【作者】 徐敏；

【导师】 刘全慧；

【作者基本信息】 湖南大学 ， 凝聚态物理， 2010， 硕士

【摘要】 传统的研究认为曲面上粒子运动的量子力学都是建立在内禀坐标的基础上的,并得到了曲面上粒子的动能算符。而物理中的曲面并不是简单的几何面,它至少有一个原子层厚度,这样就有一种研究认为要在更高一维的空间中研究曲面。这里一个显著的进展其实出现在1982年,R.C.T.da Costa指出,把几何面考虑成一定厚度的物理面时,体系动能算符多出一项与曲率有关的几何吸引势。纳米科技研究的进展促进了对这个问题的进一步思考。最近有一些研究集中在约束在曲面上运动粒子的动量问题。有意思的是,这里出现了两个不同形式的动量算符。2006年,Alexey V. Golovnev利用代数方法提出了一种笛卡尔算符；2007年,本课题组从几何角度得到了笛卡尔算符的另一个形式。这两个算符实际上是等价的。本论文主要包含两个方面的内容：一、计算球面上波包的笛卡尔动量的期待值,并研究这个算符的经典极限。论文中引入了一种特殊的经典波包——矩形波包,计算了笛卡尔坐标、动量以及它们平方的平均值。结果表明,在经典波包上笛卡尔坐标、动量的平均值为零,而它的平方的平均值随时间变化。但在经典极限下,笛卡尔坐标、动量及其平方在经典波包上的平均值都趋于经典极限。二、将含几何修正项的动能算符应用于非平庸的曲面时,没有解析求解的可能,目前得到的结果大多都是利用数值计算完成的。考虑到具有较小偏心率的旋转椭球面上粒子的能量谱,可以利用微扰法解析求解,可以定量了解几何修正的影响。本研究表明,能量的一级修正关于磁量子数对称；磁量子数为零时,能量的一级修正随角量子数增加而增加。当椭球面回到球面时,结果就回到了我们所熟知的结果。更多还原

【Abstract】 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.更多还原