the surrounding universe; however, we still know very little about the characteristics of these exoplanets themselves, particularly their atmospheres. Observatories, such as the Hubble Space Telescope and the Spitzer Space Telescope, have made some of the first observations which revealed information about the atmospheres of exoplanets but have yet to acquire complete and detailed characterizations of exoplanet atmospheres. The EXoplanet Climate Infrared TElescope (EXCITE) is a mission specifically designed to target key information about the atmospheres of exoplanets - including the global and spatially resolved energy budget, chemical bulk-compositions, vertical temperature profiles and circulation patterns across the surface, energy distribution efficiency as a function of equilibrium temperatures, and cloud formation and distribution - in order to generate dynamic and detailed atmospheric characterizations. EXCITE will use phase-resolved transit spectroscopy in the 1-4 micron wavelength range to accomplish these science goals, so it is important that the EXCITE spectrograph system is designed and tested to meet these observational requirements. For my thesis, I present my research on the EXCITE mission science goals and the design of the EXCITE spectrograph system to meet these goals, along with the work I have done in the beginning stages of testing the EXCITE spectrograph system in the lab. The primary result of my research work is the preparation of a simple optics setup in the lab to prepare a laser light source for use in the EXCITE spectrograph system - comparable to the preparation of incoming light by the EXCITE telescope system - which successfully yields an F# = 12.9 and a spot size of s = 39 ± 7 microns. These results meet the expectations of the system and convey appropriate preparation of a light source to begin the assembly and testing of the EXCITE spectrograph optics in the lab.

This is a primer on the mathematic foundation of quantum mechanics. It seeks to introduce the topic in such a way that it is useful to both mathematicians and physicists by providing an extended example of abstract math concepts to work through and by going more in-depth in the math formalism than would normally be covered in a quantum mechanics class. The thesis begins by investigating functional analysis topics such as the Hilbert space and operators acting on them. Then it goes on to the postulates of quantum mechanics which extends the math formalism covered before to physics and works as the foundation for the rest of quantum mechanics.

Introductory physics is one of the most difficult course sequences one can take as an undergraduate, due in no small part to the prerequisite knowledge of mathematics. Over the past six years, David Meltzer and his research group have developed a diagnostic meant to test students’ abilities in core mathematical concepts believed to be crucial foundations for learning physics. Concepts tested include the ability to solve systems of equations, work with trigonometric functions, manipulate fractions, and interpret information from graphs among others. With over 7000 students having taken the diagnostic, some patterns have begun to emerge, confirming work from other studies that suggest there is in fact a link between prerequisite math knowledge and success in an introductory physics course. However, most students take the diagnostic either in a classroom setting or online, so student responses are largely limited to being categorized as simply correct or incorrect. Even when students’ work is present it is impossible to assess their mindset when working through a problem without making inferences and logical leaps. In an attempt to better understand the nature of students’ misconceptions in mathematics I have conducted seven semi-formal interviews with introductory physics students just after they have completed the diagnostic where they walked me through their solutions and thought processes.

In a hypothetical Grand Unified Theory, magnetic monopoles are a particle which would act as a charge carrier for the magnetic force. Evidence of magnetic monopoles has yet to be found and based off of their relatively high mass (4-10 TeV) will be difficult to find with current technology. The goal of my thesis is to mathematically model the magnetic monopole by finding numerical solutions to the equations of motion. In my analysis, I consider four cases: kinks, cosmic strings, global monopoles, and magnetic monopoles. I will also study electromagnetic gauge fields to prepare to include gauge fields in the magnetic monopole case. Numerical solutions are found for the cosmic string and global monopole cases. As expected, the energy is high at small distance r and drops off as r goes to infinity. Currently numerical solutions are being worked towards for electromagnetic gauge fields and the magnetic monopole case.