A Study Of The Geometric Momentum On Two-dimensional Surface

Quantum mechanics on the surface has not been an excited research area.Though many problems remained unsolved for long, physics community had not beencared about too much. Recently, along with rapid development of the nano-scienceand–technology, we are confronted with many problems such as the quantum motionof electrons moving on the Carbon60or its rotation, etc.; so a complete formalism ofthe quantum mechanics on the surface becomes imperative. In wading through thescattered literature, physicists finds some evidence that some results, based ondifferent aspects and developed independently, are in fact interrelated with each other.More importantly, Dirac outlined a theoretical framework called quantum mechanicsfor constrained motion, but it needs improving.Geometric momentum is a novel, fundamental physical quantity in representingquantum motion on surface. It was proposed independently, and was found that itcould be formulated into the Dirac’s theory on constrained systems. When theconstraints are free, geometric momentum assumes its usual form.In quantum mechanics, a quantity that can be measured must have spectralrepresentation. In other words, its eigenfunctions form a complete set and theireigenvalues are real. For two-dimensional surface, geometric momentum has threecomponents. The present dissertation mainly studies these operators, their eigenvalues,eigenfunctions and their relations between, and is organized into four parts.First, through solving partial differential equations, the eigenvalue problem isdirectly settled down. Results show that all eigenvalues are continous, and threeeigenfunctions are normalized to Dirac delta function. In contrast to the usualeigenfunction whose normalization is constant, the normalization factors may containarbitrary function of some variables.Second, in order to the relations between three components of the geometricmomentum, the symmetry under rotation is explored. Results show that the geometricmomentum is really vector operator that means that these three operators areequivalent under coordinate rotations.Third, in principle the eigenfunctions determined by three components of thegeometric momentum can be transformed from one to another by unitarytransformations, but the transformation cannot be explicitly carried out. However, asimple coordinate transformation fulfils such a task. Results show that the arbitraryfunctions in the normalization factors come as a consequence of the transformation.Fourth, a mapping is pointed out that the eigenfunctions of the geometricmomentum can be transformed into the form of the plane wave, implying a new meanto project of the sphere into plane.In sum, as long as the two-dimensional spherical surface is concerned, geometricmomentum is perfect in mathematics but also a measurable quantity.