A Study Of The Geometric Momentum And Quantization Of A Constrained System In Phase Space

To examine the motion of a particle on a two-dimensional curved surfaceembedding in the three-dimensional Cartesian space, we identify a new physicalquantity-Geometric Momentum. It is a new fundamental quantity, which dependson the mean curvature of the surface. This finding stimulates a scrutinization of theconcepts such as generalized or canonical momentum in quantum mechanics. As aresult, the canonical momentum is demonstrated a more mathematical concept than aphysical one. In contrast, the Geometric Momentum offers a proper description ofmomentum.Yet, people only have an elementary understanding of the Geometric Momentum.Majority of its properties may stay unknown. For example, is it a purely item inquantum mechanics or explicitly relative to a classical quantity? This thesis mainlyexplores the classical correspondence of Geometric Momentum on two kinds ofconstraints-sphere and torus. This classical correspondence is studied in theframework of Dirac’s theory of second class constrains in classical and quantummechanics. Moreover, the traditional geometric constraints will be replaced by adynamical one in phase space. Then a quantity that is similar to GeometricMomentum is found in classical mechanics.First, in a three-dimensional Cartesian system, a particle in the phase spaceconstraint corresponding to a sphere is studied by Dirac’s theory, with utilization ofthe canonical quantization. Results show that Geometric Momentum satisfies Dirac’stheory, where the symmetrization technique is used.Secondly, in a three-dimensional Cartesian frame, a particle in the phase spaceconstraint corresponding to a torus is studied by Dirac’s theory, also with utilization ofthe canonical quantization. Results show that Geometric Momentum satisfies Dirac’stheory. The structure of torus is more complicated, the calculation about torus isuniversally applicable. It is not sufficient to use the symmetrization technique here;the dummy factor must be taken into consideration. By dummy factor we mean afactor that plays an essential role in quantum mechanics but vanishes in classicalmechanics.Thirdly, when there is an arbitrary vector potential on the sphere, the methods ofquantization, especially how to fix the vector potential, is discussed.This thesis presents an intensive research of geometric momentum on twodifferent topological systems, sphere and torus. Results show that there is classicalcorrespondence of Geometric Momentum. These researches deepen the understandingof the Dirac’s theory, which includes the possibly universal principle——the relationbetween the classical correspondence of quantum mechanics, and the transition ofclassical mechanics to the quantum one, is asymmetric.