Topological Generalizations of the Heisenberg Uncertainty Relation

Abstract

It is well known that the standard canonical uncertainty relation does not apply to the angular variable ? and its conjugate LZ. That is, the relation ? ø ? L Z > h/2 is false. The break down of the result has to do with difference in topology between the line and the circle. It is thus desirable to generalize the standard uncertainty relation topologically and find satisfactory results for the non-Euclidean spaces. This problem is intimately related to the issue of finding a consistent definition for quantum mechanics on "curved spaces". Just as the Heisenberg uncertainty relation was pivotal in understanding the basic structure of standard quantum mechanics, a solution to this problem should shine some light onto the proper conduct of quantum mechanics on general topological spaces. In this study we explore in detail how the standard uncertainty relation may breakdown. We also address the importance of topological considerations in quantum mechanics in general - we shall show how a change in topological character can change the nature of the quantum mechanics for a system and how the consideration of the topology of a system can greatly organize the solution of a problem and in some cases even be necessary for a. full understanding of the problem. We then discuss the derivation of satisfactory uncertainty relations for the compact, homogeneous spaces of the circle, the n-torus and the n-sphere. Finally, we draw out any implications to the issue of properly defining quantum mechanics on the non- Euclidean spaces.

Notes

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Thesis Completion

2006

Semester

Spring

Advisor

Efthimious, Costas

Degree

Bachelor of Science (B.S.)

College

College of Arts and Sciences

Degree Program

Physics

Subjects

Dissertations, Academic -- Sciences; Sciences -- Dissertations, Academic

Format

Print

Identifier

DP0022002

Language

English

Access Status

Open Access

Length of Campus-only Access

None

Document Type

Honors in the Major Thesis

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