量子力学专题课程详细信息

课程号 00432206 学分 2
英文名称 Advanced topics in Quantum Mechanics
先修课程 量子力学
中文简介 目前,国内现有的量子力学课程着重讲授非相对论性量子力学的基本理论和方法,对其路径积分形式的表述和实际具体的计算方法、解决具体实际问题的应用等大多不予深入讨论、甚至根本不提及,对于从量子力学到量子场论的过渡大多也不予讨论。该专题课程针对解决这些问题而开设,从而帮助同学打下从量子力学的形式理论过渡到开展具体物理研究的坚实基础。
本课程主要讲授非相对论性量子力学在描述一些基本量子现象的应用(比如能谱、量子动力学、散射理论、隧穿理论、AB效应和Berry相位等)和量子力学中的路径积分、格林函数及其它一些基本问题的实际计算方法(比如半经典近似、变分法、数值计算方法、一些精确可解模型、 微挠的费曼图展开理论等等)。课程将采用新的理论框架(路径积分和其相关计算技术)来描述量子现象,从而为同学们进一步学习量子多体理论等后续课程打下扎实的理论基础。该课程将在突出清晰的物理概念的同时,强调具体计算技能(包含解析和数值两方面),并结合当前的研究实例来展开。
英文简介 The course provides a deeper understanding of general phenomena in systems that behave essentially quantum mechanically and practical introductions to methods used to describe the behavior.
The phenomena discussed include the nature of spectrum, quantum dynamics, scattering and tunneling. In addition to a survey of exactly solvable problems, a variety of approximations are introduced: from perturbation theory, semiclassical, variational to numerical simulation. General tools of the Green’s functions and path integrals are also  introduced and utilized.
开课院系 物理学院
通选课领域  
是否属于艺术与美育
平台课性质  
平台课类型  
授课语言 英文
教材 “Quantum Mechanics”,A. Konishi and G. Paffuti,Oxford University Press, London.,2009;
Topological insulators and topological superconductors,B. A. Bernevig,Princeton University Press, Princeton,2013;
Quantum Mechanics: An Introduction,Greiner,Springer,2008,Quantum Mechanics. Symmetries,Greiner,Springer,2008,Quantum Mechanics and Path Integrals,Feynman et.al.,Dover Publications,2010,Quantum Mechanics,Claude Cohen –Tannoudji et.al.,Wiley-Interscience,2006,Principles of Quantum Mechanics,R. Shankar,Springer,1994,Essential Quantum Mechanics,Bowman,Oxford University Press,2008,Introduction to Quantum Mechanics,Griffiths,Benjamin Cummings,2004,Quantum Mechanics Non-Relativistic  Theory,Landau, Lifshitz,Butterworth-Heinemann;,1981,Schaum`s Outline of Quantum Mechanics,Yoav Peleg et.al.,McGraw-Hill,2010,Modern Quantum Mechanics,Sakurai,Addison Wesley,2010,
参考书
教学大纲 课程将采用新的理论框架(路径积分和其相关计算技术)来描述量子现象,从而为同学们进一步学习量子多体理论等后续课程打下扎实的理论基础。该课程将在突出清晰的物理概念的同时,强调具体计算技能(包含解析和数值两方面),并结合当前的研究实例来展开。
Instead of introduction: another look at QM spectrum

1. Wave function and its probabilistic interpretation.
2. Extreme quantum limit: origin of the discrete spectrum.
3. The opposite extreme: purely continuous.
4. Mixed spectrum.
5. Methods to tackle a QM problem.

I. Easily solvable models: generic bosonic and fermionic degrees of freedom

1. Harmonic oscillator as a paradigmatic bosonic degree of freedom.
3. Generalization to supersymmetric potentials.
4. Local gauge invariance and the Landau - Bronshtein quantization in magnetic field.
5. Hilbert space truncation and the paradigmatic fermionic degree of freedom. Spin.

II. Not that easily solvable QM: symmetry, discretization and numerical simulation

1. Continuous symmetries and their projective representation in QM.
2. Discrete symmetries.



III. Discretization and numerical simulation

1. Tight binding models: 1D chain.
2. Topological transition in polyacetylene.
3. Discretization of a continuous variable and numerical simulation of time independent SE.

IV. Level crossing in D>1: Dirac points and their stability.

1. Tight binding spectrum of graphene. Emergence of Dirac points.
2. How to create a gap in graphene?
3. Some generalizations.

V. Quantum dynamics

1. Few solvable cases: small spin rotation to the Landau-Zener tunneling.
2. Coherent states of harmonic oscillator and the wave packet motion.
3. Adiabatic evolution and the Berry phase.
4. Simulation of the quantum evolution.

VI. Magnetic translations and the Landau band splitting

1. Magnetic translations in continuum.
2. Quasi-momentum eigenstates for unit and fractional filling factor.
3. Magnetic field on the lattice: Wilson links and the Aharonov-Bohm phase.
4. WHE edges, inhomogeneous fluxless field and the Haldane model.
5. Fractionalization of the magnetic bands.


VII. Topological band theory and edge states

1.   Carbon chain with Majorana end states.
2.   Graphene flakes and their edges.
3.   Brillouin zone topology: Berry phases and edges.
4.   Chern class and QHE.
5.   Symmetries and stability of crossing (Dirac) points
多媒体.强调互动,课堂上希望同学能带笔记本,并用mathematica软件即时计算
平时成绩40%,笔试 60%
教学评估 B. Rosenstein:
学年度学期:16-17-3,课程班:量子力学专题1,课程推荐得分:3.75,教师推荐得分:3.75,课程得分分数段:80及以下;
学年度学期:17-18-3,课程班:量子力学专题1,课程推荐得分:3.75,教师推荐得分:3.33,课程得分分数段:80-85;
学年度学期:18-19-3,课程班:量子力学专题1,课程推荐得分:0.0,教师推荐得分:7.5,课程得分分数段:80-85;
学年度学期:22-23-3,课程班:量子力学专题1,课程推荐得分:null,教师推荐得分:null,课程得分分数段:80及以下;