Abstract

In this work, we discuss the topological transformation of quantum dynamics by showing the wave dynamics of a quantum particle on different types of topological structures in various dimensions from the fundamental polygons of the corresponding universal covering spaces. This is not the view from different perspectives of an observer who simply uses different coordinate systems to describe the same physical phenomenon but rather possible geometric and topological structures that quantum particles are endowed with when they are identified with differentiable manifolds that are embedded or immersed in Euclidean spaces of higher dimension. We present our discussions in the form of Bohr model in one, two and three dimensions using linear wave equations. In one dimension, the fundamental polygon is an interval and the universal covering space is the straight line and in this case the standing wave on a finite string is transformed into the standing wave on a circle which can be applied into the Bohr model of the hydrogen atom. In two dimensions, the fundamental polygon is a square and the universal covering space is the plane and in this case, the standing wave on the square is transformed into the standing wave on different surfaces that can be formed by gluing opposite sides of the square, which include a 2-sphere, a 2-torus, a Klein bottle and a projective plane. In three dimensions, the fundamental polygon is a cube and the universal covering space is the three-dimensional Euclidean space. It is shown that a 3-torus and the manifold K × S1 defined as the product of a Klein bottle and a circle can be constructed by gluing opposite faces of a cube. Therefore, in three-dimensions, the standing wave on a cube is transformed into the standing wave on a 3-torus or on the manifold K × S1. We also suggest that the mathematical degeneracy may play an important role in quantum dynamics and be associated with the concept of wavefunction collapse in quantum mechanics.

Highlights

  • Introductory SummaryIn our previous works on spacetime structures of quantum particles, we showed that quantum particles can be endowed with various geometric and topological structures of differentiable manifolds and classified according to the mathematical structures that are determined by the wavefunctions that are used to express the geometrical objects associated with the quantum particles, such as the Gaussian curvature and the Ricci scalar curvature

  • In this work, we discuss the topological transformation of quantum dynamics by showing the wave dynamics of a quantum particle on different types of topological structures in various dimensions from the fundamental polygons of the corresponding universal covering spaces

  • The fundamental polygon is an interval and the universal covering space is the straight line and in this case the standing wave on a finite string is transformed into the standing wave on a circle which can be applied into the Bohr model of the hydrogen atom

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Summary

Introductory Summary

In our previous works on spacetime structures of quantum particles, we showed that quantum particles can be endowed with various geometric and topological structures of differentiable manifolds and classified according to the mathematical structures that are determined by the wavefunctions that are used to express the geometrical objects associated with the quantum particles, such as the Gaussian curvature and the Ricci scalar curvature. Since an n-sphere can degenerate itself into a single point, the mathematical degeneracy may be related to the concept of wavefunction collapse in quantum mechanics where the classical observables such as position and momentum can only be obtained from the collapse of the associated wavefunctions for physical measurements This consideration suggests that quantum particles associated with differentiable manifolds may possess the more stable mathematical structures of an n-torus rather than those of an n-sphere, as a brief investigation into different methods of embeddings of differentiable manifolds in Euclidean spaces, we will examine the geometric and topological structures of the familiar 2-torus and how it can be isometrically embedded in the ambient three-dimensional Euclidean space R3. The purpose of this work is to discuss the topological transformation of quantum dynamics of quantum particles in the following we will focus only on linear wave equations on different topological structures that can be formed from the fundamental polygons of their corresponding universal covering spaces in one, two, and three dimensions

Geometric and Topological Transformation of Bohr Model of the Hydrogen Atom
R d2R dr 2
Geometric and Topological Transformation of a Two-Dimensional Wave Dynamics
H d2H d2η
Geometric and Topological Transformation of a Three-Dimensional Wave Dynamics
Conclusion
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