Undergraduate Honors Thesis


Classical and Quantum Reduction of the Hydrogen Atom Public Deposited

  • In this thesis we lay out an overview of the mathematics required to understand the reduction of symmetry of the classical Kepler problem and its quantum counterpart the hydrogen atom within the framework of symplectic geometry and deformation quantization respectively. In order to do so, we must cover the mathematical generalization of Hamiltonian mechanics, symplectic geometry. We will show how conservation of energy and Liouville's theorem manifest themselves within this generalization using the tools of differential geometry. After introducing differential and symplectic geometry, we give brief introduction to Lie Groups, their actions on smooth manifolds, and moment maps. Lie groups allow us to formalize the very physical ideas behind continuous symmetry, and they play a principal role in the reduction of symmetry, along with moment maps, as we will see in the latter half of this thesis. After having developed the prerequisite theory, we then tackle the reduction of the Kepler problem, otherwise known as the two body problem. This section makes mathematically rigorous the method by which the equations of motion for the two body problem are obtained in a typical undergraduate analytical mechanics course. After the classical reduction is finished we motivate the idea of quantization as P.A.M. Dirac does in his 1925 paper. We then cite the Gronewold van Hove no go theorems to show that such a quantization scheme can not be strictly satisfied even on $\R^{2n}$ and use this to motivate the definitions underlying deformation quantization. After defining deformation quantizations we give a few examples of such structures on $\R^{2n}$ and reduce one such example, the Weyl-Moyal product using the classical reduction tools we developed for the Kepler problem.

Date Awarded
  • 2022-04-04
Academic Affiliation
Committee Member
Granting Institution
Last Modified
  • 2022-04-18
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