高等统计物理

本课程在大学本科物理专业热力学统计物理及量子力学的基础上，使用量子力学的语言，讲述量子统计物理的基本原理及其应用。

Statisticalphysicsestablishesabridgefromthemacroscopicworldtostudythemicroscopicworld.Thisisatheorywiththefewestassumptionsandthebroadestconclusions.Uptonowthereisnoevidencetoshowthatstatisticalphysicsitselfisresponsibleforanymistakes.Statisticalphysicshasbecomeanimportantbranchofmoderntheoreticalphysicsandthiscoursehasbecomeoneofthecommonfundamentalcoursesofgraduatestudentsindifferentmajorsinphysicsdepartments.Statisticalphysicsisabranchofscienceengagedinstudyingthelawsofthermalmotionofmacroscopicsystems.Theadvancedstatisticsforgraduatestudentsmainlystudiesquantumstatistics.Thefirstfourchaptersofthisbookarefundamental,andshouldbewellknown.Thelastfivechaptersarerecentdevelopments,includingthestudiesonBose-Einsteincondensation,aclassofinverseproblemsinquantumstatistics(theirChen'sexactsolutionformulas,Dai'sexactsolutionformulas,asymptoticbehaviorcontroltheory,andconcreterealizationsoftheinversiontheories),anintroductiontothetheoryofGreen'sfunctionsinquantumstatistics,theunifieddiagonalizationtheoremforHamiltoniansofquadraticform,andanintroductiontothethirdformulationofquantumstatisticsandthefunctionalintegralapproach.Thiscoursewaseditedbyrevisingthelecturenotesoftheauthor,fromcoursesofquantumstatisticsandadvancedstatisticsforgraduatestudents,since1978.Atthesametime,thisworkcontainstheresearchresultsofsomerelatedprojects,supportedbytheNationalNaturalScienceFoundationofChina.

戴显熹，1938年5月生于温州。1961年7月毕业于复旦大学物理系。1985年起任复旦大学物理系教授，1986年起任博士生导师。长期从事量子统计和理论物理方法研究，发表学术论文100多篇。

自1978年以来，从事研究生的量子统计与高等统计课程教学，以及本科生的电动力学、量子力学、数理方法、超导物理、理论物理方法等课程的教学。曾获得杨振宁教授授予的GloriousSun奖金，曾以物理学中奇性问题研究获教育部授予的科学进步奖(二等)等。1980年来应邀访问过美国的休斯顿大学、纽约州立大学理论物理(杨振宁)研究所、德克萨斯超导中心、杨伯翰大学等，曾任杨伯翰大学客座教授。在量子统计、物理学中奇性问题、一些逆问题的严格解及其统一理论和渐近行为控制理论等方面作过较为系统的研究，首次由一材料的比热实际数据中反演出声子谱。

Chapter1FundamentalPrinciples

1.1Introduction:TheCharactersofThermodynamicsandStatisticalPhysicsandTheirRelationship

1.2BasicThermodynamicIdentities

1.3FundamentalPrinciplesandConclusionsofClassicalStatistics

1.3.1MicroscopicandMacroscopicDescriptions，StatisticalDistributionFunctions

1.3.2LiouvilleTheorem

1.3.3StatisticalIndependence

1.3.4MicroscopicalCanonical，CanonicalandGrandCanonicalEnsembles

1.4BoltzmannGas

1.5DensityMatrix

1.5.1DensityMatrix

1.5.2SomeGeneralPropertiesoftheDensityMatrix

1.6LiouvilleTheoreminQuantumStatistics

1.7CanonicalEnsemble

1.8GrandCanonicalEnsemble

1.8.1FundamentalExpressionoftheGrandCanonicalEnsemble

1.8.2DerivationoftheFundamentalThermodynamicIdentity

1.9ProbabilityDistributionandSlaterSum

1.9.1MeaningoftheDiagonalElementsoftheDensityMatrix

1.9.2SlaterSummation

1.9.3Example:ProbabilityoftheHarmonicEnsemble

1.10TheoryoftheReducedDensityMatrix

Chapter2ThePerfectGasinQuantumStatistics

2.1IndistinguishabilityPrincipleforIdenticalParticles

2.2BoseDistributionandFermiDistribution

2.2.1PerfectGasesinQuantumStatistics

2.2.2BoseDistribution

2.2.3FermiDistribution

2.2.4ComparisonofThreeDistributions;GibbsParadoxAgain

2.3DensityofStates，ChemicalPotentialandEquationofState

2.3.1DensityofStates

2.3.2VirialEquationforQuantumIdealGases

2.4Black-bodyRadiation

2.4.1ThermodynamicQuantitiesfortheBlack-bodyRadiationField

2.4.2ExitanceandVarietyofDisplacementLaws

2.4.3WavebandRadiantExitanceandWavebandPhotonExitance

2.5Bose-EinsteinCondensationinBulk

2.5.1BoseCondensation，DynamicalQuantitieswithTemperatureLowerThantheλPoint

2.5.2DiscontinuityoftheDerivativesofSpecificHeatandλPhenomena

2.5.3Two-FluidTheory

2.5.42-DCase

2.6DegenerateFermiGasesandFermSphere

2.6.1PropertiesofFermiGasesatAbsoluteZero

2.6.2SpecificHeatofFreeElectronGases

2.6.3StateEquation，HeatCapacityatConstantPressure，andHeavyFermions

2.7FermiIntegralsandtheirLowTemperatureExpansion

2.8MagnetismofFermiGases

2.8.1SpinMagnetism:Paramagnetism

2.8.2EnergySpectraandStationaryStatesofElectronsinaHomogeneousMagneticField

2.8.3DiamagnetismofOrbitalMotionofFreeElectrons

2.9PeierlsPerturbationExpansionofFreeEnergy

2.9.1ClassicalCase

2.9.2QuantumCase

2.9.3ExpansionofFreeEnergyofanIdealGasinanExternalField

2.10Appendix

Chapter3SecondQuantizationandModelHamiltonians

3.1NecessityofSecondQuantization

3.2SecondQuantizationforBoseSystem

3.3SecondQuantization·FermiSystem

3.4SomeConservationLaws

3.5SomeModelHamiltonians

3.6ElectronGaseswithCoulombInteraction

3.6.1CompletelyIonizedGases—theHighTemperaturePlasma

3.6.2TheDegenerateElectronGaswithCoulombInteraction（MetalPlasma）

3.7AndersonModel

Chapter4LeastActionPrinciple，FieldQuantizationandtheElectron-PhononSystem　4.1ClassicalDescriptionofLatticeVibrations

4.2ContinuousMediaModelofLatticeVibration（Classical）

4.3TheLeastActionPrinciple，Euler-LagrangeEquationandHamiltonEquation

4.4LagrangianandHamiltonianofContinuousMedia

4.5QuantizationoftheLatticeVibrationField

4.6DebyeTheoryofSpecificHeatofSolids

4.7TheElectron-PhononSystemandtheFrohlichHamiltonian

Chapter5Bose-EinsteinCondensation

5.1SpatialandMomentumDistributionsofBose-EinsteinCondensationinHarmonicTrapsandBlochSummation

5.1.1Introduction

5.1.2GeneralizedExpressionforParticleDensity

5.1.3DistributionsforIdealSystems

5.1.4NewExpressionwithClearPhysicalPicture

5.1.5MomentumDistributions

5.1.6ResultsofNumericalCalculations

5.1.7DiscussionandConcludingRemarks

5.1.8MomentumDistributionofBEC

5.2BECinConfinedGeometryandThermodynamicMapping

5.2.1Introduction

5.2.2ConfinementEffects

5.2.3ThermodynamicMapping

5.2.4MappingRelationforConfinedBEC

5.2.5DeterminationoftheCriticalTemperature

5.2.6Discussion

Chapter6SomeInverseProblemsinQuantumStatistics

6.1Introduction

6.2SpecificHeat-PhononSpectrumInversion

6.2.1TechniqueforEliminatingDivergences

6.2.2UniqueExistenceTheoremandExactSPIESolution

6.2.3Summary

6.3ConcreteRealizationofInversion

6.3.1TheSpecificHeat-PhononSpectrumInversionProblem

6.3.2ResultsandConcludingRemarks

6.4MobiusInversionFormula

6.4.1RiemannζFunctionandMobiusFunction

6.4.2MobiusInversionFormula

6.4.3TheModifiedMobiusInversionFormula

6.4.4ApplicationsinPhysics

6.5UnificationoftheTheories

6.5.1Introduction

6.5.2DerivingChen'sFormulafromDai'sExactSolution

6.5.3ConcludingRemarks

6.6Appendix

Chapter7AnIntroductiontoTheoryofGreen'sFunctions

7.1Temperature-TimeGreen'sFunctions

7.1.1DefinitionofTemperature-TimeGreen'sFunctions

7.1.2TheEquationofMotionofDouble-TimeGreen'sFunctions

7.1.3TimeCorrelationFunctions

7.2SpectralTheorem

7.2.1SpectralRepresentationofTimeCorrelationFunctions

7.2.2SpectralRepresentationsofRetardedandAdvancedGreen'sFunctions

7.2.3SpectralRepresentationofCausalGreen'sFunctions

7.3Example:IdealQuantumGases

7.4TheoryofSuperconductivitywithDouble-TimeGreen'sFunctions

7.5Higher-OrderSpectralTheorem，SumRulesandUniqueness

Chapter8AUnifiedDiagonalizationTheoremforQuadraticHamiltonian

8.1AModelHamiltonian

8.2DiagonalizationTheoremforFermiQuadraticForms

8.3Conclusion:AUnifiedDiagonalizationTheorem

Chapter9FunctionalIntegralApproach:AThirdFormulationofQuantumStatisticalMechanics

9.1Introduction

9.1.1Hubbard'sMethod

9.1.2Difficulties

9.2AnOperatorIdentity

9.3FunctionalIntegralFormulationofQuantumStatisticalMechanics

9.4RealityandMethodofSteepestDescents

9.5DiscussionandConcludingRemarks

9.6SomeRecentDevelopments

9.7Application:AnExactSolution

References

Index