薄壁量子化方法及应用

发布时间：2018-07-12 13:25

本文选题：薄壁量子化方法 + 几何势　；参考：《南京大学》2016年博士论文

【摘要】：薄壁量子化方法通过引入压缩势将粒子的运动约束在二维曲面。虽然薄壁量子化方法可能不能严格满足不确定关系,但是对约束在二维曲面上粒子的量子化,薄壁量子化方法却非常有效。薄壁量子化方法已取得较大发展,可是该方法的计算基本框架仍没有明确给出。当研究系统包含电磁场时,计算基本框架就显的尤为重要。遗憾的是基本框架在薄壁量子化方法的原文中没有给出。在第2章,我们清楚给出薄壁量子化方法的计算基本框架：(1)所研究系统的动力学方程初始定义在三维空间；(2)通过引入一新波函数,该波函数能够解析地分离为法向部分和面部分,基于定义在三维空间的度规张量Gij,对初始动力学方程中所有微分算符进行展开运算；(3)借助求极限q3→0,将原动力学方程解析地分离为法向动力学方程和面动力学方程。为了方便读者,在本章开始我们简要地回顾了本论文中用到的相关数学公式和概念。基于计算基本框架,我们重新回顾了约束在曲面上单粒子和多粒子的量子化结果。在第3章,在有电磁场情况下,我们重新考虑了约束在二维曲面上的带电荷粒子。根据薄壁量子化方法的计算基本框架,我们实现了电磁场与曲面曲率的解耦,解析地分离了法向动力学方程和等效面动力学方程。为了考虑曲面厚度的影响,我们对薄壁量子化方法进行扩展,将与等效面动力学方程有共源项的q3(为垂直于曲面的法向变量)一次幂项重新放回等效面动力学方程,从而得到被曲面厚度修改的几何势和动能项。借助扩展薄壁量子化方法,我们对圆环面上粒子的量子化进行讨论,得到厚度修改的几何势和动能项。在第4章,我们考虑电磁场中约束在二维曲面上带电荷有自旋粒子,并对其进行量子化,推导得等效面泡利方程。在经过旋转后的自旋算符表象中,通过薄壁量子化方法将粒子约束在二维曲面,我们求得几何势和旋转相因子e-iφ,曲线坐标微商算符作用于该相因子将产生自旋联络几何势。同时,如果研究系统不含有法向电流源,我们发现两基本判据依然正确。其一,电磁场与曲面曲率间无耦合,不依赖于曲面形状、电磁场表达形式和规范条件选取；其二,通过选取合适的规范条件,动力学方程能够解析地分离为法向动力学方程和等效面动力学方程。最后,我们将等效面泡利方程应用于讨论球面、柱面和圆环面上粒子运动,得到预期的几何势和自旋联络几何势,并发现面泡利方程只含有法向泡利矩阵。在第5章,一含有周期的瓦楞形砷化镓薄膜的模型被用来研究纳米结构中瓦楞曲面对透射率的影响。我们发现透射间隙和共振透射区完全由瓦楞的形状决定,共振透射区中的共振劈裂峰和谷由边界条件决定,边界条件是由拥有不同电子等效质量的相邻区域构成。共振劈裂峰和谷还受到薄壁厚度的轻微影响。这些结果为设计曲率调控滤波器以理论指导。
[Abstract]:Thin wall quantization is used to restrict the motion of particles to a two-dimensional surface by introducing the compression potential. Although the thin-walled quantization method may not strictly satisfy the uncertainty relation, the thin wall quantization method is very effective for the quantization of the particles on a two-dimensional surface, and the thin-walled quantum method has made great progress, but the method has been developed. The basic framework is still not given clearly. When the research system contains electromagnetic fields, the basic framework is particularly important. Unfortunately, the basic framework is not given in the source text of the thin-walled quantization method. In the second chapter, we clearly give the basic framework for the calculation of the thin wall quantization method: (1) the dynamic equation of the system. The initial definition is in the three-dimensional space; (2) by introducing a new wave function, the wave function can be separated into the normal part and the surface part analytically. Based on the definition of the metric tensor Gij in the three-dimensional space, all the differential operators in the initial dynamic equation are expanded to carry out the expansion operation. (3) the original dynamic equation is analytically separated to the limit of the ultimate dynamic equation. In this chapter, we briefly review the relevant mathematical formulas and concepts used in this paper. Based on the basic framework of calculation, we re review the quantized fruit of the single and multi particle constraints on the surface. In the third chapter, in the case of electromagnetic fields, we weigh According to the basic frame of the thin-walled quantization method, we realized the decoupling of the electromagnetic field and the curvature of the surface, separated the normal dynamic equation and the equivalent equation of the equivalent surface analytically. In order to consider the shadow of the surface thickness, we extend the method of thin wall quantization. The Q3, which has the common source of the equivalent surface dynamic equation, is replayed back to the equivalent surface dynamic equation by a power term which is the normal variable perpendicular to the surface. Thus, the geometric potential and kinetic energy terms modified by the surface thickness are obtained. With the help of the expanded thin-walled quantization method, we discuss the quantization of the particles on the circular torus, and get some of the thickness modification. In the fourth chapter, we consider the constraint on a two-dimensional surface with the charge of a spin particle on a two-dimensional surface and quantize it. The equivalent surface Pauli equation is derived. In the spin operator image after rotation, the particle is restricted to the two-dimensional surface by the thin-walled quantization method. We get the geometric potential and the rotation phase factor. E-I phi, the curvilinear coordinate micro quotient operator acts on the spin Contact Geometric potential. At the same time, if the research system does not contain the normal current source, we find that the two basic criterion is still correct. The dynamic equation can be analytically separated into the normal dynamic equation and the equivalent surface dynamics equation. Finally, we apply the equivalent surface Pauli equation to the particle motion on the spherical surface, the cylinder and the circular torus, and obtain the expected geometric potential and the spin coupling geometric potential, and find that the surface Pauli equation only contains the normal direction. In the fifth chapter, a model of a periodic corrugated gallium arsenide film is used to study the influence of the transmissivity of the corrugating elements in the nanostructures. We find that the transmission gap and the resonant transmission area are entirely determined by the shape of the corrugated. The resonant splitting peaks and valleys in the resonant transmission area are determined by the boundary conditions, and the boundary conditions are supported by the flocking. The resonant splitting peaks and valleys are also slightly affected by the thickness of the thin wall. These results provide theoretical guidance for the design of the curvature control filter.
【学位授予单位】：南京大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O413.1

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