Preface
AsInowrealizeit,thisbookwasuniquelyinspiredbyProfessorKrainov’s courseonqualitativemethodsinphysicalkineticsthatIattendedatthe MoscowEngineeringPhysicsInstitute(NationalResearchNuclearUniversityMEPhInowadays)thirtyyearsago.Aswestudentswouldlearnin amorerigorousclasstofollow,inphysicalkinetics,eventhemostbasic resultsrequirelaboriousmulti-pagederivations.ButKrainov’scourseand hisbookpublishedlaterbytheAmericanInstituteofPhysicstaughtus thatifoneis not interestedintheexactvaluesofprefactors,thentenpages ofcalculationscanbereplacedbytwoshortlinesonthe backofanenvelope; andinsomecases,evenapostalstampwouldsuffice.
Thebookyouareabouttoreadisbasedontheproblemsassigned inagraduatecourseinquantummechanicsthatIhavebeenteachingat theUniversityofMassachusettsBostonformanyyears.Similarlytothe physicalkineticsclassesIattendedattheMEPhI,thediscussiononany newtopicinmyclasswouldinvariablystartfromaseriesofqualitative problems.WhenIrealizedIhadmorethanfiftyofthem,Idecidedto assembletheminabook.
Inthisbook,Iclearly distinguishbetweenthedimensionalandtheorderof-magnitudeestimates.Dimensionalanalysisisapowerfulmethodto analyzenewunexploredequations,butitfailswhentherearetoomany dimensionlessparametersinvolved.Inanorder-of-magnitudeestimate— acalculationwhereallanglesare 90◦,allnumbersareunity,andallintegralsarejust“heighttimeswidth”—oneneedstounderstandthephysics behindtheprocessreallywell;asareward,themethodisnearlyuniversal.
Approximatelyhalfofthebookisdevotedtotheestimatesbased oneithersemi-classicalapproximationoronperturbationtheoryexpansionsin elementaryquantummechanics.Thankstoareducednumberof
viii Back-of-the-EnvelopeQuantumMechanics
independentdimensionfulparametersinthedomainsofapplicabilityof thesetheories,bothdimensionalandorder-of-magnitudeapproachesare ideallysuitedthere.
Asequenceofvariationalproblemsisalsoincluded.Thebreadthand eleganceofvariationalreasoningmakesitavaluabletoolinapreliminary analysisofaproblem;determinationoftheparityofthegroundstateina wellisagoodexample.Furthermore,eventhequantitativeresultsobtained fromsimpleone-parametricvariationalansatzesstillfitonanenvelope.
Similarly,Icouldnotresistincludingseveralpowerfulresultsproduced byapplyingtheHellmann-Feynmantheoremto integrablemany-bodyquantumsystems.Unlikeothermethodsconsidered,itproducesexactanswers; thosecanalsobeobtainedinafewlines.
The integrablepartialdifferentialequations serveasanexampleofafield wheretherearenoinnatemeasurementunits,andyetdimensionalanalysis canbedeployed;thedependenceofthesizeofaKoteweg-deVriessoliton onitsspeedisatypicalapplicationofthemethod.
Thisbookcontainsbothsolvedproblemsandexercises.Theorderofthe solvedproblemsisimportant:thesequencegraduallypreparesthereader fortheproblemswithoutsolutions.Minimaltheoreticalbackgroundisprovidedaswell.Severallesserknowntheoreticalfactsareattachedtotherespective“Background”sectionsas“Problemslinkedtothe‘Background’”. Variousapproximateandqualitativemethodsarecomparedinthreecase studies:ofahybrid,harmonic-quarticoscillator,ofa“halved”harmonic oscillator,andofagravitationalwell.
Thisbookwouldnothavebeenpossiblewithoutinputfromallthe studentsIhavetaughtinmyquantummechanicscoursesatUMassBoston andattheUniversityofSouthernCaliforniabeforeit.Specialthanksto VladimirPavlovichKrainovforintroducingmetoqualitativemethods,first asaprofessorand,lateron,asmyfirstresearchprojectadviser.Further interactionswithmymentors,VladimirMinoginandYvanCastin,inspired manynewproblemsforthebookandshapeditsstructure.
AgoodhalfofthisbookwascompiledduringquietMediterranean nights,profitingfromthefreeinternetinthelobbyoftheGalilHotelin Netanya,Israel.Manythankstoitsstuffforthecookiestheywere incessantlyfeedingmethroughoutthosenights.
Thisisanappropriateplacetothankmyfriends—VincentLorent,Lana Jitomirskaya,VanjaDunjko,LenaDotsenko,andPaulGron—forstanding bymysideingoodandinbadtimes.
IamimmenselygratefultoZaijongHwangandVanjaDunjkofora thoroughcriticalreadingofthemanuscript.
Finally,IwouldliketothankmywifeMilenaGueorguievaforcorrecting commas,articles,andawkwardsentencesandmysonMarkOlchanyifor producingthecoverart.
MaximOlshanii
Boston,Massachusetts January14,2013
SecondEdition:anupdate. MisprintsoftheFirstEditionhavebeencorrected.IamespeciallygratefultoRobertBarrandMaryFriesforspotting mostofthem.
Iamindebtedtoallthestudentsinmy EstimatesinPhysics class,at UMassBoston:theywerethefirstreadersandthefirstcriticsofalmostall thenewproblems.
Again,IamimmenselygratefultoZaijongHwangandVanjaDunjko forathoroughcriticalreadingofthemanuscript,botheditions.
TheSecondEditionfeaturestwonewChapters:Chapter2, AdimensionalestimateforPlanck’senergy:aCaseStudy andChapter12, Rare andexoticmethodsinelementaryquantummechanicsandbeyond Chapter2complementstheChapter1;thecombinationofthetwo servesasanintroductiontovariostypesofestimates.ThenewChapteris alsointendedtodemonstratethatunlikeforanyothermethodinphysics, dimensionalanalysisdoesnotrequireanyknowledgeinthefieldtowhich themethodisapplied.
TheFirstEditionofthebookwasdevotedtothe generallyapplicable shortcutsthatallowonetobypasssolvingdifferentialequations,computing complicatedintegrals,anddiagonalizinginfinite-sizematrices.Theseshortcutsincluded:order-of-magnitudeestimates,dimensionalanalysis,and variationalmethods.TheChapter12,whichmakesitsfirstappearance intheSecondEdition,complementsthislist:itfocussesonthemethods whose applicabilityislimited toanarrowclassofproblems.
MaximOlshanii
Boston,Massachusetts March15,2023
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Preface vii
1.GroundStateEnergyofaHybridHarmonic-Quartic Oscillator:aCaseStudy1
1.1Solvedproblems........................1
1.1.1Dimensionalanalysisandwhyitfails inthiscase......................1
1.1.1.1Sidecomment:dimensionalanalysis andapproximations............6
1.1.1.2Sidecomment:howtorecastinput equationsinadimensionlessform....7
1.1.2Dimensionalanalysis:theharmonicoscillator alone.........................9
1.1.3Order-of-magnitudeestimate:fullsolution....10
1.1.3.1Order-of-magnitudeestimatesvis-a-vis dimensionalanalysis...........10
1.1.3.2Harmonicvs.quarticregimes......11
1.1.3.3Theharmonicoscillatoralone......12
1.1.3.4Thequarticoscillatoralone.......12
1.1.3.5Theboundarybetweentheregimesand thefinalresult...............13
1.1.4Anafterthought:boundarybetweenregimesfrom dimensionalconsiderations.............13
1.1.5AGaussianvariationalsolution..........14
2.AdimensionalestimateforthePlancktemperature: aCaseStudy 17
2.1Solvedproblems........................17
2.1.1EstimatingthePlancktemperature........17
3.Bohr-SommerfeldQuantization 25
3.1Solvedproblems........................25
3.1.1Groundstateenergyofaharmonicoscillator...25
3.1.2Spectrumofaharmonicoscillator.........26
3.1.3WKBtreatmentofa“straightened”harmonic oscillator.......................28
3.1.4Groundstateenergyofpower-lawpotentials...30
3.1.5Spectrumofpower-lawpotentials.........31
3.1.6Thenumberofboundstatesofadiatomic molecule.......................32
3.1.7Coulombproblematzeroangularmomentum..34
3.1.8Quantizationofangularmomentum fromWKB......................38
3.1.9FromWKBquantizationof4Dangularmomentum toquantizationoftheCoulombproblem......39
3.1.10Groundstateenergyofalogarithmicpotential,a WKBanalysis....................41
3.2Problemswithoutprovidedsolutions............42
3.2.1SizeofaneutralmesoninSchwinger’stoymodel ofquarkconfinement................42
3.2.2Bohr-Sommerfeldquantizationforperiodic boundaryconditions.................43
3.2.3Groundstateenergyofmulti-dimensional power-lawpotentials.................43
3.2.41Dboxasalimitofpower-lawpotentials.....43
3.2.5Groundstateenergyofalogarithmicpotential, anestimate......................44
3.2.6Spectrumofalogarithmicpotential........45
3.2.7Closestapproachtoalogarithmichillandto power-lawhills....................45
3.2.8Spin-1/2 inthefieldofawire............46
3.2.9Dimensionalanalysisofthetime-dependent Schrödingerequationforahybrid harmonic-quarticoscillator.............46
3.3Background..........................47
3.3.1Bohr-Sommerfeldquantization...........47
3.3.2Multi-dimensionalWKB..............48
3.4Problemslinkedtothe“Background”............49
3.4.1Bohr-Sommerfeldquantizationforonesoftturning pointandahardwall................49
3.4.2Bohr-Sommerfeldquantizationfortwo hardwalls......................51
4.“Halved”HarmonicOscillator:aCaseStudy53
4.1Solvedproblems........................54
4.1.1Dimensionalanalysis................54
4.1.2Order-of-magnitudeestimate............54
4.1.3Anotherorder-of-magnitudeestimate.......55
4.1.4StraightforwardWKB................56
4.1.5Exactsolution....................56
5.Semi-ClassicalMatrixElementsofObservablesand PerturbationTheory
5.1Solvedproblems........................59
5.1.1Quantumexpectationvalueof x6 inaharmonic oscillator.......................59
5.1.2Expectationvalueof r2 foracircular Coulomborbit....................60
5.1.3WKBapproximationforsomeintegralsinvolving sphericalharmonics.................62
5.1.4Groundstatewavefunctionofa one-dimensionalbox.................64
5.1.5Eigenstatesoftheharmonicoscillatorattheorigin: howafactoroftwocanrestoreaquantum-classical correspondence...................65
5.1.6Probabilitydensitydistributionina“straightened” harmonicoscillator.................68
5.1.7Eigenstatesofaquarticpotentialattheorigin..70
5.1.8Perturbationtheorywithexactandsemi-classical matrixelementsforaharmonicoscillator perturbedbyaquarticcorrectionor.......71
5.1.9...orbyacubiccorrection.............73
5.1.10Shiftoftheenergyofthefirstexcitedstate....75
5.1.11Impossiblepotentials................76
5.1.12Correctiontothefrequencyofaharmonicoscillator asaperturbation..................79
5.1.13Outerorbitalofsodiumatom............82
5.1.14Relativecontributionsoftheexpectationvaluesof theunperturbedHamiltonianandtheperturbation tothefirstandthesecondorderperturbation theorycorrectiontoenergy.............86
5.2Problemswithoutprovidedsolutions............88
5.2.1Aperturbationtheoryestimate...........88
5.2.2Eigenstatesofatwo-dimensionalharmonic oscillatorattheorigin................89
5.2.3ApproximateWKBexpressionsformatrix elementsofobservablesinaharmonic oscillator.......................91
5.2.4Off-diagonalmatrixelementsofthespatial coordinateforaparticleinabox..........91
5.2.5Harmonicoscillatorperturbedbya δ-potential,...91
5.2.6...andbyauniformfield..............91
5.2.7Perturbativeexpansionoftheexpectationvalueof theperturbationitselfandthevirialtheorem...92
5.2.8Alittletheorem...................93
5.3Background..........................93
5.3.1MatrixelementsofoperatorsintheWKB approximation....................93
5.3.2Perturbationtheory:abriefsummary.......96
5.3.3Non-positivityofthesecondorderperturbation theoryshiftofthegroundstateenergy......97
6.VariationalProblems99
6.1Solvedproblems........................99
6.1.1Insertingawall...................99
6.1.2Parityoftheeigenstates...............100
6.1.3Simplevariationalestimateforthegroundstate energyofaharmonicoscillator...........100
6.1.4Apropertyofvariationalestimates........101
6.1.5Absenceofnodesinthegroundstate.......103
6.1.6Absenceofdegeneracyofthegroundstate energylevel......................105
6.2Problemswithoutprovidedsolutions............106
6.2.1Dostrongerpotentialsalwaysleadtohigher groundstateenergies?................106
6.2.2Variationalanalysismeetsperturbationtheory..106
6.2.3Anothervariationalestimateforthegroundstate energyofaharmonicoscillator............106
6.2.4...andyetanother..................106
6.2.5Gaussian-andwedge-variationalgroundstate energyofaquarticoscillator............107
6.3Background..........................107
6.3.1Variationalanalysis.................107
6.4Problemslinkedtothe“Background”............109
6.4.1Complexvs.realvariationalspaces........109
6.4.2Aproofthatthe (ψ )2 energyfunctionaldoesnot haveminimawithdiscontinuousderivatives....111
7.GravitationalWell:aCaseStudy113
7.1Solvedproblems........................113
7.1.1Bohr-Sommerfeldquantization...........113
7.1.2AWKB-basedorder-of-magnitudeestimateforthe spectrum.......................114
7.1.3AWKB-baseddimensionalestimate forthespectrum...................115
7.1.4Aperturbativecalculationoftheshiftofthe energylevelsunderasmallchangeinthe couplingconstant.Thefirstorder.........116
7.1.5Adimensionalestimatefortheperturbative correctiontothespectrum.............117
7.1.6Aperturbativecalculationoftheshiftofthe energylevelsunderasmallchangeinthe couplingconstant.Thesecondorder........119
7.1.7Asimplevariationaltreatmentofthegroundstate ofagravitationalwell................121
8.Miscellaneous123
8.1Solvedproblems........................123
8.1.1Adimensionalapproachtothequestionofthe numberofboundstatesin δ-potentialwell.....123
8.1.2...andinaPöschl-Tellerpotential.........124
8.1.3Existenceoflosslesseigenstatesinthe 1/x2-potential....................125
8.1.4Ontheabsenceoftheunitarylimitintwo dimensions......................126
9.TheHellmann-FeynmanTheorem129
9.1Solvedproblems........................129
9.1.1Lieb-Linigermodel..................129
9.1.2Expectationvaluesof 1/r2 and 1/r inthe Coulombproblem,usingtheHellmann-Feynman theorem........................131
9.1.3Expectationvalueofthetrappingenergyinthe groundstateoftheCalogerosystem........133
9.1.4VirialtheoremfromtheHellmann-Feynman theorem........................134
9.2Problemswithoutprovidedsolutions............136
9.2.1Virialtheoremforthelogarithmicpotentialandits corollaries.......................136
9.3Background..........................137
9.3.1TheHellmann-Feynmantheorem..........137
10.LocalDensityApproximationTheories139
10.1Solvedproblems........................139
10.1.1AThomas-Fermiestimatefortheatomsizeand totalionizationenergy................139
10.1.2Thesizeofanion..................140
10.1.3Time-dependentThomas-Fermimodelfor coldbosons......................142
10.2Problemswithoutprovidedsolutions............144
10.2.1Thequantumdot..................144
10.2.2Dimensionalanalysisofanatombeyondthe Thomas-Fermimodel................144
11.IntegrablePartialDifferentialEquations 145
11.1Solvedproblems........................145
11.1.1SolitonsoftheKorteweg-deVriesequation....145
11.1.2BreathersofthenonlinearSchrödinger equation.......................147
11.1.3Healinglength....................148
11.1.4Dimensionalanalysisoftheprojectile problemasapreludetoadiscussiononthe Kadomtsev-Petviashvilisolitons..........150
11.1.5Kadomtsev-Petviashviliequation..........152
11.1.6Thenonlineartransportequation.........154
11.1.7Burgersequation...................157
11.2Problemswithoutprovidedsolutions............161
11.2.1StationarysolitonsoftheBurgersequation....161
11.2.2StationarysolitonsofthenonlinearSchrödinger equation.......................161
11.2.3Solitonsofthesine-Gordonequation........162
12.Rareandexoticmethodsinelementaryquantum mechanicsandbeyond 163
12.1Solvedproblems........................163
12.1.1Quantum-mechanicalsupersymmetry (QM-SUSY):Pöschl-Tellerasanexample.....163
12.1.2Whatthesupersymmetricstructurealoneimplies forthescatteringstatesofthePöschl-Teller potential.......................165
12.1.3Power-indexmethod.Exampleofthenonlinear Schrödingerequation,withthePöschl-Teller problemasabyproduct...............166
12.1.4Astationary-kinksolutionoftheBurgersequation throughthepower-indexmethod..........169
12.1.5Scaleinvariance:quantumCalogeropotentialas anexample......................169
12.1.6ClassicalCalogeropotential:aposteriori manifestationsofthescaleinvariance.......172
12.1.7ClassicalCalogeropotential:apriorimanifestationsofscaleinvarianceattheMaupertuis-Jacobi level.Findingthezero-energyorbit from symmetriesalone ..................175
12.1.8Circleinversion,quantum:zero-energyeigenstates ina 1/r4 potential..................177
12.1.9Self-similartilings:momentofinertiaofan equilateraltriangleasaparadigm.........179
12.1.10Propertiesofthespectrumofanequilateral triangularquantumbilliardsthatfollowfromthe tilingself-similarity.................181
12.1.11Centerofmassofafilled“goldenb”shape,through self-similartilings..................182
12.2Problemswithoutprovidedsolutions............186
12.2.1Power-indexmethodappliedtothePöschl-Teller problemdirectly...................186
12.2.2Quasi-integrability:theoverallideaandthe exampleofthesixticoscillator...........186
12.2.3Aquasi-integrableinstanceoftwononlinearly coupledoscillators..................187
12.2.4Momentofinertiaofafilled“goldenb”shape, throughtheself-similartilings...........188 12.3Background..........................188
12.3.1Quasi-integrability:theoverallideaandthe exampleofthesixticoscillator...........188
12.3.2Relationshipbetweenvariousmethods.......190
12.3.3Furtherreading:Betheansatz...........192
Somenotations
[a] Unitsinwhichanobservable a ismeasured
[L], [T ], [M], [E ],...Unitsoflength,time,mass,energy,...
L, T , M, E,...Lengthscale,timescale,massscale,energyscale,...
P1, P2,...Independentdimensionlessparametersofagivenproblem
A[ψ( )] Afunctional A actingonawavefunction ψ(x) V Avariationalspace V
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GroundStateEnergyofaHybrid Harmonic-QuarticOscillator: aCaseStudy
Introduction
ConsidertheSchrödingerequationforaone-dimensionalparticlemovingin acombinationofharmonicpotentialoffrequency ω andaquarticpotential ofstrength β:
where m istheparticle’smass.Wewillbemainlyinterestedindetermining thegroundstateenergy.TheEq.(1.1)doesnotallowforanexactsolution. However,themajorfeaturesofthedependenceofthegroundstateenergy onthesystemparameterscanbedeterminedviaelementarymethods,such asdimensionalanalysis,order-of-magnitudeestimates,andsimplevariationalbounds.ThegoalofthisChapteristoillustratetheapplicationof thesemethodsusingthegroundstateproblem(1.1)asanexample.
1.1Solvedproblems
1.1.1 Dimensionalanalysisandwhyitfailsinthiscase
Theassignmentis:performdimensionalanalysisoftheproblemandshow thatfromadimensionalpointofviewtheproblemisunderdetermined:no estimateforthegroundstateenergycanbeproduced.However,someinformationaboutthestructureoftheexpressionforthegroundstateenergy canstillbeextracted,onpurelydimensionalgrounds.
Solution: Thedimensionalprocedureforfindingthegroundstateenergy Eg.s. (orassessingtheimpossibilityofacompletedimensionalsolution)is asfollows:
Back-of-the-EnvelopeQuantumMechanics
–Beginbyidentifyingthe principalunits ofmeasurementforthe problem,i.e.theminimalsetofunitssufficienttodescribeallinputparametersoftheproblem.Forstationaryproblemsinquantummechanics,theunitsoflength, [L],andenergy, [E ],havebeen proventoprovideahandyset;
–Identifythe inputparameters andunitsusedtomeasurethem; –Determinethe maximalsetofindependentdimensionlessparameters:thesetwillincludeonlytheparametersthataregenerally eithermuchgreaterormuchlessthanunity.Theseincludeboth thedimensionlessparameterspresentintheproblemapriori(such asthequantumnumber n),andthedimensionlesscombinations ofthedimensionfulinputparameters.Ifthesetisempty,theunknownquantitiescanbedeterminedalmostcompletely,i.e.upto anumericalprefactoroftheorderofunity.Ifsomedimensionless parametersarepresent,theclassofpossiblerelationshipsbetween theunknownsandtheinputparameterscanbenarroweddown, buttheorderofmagnitudeoftheunknownquantitiescannotbe determined.
–Foreachoftheprincipalunits,choosea scale:acombinationof theinputparametersmeasuredusingtheunitinquestion;
–Expresstheunknownquantitiesasamulti-power-lawofprincipal scales,timesanarbitraryfunctionofalldimensionlessparameters, ifany.Ifnodimensionlessparametersarepresent,thearbitrary functionisreplacedbyanarbitraryconstant,presumedtobeof theorderofunity.
Inourcase,theaboveproceduregives:
–Theprincipalunits—theunitsoflengthandtheunitsofenergy : [L] , [E ];
–Theinputparametersandtheirunits : [η]=[L]2 [E ] [Υ]=[L] 2 [E ] [β]=[L] 4 [E ] , where η ≡ 2/m,and Υ ≡ mω2; –Thesetofindependentdimensionlessparameters =
GroundStateEnergyofaHybridHarmonic-QuarticOscillator:aCaseStudy 3
Itisrepresentedbyasingleelement.Letusprovethat.First,there arenoaprioridimensionlessparametersinthisproblem.Assume nowthat P isadimensionlessparameterthatisderivedfromthe dimensionfulinputparameters.Itmustberepresentedasamultipowerlawoftheinputparameters:
Unitsfor P aregivenby
Ontheotherhand, P issupposedtobedimensionless:
Thus,thepowers
mustobeythefollowingsystemoflinear homogeneousalgebraicequations:
Thenumberofindependentdimensionlessparameterswillbegiven by (#ofindependentdimensionlessparameters) =(#ofindependent a-priori-dimensionlessparameters)
+(#ofindependentdimensionfulparameters)
Thedimensionlessparameter P1 canbefoundbysolvingthesystem (1.3).Itgives
leadingto
Back-of-the-EnvelopeQuantumMechanics
–Theprincipalscales—thelengthscaleandtheenergyscale, examplesof :
Theprincipalscalesabovearedefinedasexamplesofobservables measuredinprincipalunits, [L] and [E ] inourcase.Toderivethe aboveexpressionforthelengthscale,letusrepresentthisscaleas
Thecorrespondingunitsarerelatedas
Thepowers
obviouslyobeyasystemoflinearinhomogeneousalgebraicequationsgivenby
where M isgivenbytheexpression(1.4).Anyparticularsolution ofEq.1.6(andinthisparticularcasewehaveaone-dimensional familyofthem)canbechosentorepresentalengthscale;this choiceisamatterofconvenience.Wechoosethescaleassociated uniquelywiththeharmonicoscillator,
or,forexample,
GroundStateEnergyofaHybridHarmonic-QuarticOscillator:aCaseStudy 5
Theenergyscale E,givenbytheexpression(1.6),canbeobtained thesameway.Theonlydifferenceistherighthandsideofequation (1.6):itshouldread 1 0 ; –Solutionfortheunknown :
Thissolutiondoesnarrowtheclassofpossibleexpressionsfortheground stateenergy,butdoesnotallowonetodetermineit,notevenitsorderof magnitude.
Ifneeded,analogousexpressionsforotherobservablescanbereadily obtained.Fortheobservablesmeasuredincombinationsofprincipalunits only,oneshouldcombinetheprincipalscalestoformascalefortheobservableofinterest.Thefulldimensionalpredictionforthisobservablewillbe given,asbefore,asaproductofanarbitraryfunctionofalldimensionless parametersandthescale.Forexample,ther.m.s.(rootmeansquare)force actingonourparticleinthegroundstatewillbegivenby
istheforcescale,and Φ2(P )isanotherarbitraryfunction.
Fortheobservablesmeasuredinunitsthatdonotbelongtotheprincipal set(theminimalsetofunitstodescribeallinputparameters),otherscales mustbeinventedifneeded.Forexample,theinverseharmonicfrequency, 1/ω,providesausefultimescale:
Back-of-the-EnvelopeQuantumMechanics
Inmechanicsproblems,bothclassicalandquantum,nomorethanthree independentscalesareevernecessary.Forexample,ther.m.s.(rootmean square)groundstatevelocityisgivenby
with
beingthevelocityscale.
1.1.1.1
Sidecomment:dimensionalanalysisandapproximations
Often,whenanexacttheoryisreplacedbyanapproximateone,thenumber ofindependentdimensionfulparametersdecreases,thusshiftingthecounting(1.5)infavorofaccurateup-to-a-prefactordimensinalpredictions.In whatfollows,wewillencounterseveralexamplesofsuchreduction.For example,insemi-classicaltheory,consideredinChapter3,thePlankconstant andthelevelindex n fuseintoasingleentity,theclassicalcanonical action ˜
Thisreductionisaconsequenceoftheabsenceofthequantizationofactioninclassicalmechanics.AnotherexampleistheThomas-Fermitheory, Chapter10.There,thenumberofelectrons Z andtheelectroncharge |e| uniteandformatotalcharge Q,neverappearingseparately:
Thephysicalreasonforsuchamergeristhatinmean-fieldtheories—suchas theThomas-Fermitheory—thenumberofelectronsisnolongerquantized,
GroundStateEnergyofaHybridHarmonic-QuarticOscillator:aCaseStudy 7
andacontinuousfieldofelectronnumberdensity n(r) isusedinsteadofthe individualelectronpositions.Asimilarreductionhappenswhensolutionsof amany-bodySchrödingerequationforanensembleofbosonicparticlesare approximatedusingsolutionsofaone-bodynonlinearSchrödingerequation (SeeProblem11.1.3).
Yetanotherexampleofareductionofthenumberofinputparameters underanapproximationisprovidedbyperturbationtheory(Chapter5). Here,inthe n-thorderofperturbationtheory,thepowerofadimensionful prefactorinfrontoftheperturbationisfixedto n,effectivelyremoving theprefactorfromthelistofindependentparameters.See,specifically, Problems5.1.8and5.1.9.
1.1.1.2 Sidecomment:howtorecastinputequationsin adimensionlessform
Priortoinvolvedanalyticalornumericalcalculations,equationsareoften expressedina“dimensionlessform”,theadvantagebeingareducednumber ofparametersandanabsenceofnumbersthataretoolargeortoosmall. Therecipeisasfollows:
-Severalinputparameters,asmanyasthereareprincipalunits, arereplacedbyunity;
-Theremaininginputparametersretaintheirnotations,buttheir numericaldimensionfulvaluesarereplacedbydimensionlessnumbersgivenbytheratiosbetweentheoriginalvaluesoftheparametersandthecorresponding“scales”;
-Likewise,intheendthenumericalvaluesoftheanswersaremultipliedbythecorrespondingscales.
Awell-definedformalprocedureishiddenbehind.Itconsistsoftwoelements.(a)Theparameterstobesettounityarechosenasprincipalscales; (b)thesescalesandtheirmulti-powercombinationsareusedasnewunits ofmeasurement.
Inourexample,wecanchoose [η] and [Υ] asthe“principalunits”and η and Υ themselvesasthe“principalscales”.Obviously,inthissystem ofunits, η and Υ assumeunitvalues.Also,formoderatevaluesofthe remainingdimensionlessparameters,allanswerswegetbecomeoftheorder ofunity.
Back-of-the-EnvelopeQuantumMechanics
Conventionally,thissystemofunitsisdenotedas
orevenmoreoften
Theappearanceofthreescalesisnotanaccident.Therecipe(1.8)(a)does notleadtoanyambiguitiesintime-independentproblems;(b)allowsone tofixallthescales,notonlytheprincipalones;(c)preparesgroundfor time-dependentproblems.
Accordingtotherecipe(1.8),theoriginalSchrödingerequation(1.1) becomes
Inshort,accordingtothisrecipe,insteadoftheoriginalEq.(1.1)wedeal withitsdimensionlessform(1.9),readyforanalyticornumericalwork:the waytoobtain(1.9)istoreplaceeachoftheparameters 2/m and mω2 by unity,andreplacetheparameter β by β m2ω3 ;thelatterwillfurtherbecome anumber,say 2 74,ifsomenumericalanswersarerequired:
ImaginethatwesolvednumericallytheEq.1.9forsomevalueof β and obtained Eg.s. =1.14 ....Toreturntotheusualsystemofunits,wehaveto simplymultiplythisresultbytheenergyscale ω—i.e.theonlyparameter withunitsofenergythatcanbeconstructedoutof , m,and ω:
Thetruereasonwhythewholeprocedurelooksmysteriousatfirstisthat formallyspeakingitseverelyabusesnotations:forexample, Eg.s. =1 14 and Eg.s. =1 14 × ω shouldbedenotedbydifferentsymbols—butthey are not.Thepracticaladvantagesofthisconvention,however,compensate forthedifficultiesexperiencedatthelearningstage.
GroundStateEnergyofaHybridHarmonic-QuarticOscillator:aCaseStudy 9
1.1.2 Dimensionalanalysis:theharmonicoscillatoralone
Now,letustrytoproduceadimensionalsolutionforthegroundstateenergy oftheharmonicoscillatoralone:
Solution: Theproceduregoesasfollows:
–Theprincipalunits—theunitsoflengthandtheunitsofenergy : [L] , [E ];
–Theinputparametersandtheirunits :
]=[L]2 [E ] [Υ]=[L] 2 [E ] ,
whereagain η ≡ 2/m,and Υ ≡ mω2;
–Thesetofindependentdimensionlessparameters = ∅.Indeed, assumethereexistsadimensionlessparameter P expressedasa productofpowersofprincipalscales:
Itsunitsarenow
TheanalogueofEq.(1.3)is
Now,accordingtotherule(1.5),thisproblemhas nodimensionless parametersatall.Thisisexactlythesituationwheredimensional analysisproducesthemostcompletesolutions,accurateuptoan unknownnumericalprefactor;
