Dirac quantisation of motion on a helicoid

Goh, Yong Shen (2022) Dirac quantisation of motion on a helicoid. [Project Paper] (Submitted)

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Abstract

The quantisation of motion on curved manifolds through canonical quantisation has always been a tricky issue. It is well-established in the literature that Dirac's quantisation procedure requires the use of the Cartesian coordinate system, which is problematic for curved manifolds as there is no global Cartesian coordinate system for these surfaces. In other to overcome this problem, the canonical quantisation scheme is enhanced to include additional requirements. This thesis aims to determine the best geometric approach to take in the canonical quantisation for curved manifolds and identify the geometric potential and geometric momentum from the best approach with the help of the enhanced canonical quantisation (ECQ) scheme. To achieve the objective of this thesis, the motion of a particle on a helicoid is quantised using Dirac’s canonical quantisation procedure for constrained systems and the ECQ scheme. The problem will be approached through the intrinsic geometry approach via the system's local coordinates and the submanifold approach via the Cartesian coordinate system. The system will first undergo a classical mechanical treatment to determine its Dirac brackets, which will be followed up with the quantum mechanical treatment to determine its commutators through canonical quantisation. The results have shown that the quantisation of motion on a helicoid cannot be achieved through the intrinsic geometry approach due to a breakdown in algebraic structures. However, it can be achieved through the submanifold approach, and a self-consistent description of the problem is obtained. These results have validated the importance of the Cartesian coordinate system in canonical quantisation and have shown the importance of using the enhanced canonical quantisation scheme for the quantisation of motion on curved manifolds.

Item Type:	Project Paper
Faculty:	Fakulti Sains
Depositing User:	Ms. ROHANA ALIAS
Date Deposited:	18 Dec 2024 08:14
Last Modified:	18 Dec 2024 08:14
URI:	http://psaspb.upm.edu.my/id/eprint/1992

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