【作者】 赖玫玫；

【导师】 刘全慧；

【作者基本信息】 湖南大学 ， 理论物理， 2007， 硕士

【摘要】 约束在量子点上的量子运动是目前凝聚态物理研究的一个热点。一个很有意思的问题是量子点的几何形状,除了典型的球形和椭球形外,还有面包圈和M?bius环等形状。这些研究提出了一个基础性的研究课题:在经典力学中,对约束在曲面上粒子运动的描述可以在内部坐标即曲面局部坐标下进行,也可以在外部坐标即在笛卡尔坐标下进行。在微分几何中,对曲面的研究可以利用内禀(intrinsic)几何和外部(extrinsic)几何这两种互补也是基本的基本方法进行。但是量子力学的传统(其实也是整个现代物理例如广义相对论和规范场的研究传统)是只用内部坐标即曲面局部坐标,或者说从内禀的角度,对运动进行描述。那么是否可以也从三维笛卡尔坐标下进行?这个问题的数学表述如下:对于一个二维正则曲面,需两个参量来对曲面进行曲面化(例如u, v),这样,曲面方程就是: r =( x (u , v ), y (u , v ), z (u , v))此时的厄密动量算符p_x , p_y ,p_x称为笛卡尔动量算符。它和曲面上的正则动量完全不同。这一正则动量一般来说不是有界算符(例如径向动量,而且由于坐标系的转换而改变,一直受到数学物理界的责难。从2003年已经开始,我们开始研究笛卡尔动量算符。在有关研究的基础上,本论文证明了笛卡尔动量算符具有如下形式,其中Hn是一个几何不变量,称为平均曲率矢量场。这是一个不随坐标系的转换而改变的量。进一步,体系Hamilton量似应为,。其实这是不行的,正确的哈密顿应为或:或:其中f_x , f_y ,f_z为u, v的非平凡函数。这是一类新的算符次序问题。本文还给出了这些函数满足的微分方程,并在一些具体例子中给出了方程的解。在笛卡尔坐标下研究约束体系物理图象更清晰,经典对应更直接。本研究的结果加深了对量子力学与经典力学之间关系的理解。更多还原

【Abstract】 The quantum motion constrained on quantum dots is currently a hot research topic in condensed matter physics. A very interesting issue is the geometrical shapes of quantum dots. There are toroidal surface and M?bius surface etc besides the typical sphere and spheroid. In classical mechanics, for a particle moving on the curved surface embedded in three dimension Cartesian coordinates, the local curved coordinates on the surface and the Cartesian coordinates play equal roles in the description of its classical motion. In the differential geometry, we can use two complementary and fundamental methods such as intrinsic geometry and extrinsic geometry as well. However, traditional quantum mechanics, also the tradition of the modern physics research such as general relativity and gauge field, uses the local coordinate system only. Then can we use the three dimension Cartesian coordinates to describe motion constrained on the two dimensional curved surfaces?Expressing this problem in mathematical language, we have that two variables are needed to describe a two dimensional regular curved surface, for example (u , v ); and the surface equation is r =( x (u , v ), y (u , v ), z (u , v)), where the Hermitian momentum operators p_x , p_y ,p_x are the so called Cartesian momentum operators. They are entirely different from the canonical momentum operators on the curved surface. Generally speaking, these canonical momentum operators are not bounded operators, for example radial momentum operators , and their forms change with the coordinates transformations. This problem has been criticized by the mathematics and the physics circles.The research of the Cartesian momentum operators began in 2003. This present work proves that the Cartesian momentum operators take the following forms: where Hn is a geometrical invariant, which is called mean curvature vector field. This is an invariant under the transformations of coordinates.Furthermore, the Hamilton of the system seems to be but it is not the case. The correct Hamilton should be, where the f_x , f_y ,f_z are the nontrivial functions of u ,v . This is a new kind operator ordering problem. This paper also gives the differential equations determining these functions. Moreover, some explicit solutions are given for some concrete examples.The physical picture with Cartesian operators is much clear and classical correspondence is straightforward. Our research casts a new insight into the understanding of the classical correspondence of quantum mechanics.更多还原