二维氢原子中的基态奇异特性数值精确对角化法  

Numerical exact diagonalization of singularity in the ground state of two-dimensional hydrogen atom

作　　者：刘褚航 强百强 季育琛 李炜[2] 

机构地区：[1]上海科技大学物质科学与技术学院,上海201210 [2]中国科学院上海微系统与信息技术研究所,信息功能材料国家重点实验室,上海200050

出　　处：《物理学报》2017年第23期9-15,共7页Acta Physica Sinica

基　　金：国家自然科学基金(批准号:11404359);中国科学院青年创新促进会计划(批准号:2016215)资助的课题~~

摘　　要：利用数值有限差分法处理二维氢原子的基态波函数时,计算结果发现其存在着数值奇异特性.本文通过构造一套具有正交完备性的离散贝塞尔基函数,并结合基于Lanczos技术的数值精确对角化方法研究二维氢原子中的基态波函数的数值奇异特性,得到的波函数数值解及其相应的本征能量均与解析结果相一致.这套新的完备的离散贝塞尔基函数,可以在研究一些波函数具有数值奇异特性的体系中发挥至关重要的作用.With the development of computing technology, numerical exact diagonalization method plays a vital role in mod- ern computational condensed matter physics, especially in the research area of strongly correlated electron systems： it becomes a benchmark for other numerical computational techniques, such as quantum Monte Carlo, numerical renormal- ization group, density matrix renormalization group, and dynamic mean field theory. In this paper, we first numerically exactly diagonalize the three-dimensional hydrogen atom with the combination of finite-difference method, and find that the numerical wave function of ground state is in good agreement with the analytical calculations. We then turn to discuss the space dimension confinement hydrogen system, two-dimensional hydrogen atom, and notice that the numer- ical wave function is no longer in agreement with the analytical calculation, where the ground state wave function has a numerical singularity as radius approaches to zero. Compared with the case of the three-dimensional hydrogen atom, this issue mainly comes from the nature of space dimension confinement. To resolve such an issue of numerical singularity in two-dimensional hydrogen atom, we need to construct a new discrete and normalized Bessel function as a basis to study the ground state behavior of dimension confinement system based on the framework of Lanczos-type numerical exact diagonalization. The constructed normalized Bessel basis is orthogonal and discrete, and thus becomes suitable for practical calculation. Besides, these prominent properties of such a Bessel basis greatly reduce the complexity and difficulty in practical calculation, and thus makes computing work efficient. In addition, Lanczos-type numerical exact diagonalization method can extremely speed up the process of solving the eigenvalue equation. As a result, such a high efficient calculation of our method demonstrates the consistence between numerical and analytical ground state energy value, and the corresponding wave function with e

关 键 词：二维氢原子 离散贝塞尔基函数 Lanczos法 

分 类 号：O562[理学—原子与分子物理]