Cartesian Momentum Of Particles On Curved Surface

The orthodox studies of quantum motion restricted on the curved surface are based on the intrinsic point of view in geometry. The surface in physics can not be a geometric surface of no width, and it has at least one atomic layer thick. Therefore a proper treatment of the quantum motion can be embedded into higher dimensional.flat space. A significant progress appeared in 1982 when R.C.T.da Costa pointed out that the operator of kinetic energy of a particle confined on the geometrical surface attain an attractive geometrical potential.Along with the progress in technology and science at nano-scale, many fundamental problems of quantum mechanics have attracted much attention. Recently, there are studies focusing on how to get the momentum operator of the particles on the curved surface. Interestingly, there are two different momentum operators. In 2006, Alexey V. Golovnev presented a form of the Cartesian operator within algebra method. Entirely independently in 2007, our group reported another form of the Cartesian operator with the geometrical method. In fact the two operators are equivalent.This dissertation basically includes the following two studies:First, we calculate the expectations of the Cartesian momentum operator over the wave packet on sphere. By introducing the rectangular wave packet, we have obtained the mean expectation values of the coordinates, momenta, and the square of these operators. The results show that, for some particular wave packet, the average values of Cartesian coordinates, moments are zero, while the average values of the square of them changing over time. However, in the classical limit, the mean values of the Cartesian coordinates, moments and the square of moments for the classical wave packet go over to classical quantities in classical limit.Second, with noting that application to nontrivial surface, there is impossibly to develop an analytical solution of the energy spectrum. In this study, we obtain the energy eigenvalues of the system on ellipsoid with small eccentricity, up to the first approximation. The result shows that the approximation of the energy spectrum is symmetry for magnetic quantum number. When the magnetic quantum number is zero, the approximated energy increases with the increasing of argular quantum number.