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;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.

Thecourseprovidesadeeperunderstandingofgeneralphenomenainsystemsthatbehaveessentiallyquantummechanicallyandpracticalintroductionstomethodsusedtodescribethebehavior.

Thephenomenadiscussedincludethenatureofspectrum,quantumdynamics,scatteringandtunneling.Inadditiontoasurveyofexactlysolvableproblems,avarietyofapproximationsareintroduced:fromperturbationtheory,semiclassical,variationaltonumericalsimulation.GeneraltoolsoftheGreen'sfunctionsandpathintegralsarealsointroducedandutilized.

教学大纲

I.ReviewofQuantumMechanicsviaSolvableModels

A.Anotherlookatelementaryoneparticlequantummechanics

(1)SpectraofthesingleparticleSchroedingerequation:discreteversuscontinuous.

(2)Extremequantumlimit:pletelydiscretizedspectrum.

(3)Theoppositeextreme:purelycontinuous.

(4)Mixedspectrum.

(5)MethodstotackleaQMproblem.

B.SolvablelinearandSUSYModels.Bosonicdegreeoffreedom

(1)Harmonicoscillatorsolvedusingthe(boson)creationandtheannihilationoperators.

(2)Coherentstatesandfieldquantization.Harmonicoscillatorasaparadigmofbosonicdegreeoffreedom.

(3)Generalizationtogeneralsupersymmetricpotential.

(4)Landau–Bronshteinquantizationinmagicfield.

C.OthersolvableModels.Twolevelsystemasafermionicdegreeoffreedom

(1)Anothermethod:matchingsolvablemodelsatboundaries.

(2)Boundstatesinawell.

(3)Morewells.Hilbertspacetruncationandtheparadigmicfermionicdegreeoffreedom.

(4)Integraltransformationforthedeltafunctionpotentials.

II.NumericalmethodofthesolutionsofQMproblems.

(1)Discretizationofacontinuousvariable.

(2)SimulationofTimeDependentSchroedingerEquation(TDSE).

III.Majortools:Green'sfunctionsandpathintegral

A.Globalandlocaldensityofstates(DOS).

(1)DensityofstatesandtheFermi–Diracstatistics.

(2)DOSoffreeelectrongas.

(3)GeneraldefinitionofglobalandlocalDOS.

B.Green'sfunctions(GF)

(1)Feynmanpropagator:GFint-xspace.

(2)AnalogyanddifferenceswithBrownianmotion.

(3)GaussianGF:freeparticles,harmonicoscillator,particleinmagicfield.

C.Thepath(functional)integralrepresentationofGF

(1)Derivationofthepathintegral.

(2)Euclideanversionandrelationwithrandomwalksandclassicalstatisticalmechanics(polymerphysics).

IV.PerturbationtheoryandGreen'sfunctions

A.Timeindependentperturbationtheory

(1)Howtoexpandin"smallperturbation".

(2)Nexttoleadingorder.

(3)Whatcannotbecalculatedwithpertubationtheory

B.Scatteringandtimeindependentperturbationtheory

(1)Basisofscatteringstatesandexpansioninweakscatteringpotential.

(2)Thescatteringrate.

C.Diagrammaticperturbationtheory

(1)TDSEintheinteractionpictureanditsiteration.

(2)TimeorderedGF.

(3)Feynmandiagramsinx-tspace.

(4)FouriertransformedGFandtheirphysicalproperties.

(5)FeynmandiagramsinE-pspace.

V.Semiclassicalandvariationalmethods

A.Semiclassicalmethods

(1)Reminder:classicalmotionin1D

(2)Theeikonal(WKB)expansion

(3)TheBohr–Sommerfeldconditionforboundstates

(4)Thetunneleffect

(5)*.Quantumoscillationsanddecay.

B.Semiclassicalmethodsviapathintegral

(1)GFandspectrumrevisited.

(2)SemiclassicalapproximationoftheGF.

(3)Instantons.

C*.Adiabaticevolutioninsemiclassicalapproximation

(1)Adiabaticquantummotion.

(2)Berryphase.

(3)Aharonov–Bohmeffect.

(4)*.Mesoscopicring.

D.Gaussianvariationalapproach

(1)Gaussianor"squeezed"states.

(2)Variationalprincipleforanharmonicoscillatorandthedoublewell.

授课方式:多媒体

教学步骤: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

考核方式:平时成绩40%,笔试60%

返回目录

量子力学QuantumMechanicsⅡ

钱志新副教授

学分2

授课对象物理类高年级学生

先修课程量子力学Ⅰ

课程简介:

在量子力学Ⅰ的基础上,介绍He原子和多电子原子的基本理论,主要讨论量子力学中的微扰理论和变分方法,Hartree理论,Hartree-Fock理论和Thomas-Fermi理论.双原子分子,多原子分子的转动谱和振动谱.价键理论的基本概念.介绍量子力学中的Feynman路径积分方法(以自由粒子和谐振子为例).介绍WKB近似,Bohr量子化条件,势垒的隧穿.量子力学里态的相干态描述,介绍量子力学中的相位,包括Aharonov-Bohm效应的实验观察,引力导致的量子力学相位效应,Berry相和Aharonov-Anandan的介绍.讨论量子力学中的密度矩阵理论和Wigner函数的概念.

考核方式:课堂教学,期末考试(笔试)

参考资料:

1,曾谨言,量子力学,科学出版社

2,L.D.Landau等,量子力学

3,L.I.Shiff,量子力学

4,J.J.Sakurai,ModernQuantumMechanics

5,R.P.Feynman等,量子力学和路径积分

6,H.艾林等,量子化学

7,,NatureoftheChemicalBond

教师简介:

钱志新,博士,副教授.研究方向:量子多体理论.研究领域为凝聚态物理,发表论文30余篇.授课:量子力学II(2004--2016),量子统计力学(2004-2016).返回目录

地球与空间科学学院