Honors College Thesis
 

Investigating polymer quantum mechanics : mathematical formulation, the particle on a ring, and time evolution

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https://ir.library.oregonstate.edu/concern/honors_college_theses/rr172571p

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  • Loop quantum gravity is a theoretical framework which aims to quantize general relativity. One of the unique aspects of this theory is that it imposes a discrete structure on space-time. One toy model of loop quantum gravity is loop quantum cosmology, which maintains the idea of a discrete spatial structure while reducing the allowed degrees of freedom. The assumption of discrete space leads to a complication in quantization, because we are not able to take infinitesimal changes in order to define momentum on a discrete lattice. Thus the model of polymer quantum mechanics was formulated to test whether the results derived from using a discrete formulation of space could still generate verifiable results. Previous researchers have used polymer quantum mechanics on the free particle and harmonic oscillator toy problems in the momentum basis, and have found that in the continuum the results converge to those found in Schrodinger quantization which in turn confirmed the validity of polymer quantum mechanics in these situations. In this paper, I describe the construction and mathematical formulation of polymer quantum mechanics, as well as its application to the particle on a ring, and investigate time evolution of energy eigenstates. Using the derived energy eigenstates, I was able to verify that in the assumption that time is continuous, the Hamiltonian is still the generator of time evolution in polymer quantum mechanics. In addition, I constructed a dispersion relation of a localized superposition of energy eigenstates which corresponds to a single normalized position state. I then used this to calculate the expectation values of both position and momentum and found that position has an expectation value which depends on time. However we found instances which were inconsistent with Schrodinger quantum mechanics, including the normalization of distinct position eigenstates. These specific position eigenstates were formed as a linear superposition of energy eigenvalues, and serves as an interesting case example for where new and interesting results can emerge with different quantization schemes. Furthermore, when we introduce time evolution, it appears that the states transform the same way they would in the continuum, implying that there is some intrinsic connection between the discrete spatial structure and the preservation of time evolution.
  • Key Words: quantum gravity, quantum mechanics, polymer quantum mechanics, loop quantum cosmology, time
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