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二维氢原子中的基态奇异特性数值精确对角化法
引用本文:刘褚航,强百强,季育琛,李炜.二维氢原子中的基态奇异特性数值精确对角化法[J].物理学报,2017,66(23):230102-230102.
作者姓名:刘褚航  强百强  季育琛  李炜
作者单位: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
基金项目:国家自然科学基金(批准号:11404359)和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.

关 键 词:二维氢原子  离散贝塞尔基函数  Lanczos法
收稿时间:2017-07-06

Numerical exact diagonalization of singularity in the ground state of two-dimensional hydrogen atom
Liu Chu-Hang,Qiang Bai-Qiang,Ji Yu-Chen,Li Wei.Numerical exact diagonalization of singularity in the ground state of two-dimensional hydrogen atom[J].Acta Physica Sinica,2017,66(23):230102-230102.
Authors:Liu Chu-Hang  Qiang Bai-Qiang  Ji Yu-Chen  Li Wei
Institution:1. School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China;2. State Key Laboratory of Functional Materials for Informatics, Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai 200050, China
Abstract:With the development of computing technology, numerical exact diagonalization method plays a vital role in modern 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 renormalization 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 numerical 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 enough truncated basis number. Since this kind of numerical singularity occurs in many space dimension confinement systems, our finding for constructing a new discrete Bessel basis function may be helpful in studying the quantum systems with numerical singularity behaviors in wavefunctions in future. On the other hand, it should be pointed out that the Bessel basis is incorporated into the linear augment plane wave method in the density functional theory to study the electronic band structure of the condensed material and obtain high accurate results, especially in the theoretical prediction of topological insulators and in experimental realization as well.
Keywords:two-dimensional hydrogen atom  discrete Bessel basis function  Lanczos-type numerical exact diagonalization
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