The research presented in this Honors Thesis provides development in machine learning models which predict future states of a system with unknown dynamics, based on observations of the system. Two case studies are presented for (1) a non-conservative pendulum and (2) a differential game dictating a two-car uncontrolled intersection scenario. In the paper we investigate how learning architectures can be manipulated for problem specific geometry. The result of this research provides that these problem specific models are valuable for accurate learning and predicting the dynamics of physics systems.<br/><br/>In order to properly model the physics of a real pendulum, modifications were made to a prior architecture which was sufficient in modeling an ideal pendulum. The necessary modifications to the previous network [13] were problem specific and not transferrable to all other non-conservative physics scenarios. The modified architecture successfully models real pendulum dynamics. This case study provides a basis for future research in augmenting the symplectic gradient of a Hamiltonian energy function to provide a generalized, non-conservative physics model.<br/><br/>A problem specific architecture was also utilized to create an accurate model for the two-car intersection case. The Costate Network proved to be an improvement from the previously used Value Network [17]. Note that this comparison is applied lightly due to slight implementation differences. The development of the Costate Network provides a basis for using characteristics to decompose functions and create a simplified learning problem.<br/><br/>This paper is successful in creating new opportunities to develop physics models, in which the sample cases should be used as a guide for modeling other real and pseudo physics. Although the focused models in this paper are not generalizable, it is important to note that these cases provide direction for future research.

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.

With the extreme strides taken in physics in the early twentieth century, one of the biggest questions on the minds of scientists was what this new branch of quantum physics would be able to be used for. The twentieth century saw the rise of computers as devices that significantly aided in calculations and performing algorithms. Because of the incredible success of computers and all of the groundbreaking possibilities that they afforded, research into using quantum mechanics for these systems was proposed. Although theoretical at the time, it was found that a computer that had the ability to leverage quantum mechanics would be far superior to any classical machine. This sparked a wave of interest in research and funding in this exciting new field. General-use quantum computers have the potential to disrupt countless industries and fields of study, like physics, medicine, engineering, cryptography, finance, meteorology, climatology, and more. The supremacy of quantum computers has not yet been reached, but the continued funding and research into this new technology ensures that one day humanity will be able to unlock the full potential of quantum computing.

We implemented the well-known Ising model in one dimension as a computer program and simulated its behavior with four algorithms: (i) the seminal Metropolis algorithm; (ii) the microcanonical algorithm described by Creutz in 1983; (iii) a variation on Creutz’s time-reversible algorithm allowing for bonds between spins to change dynamically; and (iv) a combination of the latter two algorithms in a manner reflecting the different timescales on which these two processes occur (“freezing” the bonds in place for part of the simulation). All variations on Creutz’s algorithm were symmetrical in time, and thus reversible. The first three algorithms all favored low-energy states of the spin lattice and generated the Boltzmann energy distribution after reaching thermal equilibrium, as expected, while the last algorithm broke from the Boltzmann distribution while the bonds were “frozen.” The interpretation of this result as a net increase to the system’s total entropy is consistent with the second law of thermodynamics, which leads to the relationship between maximum entropy and the Boltzmann distribution.

The self-assembly of strongly-coupled nanocrystal superlattices, as a convenient bottom-up synthesis technique featuring a wide parameter space, is at the forefront of next-generation material design. To realize the full potential of such tunable, functional materials, a more complete understanding of the self-assembly process and the artificial crystals it produces is required. In this work, we discuss the results of a hard coherent X-ray scattering experiment at the Linac Coherent Light Source, observing superlattices long after their initial nucleation. The resulting scattering intensity correlation functions have dispersion suggestive of a disordered crystalline structure and indicate the occurrence of rapid, strain-relieving events therein. We also present real space reconstructions of individual superlattices obtained via coherent diffractive imaging. Through this analysis we thus obtain high-resolution structural and dynamical information of self-assembled superlattices in their native liquid environment.

This document is a guide that can be used by undergraduate physics students alongside Richard J. Jacob and Professor Emeritus’s Tutorials in the Mathematical Methods of Physics to aid in their understanding of the key mathematical concepts from PHY201 and PHY302. This guide can stand on its own and be used in other upper division physics courses as a handbook for common special functions. Additionally, we have created several Mathematica notebooks that showcase and visualize some of the topics discussed (available from the GitHub link in the introduction of the guide).