Conquering the physics GRE

297 Pages·2018·9.1 MB·English

by  Yoni Kahn

Conquering the Physics GRE
Third Edition
ThePhysicsGREplaysasignificantroleindecidingadmissionstonearlyallUSphysics
Ph.D.programs,yetfewexam-prepbooksfocusonthetest’sactualcontentandunique
structure.Recognizedasoneofthebeststudentresourcesavailable,thistailoredguide
hasbeenthoroughlyupdatedforthecurrentPhysicsGRE.Itcontainscarefullyselected
reviewmaterialmatchedtoallofthetopicscovered,aswellastipsandtrickstohelpyou
solveproblemsundertimepressure.Itfeaturesthreefull-lengthpracticeexams,revised
toaccuratelyreflectthedifficultyofthecurrenttest,withfullyworkedsolutionssothat
youcansimulatetakingthetest,reviewyourpreparedness,andidentifyareasinwhich
further study is needed. Written by working physicists who took the Physics GRE for
theirowngraduateadmissionstoMIT,thisself-containedreferenceguidewillhelpyou
achieveyourbestscore.
YoniKahnisatheoreticalphysicistresearchingdarkmatterandsupersymmetry.Apost-
doctoral research associate at Princeton University, he obtained his Ph.D. from MIT
in 2015 and in 2016 received the American Physical Society’s J.J. and Noriko Sakurai
DissertationAwardinTheoreticalParticlePhysics.
Adam Anderson is an experimental physicist working at the interface between cos-
mology and particle physics. He received his Ph.D. from MIT in 2015 and is now
a Lederman postdoctoral fellow at Fermi National Accelerator Laboratory, develop-
ing instruments for performing precision measurements of the cosmic microwave
background.

Conquering the
Physics GRE
Third Edition
Yoni Kahn
PrincetonUniversity,NewJersey
Adam Anderson
Fermilab,Batavia,Illinois
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education,learning,andresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781108409568
DOI:10.1017/9781108296977
(cid:2)c YoniKahnandAdamAnderson2018
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
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PreviouslypublishedbyCreateSpace,2012,2014
Thirdedition2018
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CONTENTS
Preface pageix 1.7 SpringsandHarmonicOscillators 22
HowtoUseThisBook xi 1.7.1 NormalModes 23
Resources xii 1.7.2 Damping,Driving,andResonance 24
1.7.3 FurtherExamples 25
1.7.4 Problems:Springs 27
1
1.8 FluidMechanics 27
1.8.1 Bernoulli’sPrinciple 27
Classical Mechanics 1
1.8.2 BuoyantForces 29
1.1 Blocks 1 1.8.3 Problems:FluidMechanics 29
1.1.1 BlocksonRamps 1 1.9 Solutions:ClassicalMechanics 29
1.1.2 FallingandHangingBlocks 2
1.1.3 BlocksinContact 3
1.1.4 Problems:Blocks 3 2
1.2 Kinematics 5
1.2.1 CircularMotion 5 Electricity and Magnetism 35
1.2.2 Problems:Kinematics 6 2.1 Electrostatics 35
1.3 Energy 7 2.1.1 Maxwell’sEquationsforElectrostatics 35
1.3.1 TypesofEnergy 7 2.1.2 ElectricPotential 35
1.3.2 Kinetic/PotentialProblems 8 2.1.3 IntegralFormofMaxwell’sEquations 36
1.3.3 RollingWithoutSlipping 9 2.1.4 StandardElectrostaticsConfigurations 37
1.3.4 Work–EnergyTheorem 11 2.1.5 BoundaryConditions 38
1.3.5 Problems:Energy 11 2.1.6 Conductors 40
1.4 Momentum 12 2.1.7 MethodofImages 40
1.4.1 LinearCollisions 12 2.1.8 WorkandEnergyinElectrostatics 42
1.4.2 RotationalMotionandAngularMomentum 12 2.1.9 Capacitors 43
1.4.3 MomentofInertia 14 2.1.10 Problems:Electrostatics 44
1.4.4 CenterofMass 15 2.2 Magnetostatics 45
1.4.5 Problems:Momentum 15 2.2.1 BasicTools 45
1.5 LagrangiansandHamiltonians 16 2.2.2 Ampère’sLawandtheBiot–SavartLaw 46
1.5.1 Lagrangians 16 2.2.3 StandardMagnetostaticsConfigurations 46
1.5.2 Euler–LagrangeEquations 17 2.2.4 BoundaryConditions 48
1.5.3 HamiltoniansandHamilton’sEquations 2.2.5 WorkandEnergyinMagnetostatics 48
ofMotion 18 2.2.6 CyclotronMotion 48
1.5.4 Problems:LagrangiansandHamiltonians 19 2.2.7 Problems:Magnetostatics 49
1.6 Orbits 19 2.3 Electrodynamics 49
1.6.1 EffectivePotential 19 2.3.1 Maxwell’sEquations 49
1.6.2 ClassificationofOrbits 20 2.3.2 Faraday’sLaw 50
1.6.3 Kepler’s“Laws” 21 2.3.3 Inductors 50
1.6.4 Problems:Orbits 22 2.3.4 Problems:Electrodynamics 51
vi Contents
2.4 Dipoles 52 4.1.3 ClassicalLimit 80
2.4.1 ElectricDipoles 52 4.1.4 EquipartitionTheorem 80
2.4.2 MagneticDipoles 52 4.1.5 SomeCombinatorialFacts 80
2.4.3 MultipoleExpansion 53 4.2 Thermodynamics 80
2.4.4 Problems:Dipoles 53 4.2.1 ThreeLaws 81
2.5 MatterEffects 53 4.2.2 GasesandEquationsofState 82
2.5.1 Polarization 54 4.2.3 TypesofProcesses 82
2.5.2 Dielectrics 54 4.2.4 RelationsBetweenThermodynamicVariables 84
2.5.3 Problems:MatterEffects 54 4.2.5 HeatCapacity 84
2.6 ElectromagneticWaves 54 4.2.6 ModelSystems 85
2.6.1 WaveEquationandPoyntingVector 54 4.3 QuantumStatisticalMechanics 87
2.6.2 Radiation 56 4.4 Problems:ThermodynamicsandStatistical
2.6.3 Problems:ElectromagneticWaves 56
Mechanics 88
2.7 Circuits 56
4.5 Solutions:ThermodynamicsandStatistical
2.7.1 BasicElements 57
Mechanics 90
2.7.2 Kirchhoff’sRules 57
2.7.3 EnergyinCircuits 57 5
2.7.4 StandardCircuitTypes 58
2.7.5 Problems:Circuits 58 Quantum Mechanics and Atomic Physics 92
2.8 Solutions:ElectricityandMagnetism 59
5.1 Formalism(HowToCalculate) 92
5.1.1 WavefunctionsandOperators 92
3 5.1.2 DiracNotation 94
5.1.3 SchrödingerEquation 95
Optics and Waves 63 5.1.4 CommutatorsandtheUncertaintyPrinciple 96
5.1.5 Problems:Formalism 98
3.1 PropertiesofWaves 63
5.2 HarmonicOscillator 99
3.1.1 WaveEquation 63
3.1.2 NomenclatureandComplexNotation 63 5.2.1 OneDimension 99
3.1.3 DispersionRelations 65 5.2.2 ThreeDimensions 100
3.1.4 ExamplesofWaves 65 5.2.3 Problems:HarmonicOscillator 101
3.1.5 IndexofRefraction 65 5.3 OtherStandardHamiltonians 101
3.1.6 Polarization 66 5.3.1 InfiniteSquareWell 101
3.2 InterferenceandDiffraction 67 5.3.2 FreeParticle 102
3.2.1 Double-SlitInterference 67 5.3.3 DeltaFunction 102
3.2.2 Single-SlitDiffraction 68 5.3.4 FiniteSquareWell 103
3.2.3 OpticalPathLength 68 5.3.5 ScatteringStates:ReflectionandTransmission 103
3.2.4 ThinFilmsandPhaseShifts 69 5.3.6 Problems:OtherStandardHamiltonians 104
3.2.5 MiscellaneousDiffraction 70 5.4 QuantumMechanicsinThreeDimensions 104
3.3 GeometricOptics 70 5.4.1 RadialEquationandEffectivePotential 105
3.3.1 ReflectionandRefraction 70 5.4.2 AngularMomentumandSphericalHarmonics 105
3.3.2 LensesandMirrors 71 5.4.3 TheHydrogenAtom 106
5.4.4 Problems:QuantumMechanicsinThree
3.4 AssortedExtraTopics 72
Dimensions 108
3.4.1 RayleighScattering 72
5.5 Spin 108
3.4.2 DopplerEffect 72
3.4.3 StandingSoundWaves 73 5.5.1 Spin-1/2 108
5.5.2 SpinandtheWavefunction 109
3.5 Problems:OpticsandWaves 74
5.5.3 AddingSpins 110
3.6 Solutions:OpticsandWaves 75
5.5.4 BosonsandFermions 111
5.5.5 Problems:Spin 112
4
5.6 ApproximationMethods 113
5.6.1 Time-IndependentPerturbationTheory:
Thermodynamics and Statistical Mechanics 78
FirstandSecondOrder 113
4.1 BasicStatisticalMechanics 78 5.6.2 VariationalPrinciple 114
4.1.1 EnsemblesandthePartitionFunction 78 5.6.3 AdiabaticTheorem 114
4.1.2 Entropy 79 5.6.4 Problems:ApproximationMethods 114
Contents vii
5.7 AtomicPhysicsTopics 115 7.6 Problems:LaboratoryMethods 143
5.7.1 BohrModel 115 7.7 Solutions:LaboratoryMethods 145
5.7.2 PerturbationstoHydrogenAtoms 115
5.7.3 ShellModelandElectronicNotation 116
5.7.4 StarkandZeemanEffects 116 8
5.7.5 SelectionRules 117
5.7.6 BlackbodyRadiation 117 Specialized Topics 146
5.7.7 Problems:AtomicPhysicsTopics 118 8.1 NuclearandParticlePhysics 146
5.8 Solutions:QuantumMechanicsandAtomic 8.1.1 TheStandardModel:Particlesand
Physics 119 Interactions 146
8.1.2 NuclearPhysics:BoundStates 147
6
8.1.3 SymmetriesandConservationLaws 148
8.1.4 RecentDevelopments 149
Special Relativity 123
8.2 CondensedMatterPhysics 149
6.1 RelativityBasics 123 8.2.1 CrystalStructure 149
6.1.1 Simultaneity 124 8.2.2 ElectronTheoryofMetals 150
6.1.2 TimeDilation 124 8.2.3 Semiconductors 151
6.1.3 LorentzContraction 124 8.2.4 Superconductors 151
6.1.4 VelocityAddition 125 8.3 Astrophysics 152
6.2 4-Vectors 125
8.4 RecentNobelPrizes 153
6.2.1 LorentzTransformationMatrices 125
8.5 Problems:SpecializedTopics 155
6.2.2 RelativisticDotProduct 126
8.6 Solutions:SpecializedTopics 157
6.3 RelativisticKinematics 127
6.3.1 Conservedvs.Invariant 127
6.3.2 ExploitingtheInvariantDotProduct 128
9
6.4 MiscellaneousRelativityTopics 129
6.4.1 RelativisticDopplerShift 129 Special Tips and Tricks for the Physics GRE 159
6.4.2 PythagoreanTriples 129
9.1 Derive,Don’tMemorize 159
6.5 Relativity:WhattoMemorize 129
9.2 DimensionalAnalysis 160
6.6 Problems:SpecialRelativity 130
9.3 LimitingCases 161
6.7 Solutions:SpecialRelativity 131
9.4 NumbersandEstimation 162
9.5 AnswerTypes(WhattoRememberinaFormula) 163
7
9.6 GeneralTest-TakingStrategies 165
Laboratory Methods 134 9.7 Problems:TipsandTricks 165
7.1 GraphReading 134 9.8 Solutions:TipsandTricks 166
7.1.1 DimensionalAnalysis 134
7.1.2 LogPlots 134 Sample Exams and Solutions 167
7.2 Statistics 135
SampleExam1 169
7.2.1 ErrorAnalysis 135
7.2.2 PoissonProcesses 136 SampleExam2 187
7.3 Electronics 136 SampleExam3 209
7.3.1 ACBehaviorofBasicCircuitElements 136 AnswerstoSampleExam1 227
7.3.2 MoreAdvancedCircuitElements 138 AnswerstoSampleExam2 228
7.3.3 LogicGates 138
AnswerstoSampleExam3 229
7.4 RadiationDetectionandInstrumentation 139
SolutionstoSampleExam1 230
7.4.1 InteractionofChargedParticleswithMatter 139
SolutionstoSampleExam2 243
7.4.2 PhotonInteractions 140
SolutionstoSampleExam3 254
7.4.3 GeneralPropertiesofParticleDetectors 141
7.4.4 RadioactiveDecays 141
7.5 LasersandInterferometers 141 References 267
7.5.1 GenericLaserOperation 141 EquationIndex 268
7.5.2 TypesofLasers 142 SubjectIndex 276
7.5.3 Interferometers 143 ProblemsIndex 280

PREFACE
ConqueringthePhysicsGRE representsthecombinedefforts materialsuchaslaboratorymethodsinthesametext,specif-
of two MIT graduate students frustrated with the lack of ically focused on aspects of these subjects relevant for the
decentpreparationmaterialsforthePhysicsGREsubjecttest. GRE.Exam-stylepracticeproblemsandworkedsolutionsare
When we took the exams, in 2007 and 2009, we did what included for each review chapter, giving over 150 additional
anystudentintheinternetagewoulddo–searchedthevar- GRE-stylepracticeproblemsinadditiontothe300fromthe
iousonlinebookstoresfor“physicsGREprep,”“physicsGRE exams.Theshorterchaptershavereviewproblemsatthevery
practice tests,” and so on. We were puzzled when the only end,whilethelongeroneshavereviewproblemsdistributed
results were physics practice problems that had nothing to throughoutthechapter.
do with the GRE specifically or, worse, GRE practice books Thechapteronquantummechanicsandatomicphysicsis
having nothing to do with physics. Undeterred, we headed the longest, for two reasons: the combination of these two
to our local brick-and-mortar bookstores, where we found topics makes up nearly 25% of the exam, and the formal-
a similar situation. There were practice books for every sin- ism of quantum mechanics is so different from the rest of
gle GRE subject exam, except physics. Further web searches thephysicstopicscoveredontheGREthatwefeltitworth-
unearthed www.grephysics.net, containing every problem whiletodiscussanumberofcalculationalshortcutsindetail.
andsolutionfromeverypracticetestreleaseduptothatpoint, Uniquetoourbookisachapteronspecialtipsandtricksrele-
andwww.physicsgre.com,awebforumdevotedtodiscussing vantfortakingtheGREasastandardizedmultiple-choicetest.
problemsandstrategiesforthetest.Wediscoveredthesesites Someofthestandardtest-takingwisdomstillapplies,butwe
had sprung up thanks to other frustrated physicists just like havefoundthatthestructureofthemultipleanswerchoices
us:therewasnoreviewmaterialavailable,sostudentsdidthe oftenprovidesvaluablehintsonhowtosolveaproblem:you
besttheycouldwiththemeagermaterialthatdidexist.This will not find this information in any other test-prep book,
situationisparticularlyacuteforstudentsinsmallerdepart- becauseitisbasedontechniquessuchasdimensionalanalysis
ments, who have fewer classmates with whom to study and and back-of-the-envelope estimation, which most test-prep
sharethe“warstories”oftheGRE. authors(whoarenotphysicists)aresimplyunawareof.
This book endeavors to fix that situation. Its main con- Next, a brief word on what this book is not. This is not
tribution is a set of three full-length practice tests and fully a detailed review of undergraduate physics: many of the
workedsolutions,designedtobeascloseaspossibleinstyle, more difficult subjects get an extremely abbreviated treat-
difficulty,contentdistribution,andformattotheactualGRE ment,designedtohighlightonlythoseformulasandproblem
exam.Wehavealsoincludedreviewmaterialforallofthenine typesrelevantfortheexam.Webelievethiswillhelpyousuc-
contentareasonthePhysicsGREexam:classicalmechanics, ceedonthePhysicsGRE,butifanyofthestandardsubjects
electricity and magnetism, optics and waves, thermodynam- are completely unfamiliar to you, please do not try to teach
ics and statistical mechanics, quantum mechanics, atomic them to yourself from our book. There are many excellent
physics,specialrelativity,laboratorymethods,andspecialized textsoutthererelevantforthatpurpose,andwehaveincluded
topics. To our knowledge, this is the first time that reviews alistofthemintheResourcessectionfollowingthispreface.
ofstandardundergraduatesubjectssuchasclassicalmechan- Westronglyencourageyoutoconsultthesereferences,aswe
icsandthermodynamicshavebeenpairedwithlessstandard havefoundthemusefulbothinwritingthispresenttextand