The historical and physical foundations of quantum mechanics robert golub

The Historical and Physical Foundations of Quantum Mechanics

Robert Golub

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THEHISTORICALANDPHYSICALFOUNDATIONSOFQUANTUM MECHANICS

TheHistoricalandPhysicalFoundations ofQuantumMechanics

RobertGolub

NorthCarolinaStateUniversity,Raleigh,NC,USA

StevenK.Lamoreaux YaleUniversity,NewHaven,CT,USA

GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom

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c ⃝ RobertGolubandStevenK.Lamoreaux2023

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WededicatethisbooktothememoriesofProf.J.M.Pendlebury, Dr.V.K.Ignatovich,Prof.A.Steyerl,andProf.H.G.Dehmelt.

Preface

Thereisanenormousnumberofbooksandotherwritingsconcernedwithexplaining andinterpretingquantummechanics.Standardtextstendtoconcentrateonmethodologyandapplicationstospecificproblems,whilediscussionsofinterpretationandthe historicaldevelopmenttendtocontainaminimumofmathematics.Theideabehind thisbookisthattogainarealunderstandingofthesubject,someacquaintancewith thehistoricaldevelopmentisessential;afterall,thathistoryisthenarrativeofhow humanitylearnedquantummechanics.Theideaswerenotfoundwrittenontablets onafarmine.g.,NewYorkState,butwereslowlyandpainstakinglydevelopedby peoplejustlikeus.Wealsoprovideaccompanyingdiscussionsforthevariousinterpretationsthathavebeensuggested,providingsufficientmathematicalillustrationsthat highlighttherespectivefeaturesanddifferences.

Acknowledgements

Wewouldliketothankourfriendsandfamilyfortheirunderstandingandsupportas ourattentionwasdrawnawaytothewritingofthisbook.

R.G.wouldliketothankhiswifeEkaterinaKorobkina,E.DavidDavisforcontributionstoChapter21andhelpwiththebook,andRolandG¨ahler.Inaddition,he thanksProf.ChuengJifordiscussionsattheearlystagesofthebook.

S.K.L.thankshiswife,Melissa,anddaughter,Zoe,hisFridayOwlShopcigar friendsCarlJ.FranoandJamesSurprenant,andDr.SidneyB.Cahnfortheirunfailingandunflappablemoralsupport.HealsothanksMr.EdwardS.McCatty(B.A. (Amherst),M.Div.(Yale),M.A.Lit.(UCL))foreditorialcommentsonChapter1and forprovidinginspirationthroughouttheproject.

WeespeciallythankDr.YuliaGurevich,whoseexpertediting,togetherwithher vastknowledgeofphysics,clarifiedandstrengthenedmanypartsofthisbook.Her artistictalentisevidentinmanyofthefigures.

YaleUniversityprovidedsupportforthepreparationandeditingofthemanuscript. NCSUalsoprovidedsupportwithasabbaticalforR.G.

♠♢♣♡

PARTIIAPPLICATIONSOFQUANTUMMECHANICS

PartI BasisoftheTheory

Thefirstpartofthisbookprovidesahistoricalbackgroundandbringsustothe moderntheory.

Introduction

“Youhavenothingtodobutmentionthequantumtheoryandpeoplewilltakeyourvoicefor thevoiceofscienceandbelieveanythingyousay.”GeorgeBernardShaw,19381

1.1 Overview

AsearchonAmazon.comforbookson“quantumtheory”returnsover10,000hits whilesearchingfor“quantumphysics”returnsover20,000.Thiscorrespondstoone bookadayfor30years.Thesebooksrangefromadvancedmathematicaltreatises tobookswithoutasingleequation,fromdeepphilosophicaldebatesbetweenauthors withdifferentunderstandingsofthesubjecttotextbooksteachingthemethodology andvariousapplications.Inaddition,therearevastnumbersofpapersinhistorical andphilosophicaljournalsconcernedwiththedevelopmentandphilosophicalimplicationsofthetheory.Forthoseinterested,therearealsomanyvolumesofcollected correspondenceandmanyonlinearchivesoforalandwrittenmaterial.2

Whilethereislittledisputeoverthemathematicalapparatusofthetheoryandits applicationtophysicalproblemsthereisawidespectrumofdivergentopinionsabout whatthetheoryistryingtotellusconcerningthenatureofreality.Foralongtime followingWWII,therewaslittleinterestamongphysicistsforsuchquestionsasattentionwasturnedtothefreneticdevelopmentofdifferenttechnologies.However,recent decadeshaveseen,inadditiontoanamazingrangeofapplicationsofthetheory,an ever-increasingattentiontowhatiscalledthe“interpretation”ofquantummechanics. Thereisnowabewilderingforestoftheseinterpretationseachofwhichhasagroupof supportersaswellasopponents.As,tothisdate,noneoftheinterpretationshasbeen abletoconvinceamajorityofworkingphysicists(who,itshouldbesaid,mostlyignore thesediscussions,anattitudethathasbeensummedupas“shutupandcalculate”) ofitscorrectnessornecessity.Itisalmostasifphysicsissplittingintoanumberof cultsunitingsupportersandcriticsinanever-endingembrace.

Itisstrikingthatalloftheproposedinterpretationsareconcernedwiththeoriginal formofthetheory,theSchr¨odingertheorysupplementedbytheDiractransformation theory,seeminglyignoringthemostadvancedformofthetheory,i.e.,thatinvolvingthe

1QuotedbySimon,D.R.,OnthePowerofQuantumComputation,35thAnnualSymposiumon theFoundationsofComputerScience,(1994)SantaFe,NMandatwww.greatest-quotations.com.

2Seee.g.,AmericanPhilosophicalSocietyLibrary:SourcesfortheHistoryofQuantumPhysics, 1898-1950,https://search.amphilsoc.org/collections/view?docId=ead/Mss.530.1.Ar2-ead.xml.

TheHistoricalandPhysicalFoundationsofQuantumMechanics.RobertGolubandStevenK.Lamoreaux,OxfordUniversityPress.

c ⃝ RobertGolubandStevenK.Lamoreaux(2023).DOI:10.1093/oso/9780198822189.003.0001

quantizationofthenon-relativisticSchr¨odingerequation,introducedbyJordan,with thesupportofPauliandWigneramongothers,thathecalled“secondquantization.”

Aswewillsee,thisformulationsolvesseveralproblemsassociatedwiththeoriginal formofthetheoryandalmostsuppliesitsowninterpretation,asdoesrelativistic quantumfieldtheorywherequestionsofinterpretation,essentiallywhetherparticles orwavesareprior,aremuchlessprominent.3 Thisconcentrationonanotfully maturedversionofthetheorymightbeconsideredbysomeasanindicationthatthe interpretationdiscussioniscaughtinatime-warp,devotingitsattentiontoatheory thatcouldbeviewedasalreadysuperseded.

Thepurposeofthisbookistotakeastepbackandattempttoretracethedevelopmentofthetheorybyinvestigatingoriginalsources,theoriginalpublishedpapers andletters,oftheparticipants.Thisisthepathbywhichhumanitylearnedquantum mechanicsandfollowingitmighthopefullyleadtoanimprovedunderstanding.Of course,theattemptbyphysiciststhemselvestoapproachthehistoryoftheirsubject isanexercisefraughtwithdifficulties,ashasbeenrecognizedbyseveralpractitioners. Forexample,SilvanSchweber,atheoreticalphysicistturnedhistorianofscience,recognizes4 that“thehistoryofsciencecannotescapesomeformofwhiggism.Thedata issorichthatsomeselectionmustbemade.”Awhighistoryofscienceistheview ofthescientificwinnerswhowriteasiftheirtriumphwasaninevitableresultofthe correctnessoftheirideas.Whighistoryofsciencedisplaysthehistoricaldevelopment asproceedingfromapastruledbyignorancetoagloriouspresentwithouttaking accountoftheactualstateofknowledgeinthepast.

WecanseetheresultoftryingtoovercomethelimitationmentionedbySchweberandincludeallrelevantpublicationsalongwithbiographicalinformationonthe manyactorsandexcerptsfromcorrespondence,intheheroicworkmadebyMehra andRechenberg,whohavecompletedaninevolumetreatise,“TheHistoricalDevelopmentofQuantumMechanics,”publishedbetween1982and2001.5 Thishasbeen anenormoushelpinwritingthepresentvolume

S.A.Goudsmit,thecodiscovererofelectronspin,6 wasskepticalastotheutility ofthehistoryofscience:7 “Manyhistorianshavewrittenveryprettystoriesabout howadiscoveryshouldhavebeenmade,butitisunfortunatelyveryimprobablethat thedevelopmentwasaslogicalasthesefabricatedstorieswouldindicate.Luckand randomeventsplayamuchlargerrolethanpeoplearereadytoadmit.”Inaddition hecomplainsthat:“They(thehistoriansofphysics)presentthingsasifthewholeof physicswascreatedbyahandfulofgeniuses.Thisiscompletelyunfairtothemany physicistswhoseworkenablesthegreatdiscoveriesofthegeniuses.”

3Weinberg,S. TheQuantumTheoryofFields,Vol.1,Foundations,CUP,(1995)

4Schweber,S.S., QEDandtheMenWhoMadeIt:Dyson,Feynman,SchwingerandTomonaga, Princeton,1994

5Mehra,J.andRechenberg,H., TheHistoricalDevelopmentofQuantumMechanicsVolumes 1–6,comprisingninevolumesintotalassomeofthevolumesareprintedintwoparts,SpringerVerlag1982–2001.

6Uhlenbeck,G.F.andGoudsmit,S.,Naturwissenschaften,13,953,(1925)and SpinningElectrons andthestructureofSpectra,Nature,117,264(1926)

7Goudsmit,S.A., TheDiscoveryoftheElectronSpin (inGerman),Phys.Blaetter,10,4345(1965)

ThePrehistoryofQuantumMechanics:atomism 5

Hethengoesontostate“Historiansareoftenunjustwithrespecttotheexperimentalphysicists.Eventhoughtheevolutionofideasisveryimportantforhistory,we shouldnotneglectthegeniusesamongtheexperimentalphysicistswhosediscoveries andresultsareabsolutelynecessaryfornewideasandtheirverification,”andfurther makesthepointthat“Publishedarticlesarenotveryreliableashistoricalsources.In agoodarticle,theauthortriestoconvincethereadersoheoftenchoosesadifferent trainofthoughtasthatbywhichhecameupontheidea.”Thisissomethingthatcan beattestedtobyanyexperiencedresearcher.

StevenWeinberg(op.cit.)explicitlydisdainsthehistoricalapproachtoteaching physicaltheories,preferringalogicaldevelopmentofthetheoryasitispresently understood.This,ofcourse,vitiatestheimportanceofdirectobservationofnatural phenomena,andthefactthatcurrentphysicaltheorieswereatonetimetenuous hypothesesthatrequiredtestingviathescientificmethod.Assuch,abandoningthe historicalapproachappearsasathrowbacktowardScholasticismwithitsbasisin dogmatism.

AlbertEinsteinwasalsoskepticalofahistoricalapproach:

Onlythosewhohavesuccessfullywrestledwithproblematicsituationsoftheirownagecan haveadeepinsightintothosesituations,unlikelaterhistorianswhofinditdifficulttomake abstractionsfromthoseconceptsandviewswhichappeartohisgenerationasestablishedor evenselfevident.8

Whilethereiscertainlyalargedegreeoftruthinallofthisthefactisthatthe originalpublishedpapersareclosertotheoriginalideasthanathird-generationtextbookandcanbeexpectedtoreflectsomethingofthethen-contemporaryzeitgeistas theresultoftheauthor’sstatedwishtopersuadehisreaders.Wealsomakeuseof lettersandcontemporaryaccountswhenappropriate.

Thus,inthisbook,whilebeingawareoftheseissues,wewillattempttotracethe mainlinesofthedevelopmentwiththehopethatthisreturntotherootswillcast somelightonwhataretodayconsideredthedifficultiesofthetheory.

1.2 ThePrehistoryofQuantumMechanics:atomism

Quantummechanicshasitsfundamentalbasisintheatomictheoryofmatter,which hasitsrootsinatomism.Atomismwasoriginallya philosophical theorythatmaterial objectsarediscontinuous,beingconstructedofindivisibledistincttypesofatoms— equivalently,quantizedunitsofmatter,thatserveasbuildingblocks.Atomsarenow understoodtobeoflimitedvariability(chemicalelements,isotopes,periodictable), buteachtypeofatomhasuniqueandfixedproperties,andallatomsofagiventype arenowunderstoodtobeidenticalandindistinguishable.

Theconceptofatomismhasalongcheckeredifnottortuoushistory,onethatis rarelyexpoundeduponinphysicsbooks.Wewillpresentaveryabbreviatedoverview ofthedevelopmentofmodernideas,andthesearefromaveryWesternperspective. Therewaslikelywidespreadcommunicationintheancientworldthatallowedideasto bespread,anditisnotimpossiblethatGreekatomismhaditsoriginwiththeIndian

8Einstein,A.,ReplytocriticismsinSchilpp,P.A.,ed., AlbertEinsteinPhilosopher-Scientist, Vol.II,Harper,1949,1951.

sageandphilosopherAcharyaKanad(Kashyap)whoaround600BCEspeculatedon thelimitofdivisibilityofmatterandproposedparticlesthatcouldnotbedivided further, anu or atoms.Perhapsitstandstoreasonthatanysocietywithamerchant classhasspeculatedonthedegreeofdivisibility(henceminimummarketableunit)of materialbodies;avastbodyofhistoryisneverrecorded,orlost—inthewordsofRoy Batty—liketearsintherain.

Intheteachingofphysics,inthoserareinstanceswherehistoryismentioned,Democritus(ca.450BCE)isoftencreditedwiththeoriginalformulationoftheatomic hypothesis,andthatisit,nothingmore.Thestoryisalmostinfinitelymorecomplicated,andwewillattempttoprovidesomehighlights.Democrituswasastudentof Lucretius(ca.475BCE)withwhomtheatomicideahasitsroots,whichheformulated inresponsetoParmenides’deductionthatrealityisanillusion.9

AccordingtoParmenides,foranobjecttomovefromonelocationtoanother,it wouldneedtobedestroyedatthefirstlocationandrecreatedatthenewlocation. Asthisappearstobeanimpossibility,Parmenidesmadethelogicalleapthatreality isanillusion.Thenotionthatthereisnoreality,thatallthatexistsisillusion,has comeupmanytimessincetheancientGreekphilosophers—Shakespeare’s“Allthe worldisastage,”andmorerecently,thenotionthatwearelivinginacomputer simulationisbeingtakenseriously10 andisanessentialformofIdealism.Thevarious ZenoparadoxeswereputforwardinsupportofParmenides’assertion,toshowthat thephysicaluniverseaswebelieveweareobservingitisindeedanimpossibility.Of course,itiseasytobelieveeverythingisanillusionuntilaseveretoothachestartsona Saturdaynight;realityisofteninconflictwithourbeliefs,expectations,andprejudices, thatareformedintheechochambersofourminds,colleagues,andnowadays,social media(FacebookandTwitter).

Lucretius,followedby,andembellishedby,Democritus,answeredParmenides’ claimbyinventingatoms,andequallyimportant,the void,inwhichatomsmove. Thevoidisnothingness,andtheargumentagainstitsexistencecontinuestodaybecausewearefacedwiththeproblemofinventingadescriptionforsomethingthatdoes notexist,whichisanapparentself-contradiction.Nonetheless,thecompleteatomic picturewaslaidoutbyDemocritus,inwhichobjectsareconstructedofatomsof varyingcharacteristics,andtheseatoms,collectedtogetherasobjects,movetogether freelyinthevoid.Thesearethebasictenetsofthemodernpictureoftheuniverseand matter,perhapscoincidentally,asthiswasaphilosophicaltheory.

Jumpingaheadsome100years,Aristotletookastepbackwardinhisadoption ofEmpedocles’notion(450BCE)thatthematerialworldcomprisesfourelements, earth,wind,fire,water,andfurthersurmisedthatthenaturalstateofmatterwas atrest.(TheGreeknotionofelementsmighthavealsobeenderivedfromtheHindu Veda whichexistedinoralformfrom2millenniaBCEandinwrittenformfrom1 millenniaBCE,inwhichthesamefourelementsplusafifth,theall-importantvoid, arepostulated.)

9AlthoughitistemptingtoascribethediscontinuityofmatterasassumedbyDemocritusas resultingfromalackofunderstandingofmathematicalcontinuity,however,thedevelopmentofthe earlyphilosophicaltheoriesfollowsamorecomplicatedpath.See,forexample,BernardPullman, The AtomintheHistoryofHumanThought (OxfordUniversityPress,2001).

10Bostrom,Nick(2003). AreYouLivinginaComputerSimulation?.PhilosophicalQuarterly 53, (211):243-255.doi:10.1111/1467-9213.00309.

ThePrehistoryofQuantumMechanics:atomism 7

Afifthelement,quintessence,wasintroducedastheelementfromwhichheavenly bodiesareconstructed.Bythemedievalages,anewnotion(aformofMonism)was introducedthateverythingwasaquintessence-likeelement,scrapedtogetherintoa particularform,atwhichpointthequintessenceassumedthepropertiesoftheform, e.g.,apencilsharpener,thekeyboardonwhichIamtyping,etc.Thisnotionwastaken asacentralprincipleortenetbytheCatholicChurch,andprovidesamechanismfor transubstantiation.Thistenetwasimportantenoughthatatomismwasspecifically addressedbytheCouncilofTrent(1545to1563)asanathema(heretical).

Galileoisofcourseknownforthetrialsheenduredconcerninghispromotingthe heliocentricmodelofthesolarsystem.ThemostpuzzlingaspectoftheentireGalileo affairisthathehadbeenwellreceivedbyPopeUrbanVIII,whowasfullyawareof andstudiedGalileo’swritings.TheresultsofGalileo’sfirsttrialin1616werelimited toorderstoceaseholding,teaching,ordefendingheliocentricideas.Uptothistime, GalileohadagoodrelationshipwiththeJesuits,eventhefactioninchargeofimposingchurchdoctrine,whichincludedthecanonsoftheCouncilofTrent;thisfaction wasalsoinchargeofgeneraleducation.Galileo’steachingswereatoddswithAristotleandScholasticism,sothatafactionofJesuits(forwhomAristotelianteachings wereeducationalcanon)becameincreasinglyhostiletowardGalileo;thishostilityonly increasedwiththeminimalresultsofGalileo’sfirsttrial,especiallywhenhedidnot ceasepromotinghisscientificideasandcontinuedtowritebooks.PopeUrbanVIII,to appeasetheseJesuitsinhiseffortstoconsolidatepower,accededtotheirdemandsthat Galileobeagainbroughttotrial,beforetheInquisition,forheresy.Recentdiscoveries intheVaticanrecordsshowthatmorecharges,inadditiontothoseassociatedwith heliocentrictheories,werebeingpreparedtobringupGalileo’sembracingatomism asanadditionalheresy.11 Tofurtherinflamethesituation,Simplicioin Dialogueon theTwoWorldSystems wassuggestedasmodeledonUrbanVIII.In1632,thePope orderedanotherinvestigationagainstGalileo.Thistimehewasprosecutedfollowing thenormalmethodsoftheInquisition,however,Galileowasthenofadvancedage andwasthereforenotsubjecttotortureanddeathforbeingfoundguiltyofheresy, butconsequentlywasplacedunderhousearrestfortherestofhislife.Ayounger man,GiordanoBruno,who30yearsearlier(duringthetenureofPopeClementVIII) embracedheliocentricity,atomism,andmanyotherhereticalscientificandsociologicalnotions,andtaughtthemwithabandon,wasdecreedguiltyofheresyandon17 February1600washungupsidedownnakedbeforebeingburnedatthestake.

OneinterestingandimportantasideisthatGalileo’sandBruno’swritingswere preservedintheVaticanArchives;thisisoneparticularlyastonishingaspectofthe CatholicChurchinthatthewritingsofenemieswereveryoftenpreserved,unlikemost humaninstitutionswherethememoriesofadversariesareerasedasapatheticpanacea againstfuturethreats.TheChurchdidnotinitiateacampaigntocollectupGalileo’s booksandritualisticallydestroythem,incontradistinctionto,forexample,theNazis’ handlingtheworksofenemiesofthestatebyburningbooksinwell-publicizedbonfires, ortheMemoryHoleofOrwell’s1984.Thisisnottosaythatit never happened,but itappearsthepreservationwasageneralmatterofcourse.

11PietroRedondi(RaymondRosenthal,Translator), Galileo-Heretic (PrincetonUniversityPress, 1989).

Theabove,ofcourse,ispresentedwiththecaveatthattheunderstandingofhistoricaleventsisfraughtwithdifficulties;dowehavethecompletepicture?Whatwereall participantsintheeventthinking?Whatwerethefundamentalmotivations,e.g.,consolidationofpower,controllingthemasses,etc.?RegardingGalileo,ArthurKoestler comments:

ButthereexistedapowerfulbodyofmenwhosehostilitytoGalileoneverabated:theAristoteliansattheuniversities.Theinertiaofthehumanmindanditsresistancetoinnovation aremostclearlydemonstratednot,asonemightexpect,bytheignorantmasswhichiseasily swayedonceitsimaginationiscaught—butbyprofessionalswithavestedinterestintradition andinthemonopolyoflearning.Innovationisatwofoldthreattoacademicmediocrities:it endangerstheiroracularauthority,anditevokesthedeeperfearthattheirwhole,laboriously constructedintellectualedificemightcollapse.Theacademicbackwoodsmenhavebeenthe curseofgeniusfromAristarchustoDarwinandFreud;theystretch,asolidandhostilephalanxofpedanticmediocrities,acrossthecenturies.Itwasthisthreat,notBishopDantiscus orPopePaulIIIwhichhadcowedCanonKoppernigkintolifelongsilence.InGalileo’scase, thephalanxresembledmorearearguard—butarearguardstillfirmlyentrenchedinacademic chairsandpreachers’pulpits.

asquotedinPullman,op.cit.,p128.,fromArthurKoestler.12

Onakindernote,asaphilosophicaltheory,Aristotelianismisperfectlyinternally consistent.However,thisdoesnotmeanitrepresentsreality;asBertrandRussell quipped,13

Aphilosophy[ofnature]thatisnotself-consistentcannotbeentirelycorrect,butonethatis self-consistentmaywellbecompletelyfalse.

AccordingtoJeroenvanDongen,“Kuhnhimselfmentionedakindofepiphany hehadexperiencedwhenassistingConantinteachingthehistoryofscience:Reading Aristotle,heshockinglydiscoveredthathisownNewtonianexpectationswereblocking himfromseeingtheconsistencyandintegrityofAristotle’sphysics.Thisexperience puthimonthepath”tohisfamousbookintroducingtheconceptofparadigms.14 Koestler,again,waslesskind,

Aristotelianphysicsisreallyapseudoscience,outofwhichnotasinglediscovery,invention ornewinsighthascomeintwothousandyears;norcoulditevercomeandthatwasits secondprofoundattraction.Itwasastaticsystem,describingastaticworld,inwhichthe naturalstateofthingswastobeatrest,ortocometorestattheplacewherebynaturethey belonged,unlesspushedordragged;andthisschemeofthingswastheidealfurnishingfor thewalled-inuniverse,withitsimmutablyfixedScaleofBeing.

However,thiswasnotAristotle’sfault;toagainquoteBertrandRussell, Inreadinganyimportantphilosopher,butmostofallinreadingAristotle,itisnecessary tostudyhimintwoways:withreferencetohispredecessors,andwithreferencetohis successors.Intheformeraspect,Aristotle’smeritsareenormous;inthelatter,hisdemerits areequallyenormous.Forhisdemerits,however,hissuccessorsaremoreresponsiblethanhe is.HecameattheendofthecreativeperiodinGreekthought,andafterhisdeathitwas twothousandyearsbeforetheworldproducedanyphilosopherwhocouldberegardedas approximatelyhisequal.Towardtheendofthislongperiodhisauthorityhadbecomealmost asunquestionedasthatoftheChurch,andinscience,aswellasinphilosophy,hadbecomea seriousobstacletoprogress.Eversincethebeginningoftheseventeenthcentury,almostevery seriousintellectualadvancehashadtobeginwithanattackonsomeAristoteliandoctrine;

12Koestler,A., TheSleepwalkers:AHistoryofMan’sChangingViewoftheUniverse (London, Arkana,1959);theabovequoteisapparentlyaback-translationfromFrench.

13BertrandRussell, HistoryofWesternPhilosophy (FirstpublishedbyGeorgeAllenandUnwin Ltd,London.1946).

14vanDongen,J. InEurope,PhysicsinPerspective22,3-25(2020).

ThePrehistoryofQuantumMechanics:atomism 9

inlogic,thisisstilltrueatthepresentday.Butitwouldhavebeenatleastasdisastrous ifanyofhispredecessors(exceptperhapsDemocritus)hadacquiredequalauthority.Todo himjustice,wemust,tobeginwith,forgethisexcessiveposthumousfame,andtheequally excessiveposthumouscondemnationtowhichitled.

Andfinally,Crescenzo(asquotedbyPullman 15)states, ForbothPlatoandAristotle,whowereconstantlyinsearchoftheprimecauseandultimate purpose,itisasthoughDemocritushadtoldthemtheplotofacomedywhileskippingthe firstandlastscenes.

WhatGalileobroughtforward,asbegunbyCopernicusandKepler,isthepossibilityoftheuseofmathematicstodescribephysicalsystems,andmotionordynamics, inparticular.Thisapplicabilityandeffectivenessofmathematicsinthisendeavoris thebasisofmodernphysicalsciencesandengineering,anditisnotobviousthatthis shouldbepossible.16

Newton’swork(circa1700)carriedtheapplicationofmathematicstoarevolutionarynewlevel,andwasaharbingeroftheendofAristotelian dominanceinWesternthinking.

Evensomealchemistsatthistimehadmovedonfromthenotionthateverythingis composedofthefourprimordialelements,earth,air,water,heat.Forexample,instead ofcombiningelementstoformgold,theGermanalchemistHenningBrandattempted toextractexistinggoldfromurine;hereasonedthatbecauseurineisnormallygolden, itmustcontaingold,ormightholdthekeytofindingthePhilosopher’sStone.Some timearound1669,heembarkedontheAugeantaskofboilingdown5,700litersof putrefiedhumanurine(thereisnorecordofhowheobtainedthisquantity),andthen subjectingtheresiduetoheatinthepresenceofcarbon,whichreducedphosphates toelementalphosphorus,ataskthatbringstomindMarieandPierreCurie’slater Augeantaskofextractingafractionofagramofradiumfromtonsofpitchblende.

Phosphoruswasthefirstelementtobediscoveredthatwasnotalreadyknownin ancienttimes,andtheappearanceofacontinuousglowmusthavebeenastounding andawe-inspiringtothealchemist.Brand,ofcourseintypicalalchemistfashion,kept hisdiscoverysecretbutendedupsellingtherecipe–andalsotippedoffRobertBoyle (whosoonfiguredouthisownextractionmethod)astothesourceofphosphorus.

ThefinalmajorblowtoAristotelianismcamewiththediscoverythatwatercould becreatedbycombininghydrogenandoxygen,mostnotablyasdescribedbyLavoisier in1789;hesurmisedthatwateris85%oxygenand15%hydrogenbyweight–andis thereforehardlyanelement.Lavoisier’soxygentheoryofcombustionalsobrought downthephlogistontheoryofcombustion,andledtothelawofconservationofmass.

OneofthereasonsforourdelvingintothishistoryisthatScholasticism(education basedonAristotelianprecepts)dominatedmuchofWesternEuropefrombeforethe twelfthcenturythroughtheeighteenthandwellintothenineteenthcenturyinsome regions,andhadaprofoundeffectonthedevelopmentofatomictheoryinphysics, butlesssoonchemistry.Oneofthelasttimesthatsomeoneofnotemademention oftheprimordialelementswasNapoleon,whoquipped,“Godcreatedafifthelement especiallyforPoland–mud,”afterhisarmywasmiredduringthe1806campaigninto

15Pullman, loc.cit.,p.56

16EugeneWigner, TheUnreasonableEffectivenessofMathematicsintheNaturalSciences,CommunicationsinPureandAppliedMathematics, 13,1(1960).

Poland.17 Bythemid-nineteenthcentury,Aristotelianismwasrelegatedtoajoke,in particular,Melvillein MobyDick describesthestateofthe Pequod,whenherupper deckswereoverloadedduringthesearchforaleakingwhale-oilcaskstoreddeepbelow deck,as,“Top-heavywastheshipasadinnerlessstudentwithallAristotleinhis head.”

Itisnoteworthythatteachingatomismwascontroversialandassociatedwithaprogressiveoutlookformorethanamillennium.Aslateas1624thecourtofKingLouis VIIIofFrancethreatenedtheteachingofatomismwiththedeathpenalty.General

17F.LorainePetre, Napoleon’sCampaigninPoland (SampsonLow,MarstonandCompany:London,1901).p.51.

Fig.1.1 TheAlchemistDiscoveringPhosphorus byJosephWrightofDerby(1771,reworked 1795)(PublicDomain).

questionsabouttheinfinitesimalwerefrowneduponwellintothenineteenthcentury(suchnotionswereatoddswithAristotelianismandScholasticism),forexample, BernardBolzano(1781–1848)wasviewedasaprogressiveradical,thusunacceptable totheAustrianrulers(HouseofHapsburg-Lorraine)ofBohemiaandwasejectedfrom hisuniversityposition;itwas50yearsafterhisdeaththatKarlWeierstraussfoundin hiswritingsthefoundationoftheBolzano-Weirstrausstheorem,whichisessentialto thenotionofcontinuityandoneofthetheoreticalunderpinningsofcalculus.

Aswehavealreadystated,thisbookisnotahistoryofscience,butanattemptto placethedevelopmentofquantummechanicsinahistoricalframework.Asasociety, welikenicestoriesofhowideasweredevelopedandintroduced,butthetruthisalmost alwaysmorecomplicated.Asillyexampleisthesupposedinventionofthesandwichby JohnMontague,the4thEarlofSandwich.Itishardtobelievethatinthe30millennia thatbreadexistedinoneformoranother,nobodyeverplacedasliceofmeatbetween twopiecesofbread.Whatthegoodearlaccomplishedwastomaketheconsumption ofsuchacceptableinpolitecompany.

EventherelativelyrecentworkofPlanckhasbeenhotlyarguedamongscience historians.Inparticular,MartinKleinandThomasKuhnreallycouldnotagreeon Planck’spersonality;washecarefullyconservativeorarecklessrevolutionary?18 In fact,botharecorrecttosomedegree.Humansarecomplex,andoftenexpressdifferent andseeminglyincompatibleviewsdependingonthesituation.

1.3 Religionandscience

TheseparationbetweenscienceandreligionthatdevelopedduringthisperiodinEurope,thatis,fromtheeleventhtotheeighteenthcentury,andevenintothenineteenth century,isquiteremarkableandperhapsuniqueinthedevelopmentofhumansocietiesandcultures.BythetimeofNewton,theneedforreconciliationbetweenscientific observationwiththeBiblelargelydisappearedfromscientificliterature;whatisespeciallyremarkableisthatNewtonhadaliteralistinterpretationoftheBibleandwrote extensivelyonthesubject,however,heheldhisnotionsinclosesecretashisembracingmonotheismwasatoddswiththedoctrineofTrinityCollege,whereheheldhis facultyposition,astherewerepotentialseriousconsequencesforholdingsuchviews.

Howthisseparationcameaboutremainsamystery,althoughsomecreditFrancis Baconwithdevelopingtheconceptofempiricismandwiththedevelopmentofthe scientificmethod,however,hewasacontemporaryofGalileo,andtheywerecertainly awareofeachother’swork,soitappearsdifficulttoassigncredittoeither.Bacon wasespeciallyagainstAristotle’ssyllogismandrulesofinductiveenquiry;forGalileo, Aristotelianphysicswassimplyincompatiblewithreality.19 Otherspavedtheway;recallGiordanoBruno,whoperishedforhisscienceteachingsomeyearsbeforeGalileo’s predicament.Oftentheyoungaretheonesleadingthewaytowardrevolution.Compare thesituationofBrunoandGalileowiththeanti-VietnamwarmovementintheUSA, originallyfomentedprincipallybycollegestudentsuntilthefundamentalhypocrisy

18JeroenvanDongen, InEurope,loc.cit.

19W.Mays, ScientificMethodinGalileoandBacon, IndianPhilosophicalQuarterly 1,vol.3,217 (1974).

andpathologyofthewarwerefinallyrevealedthroughtheKentStateMassacreand thePentagonPapers.

Itisalsowrongtosaythatbeforee.g.,FrancisBacontherewasnosuchthing asthescientificmethod.Testingbytrialanderrorispartofthehumanpsycheand hasexistedsincethebeginningofconsciousthought.TheEgyptianscouldnothave constructedthepyramidsandotherstructuresiftheydidnothaveasystemtostudy nature,recordobservationsandmethods,andtransmitknowledgebetweengenerations.RecordssuggestthattheEgyptianculturestagnatedinthatnewideaswerenot allowedtodevelop,andthusthesocietycouldnotkeepupwithachangingenvironment,orthreatsfromexternalpoliticalforces.

TheseparationprobablywastheresultofScholasticismdominatingmonasticmedievalteaching,acriticalmethodofphilosophicalanalysispredicateduponAristotle, withaLatinCatholictheismbeingseparateandnotsubjectedtologicalargument andanalysisbutwhichwastobeacceptedasinfallibleandinvariantdoctrine.Such curriculadominatedteachingintheEuropeanmedievaluniversitiesfromabout1100 to1700.ThoseinterestedinscientificobservationhadtoskirtaroundAristotle,and indoingso,bypassedreligiousscrutinyanddebate.Theproblemsscientistsfacedare bestillustratedbyGalileo’sinteractionwiththeChurch;aslongasGalileocalledhis observationsandconclusions“theoretical”therewasnoconflictwiththeChurchor withitsdoctrine.Inthissense,theconflictwithGalileowasabattletodecidewho getstointerpretscripture,oralternatively,whohaspoliticalcontrol.

InthehistoryofWesternscience,allofthisledtosciencebeingdoneoutsideof religiousconsiderations,originallyclearlytoavoidconflictwiththeChurch,andalso withpolitics,whichisalmostthesamething.Laterthisseparationbecameamatter ofcourseandpartofournowacceptedscientificculture.

InnationsgovernedbyShariaLaw,scienceanddoctrineareexpectedtobe,and toremain,mutuallycompatible.DuringtheSovietera,Russianscientistshadtoat leastobliquelyacknowledgedialecticalmaterialism.Awell-knownanecdotetellsof BeriaapproachingKurchatov,theheadscientistoftheSovietatomicbombproject, regardingthefallaciesofEinstein’stheoryofrelativityasitisincompatiblewith thefundamentalnotionsofdialecticalmaterialism.Kurchatovrepliedthatwithout relativity,therecannotbeanatomicbomb.Apparently,theSovietphilosopherswere abletounifyawaytheincompatibilities.

Thescientist-as-atheistisamodernWesternnotionthathaditsbeginningsfrom dancingaroundtheChurchandAristotle,butwaslateramplifiedbyDarwin’stheoryofevolution.Darwindidnotsetatimelineforevolution,becauseherealized thatestimatesfortheageoftheSun(30millionofyears),duetotheenergyreleasedfromgravitationalinteractions,werenotsufficientlylong.Biologistsarguedthat thereneededtobeanotherenergysourcefortheSun,asdidgeologistswhoneeded moretimefortheirsedimentaryrockstoform,andtheywerecorrect.20 Theageof theearthduetoOldTestamentgenealogyandtheJewishcalendarisabout6,000 years,andthisisviewedbymanyasaconflictbetweenscienceandreligionthatarises fromtakingtheancientscripturesliterally,insteadofseriously.Itisworthnoting

20Bethe,H.A., EnergyProductionintheStars,NobelLecture,Dec.11,1967.https://www. nobelprize.org/uploads/2018/06/bethe-lecture.pdf.

Birthofthemodernatomictheoryofmatter 13

thatsomeKabbalistsfromSpaininthetwelfthtothirteenthcenturiescalculatedthe Earth’sageasintherangeof1millionto2.5billionyears.21

Thefundamentalincompatibilitybetweenscience,religion,andpoliticsisthat thebasisofscienceisfalsification.Asacademics,wetendtoviewconflictsbetween doctrineandscientificobservationasbeingduetosemanticissues,e.g.,thesixdays ofcreationrefersnottodays,butperhapstovasteons;themodernreaderhasno ideaofwhattheoriginalwriterhadinmind.Thisisaboutthebestthatcanbedone underthenotionoftheinfallibilityofancientscripture.Andwhatdowemeanby falsification?Inscience,thebreadthofthisnotiongoesfromtwomeasurementsofthe samequantitybeinginconsistentduetoexperimentalerrors,toanentiretheoretical constructbeingincorrect.

InresponsetoEinstein’sfamousremark,“GoddoesnotplaydicewiththeUniverse,”Bohrsaid“Einstein,stoptellingGodwhattodo.”Perhapsthisisalesson regardingthestrictinterpretationofhistoricalandphilosophicaldocuments.

1.4 Birthofthemodernatomictheoryofmatter

ThefirstmodernkinetictheoryofgasesisduetoJamesHermann,whoin1716deduced thatthepressureexertedbyagasisproportionaltotheaveragesquaredvelocityof thegasparticlestimesthenumberdensity.

In1729Eulerattemptedtomathematicallyexplainthebehaviorofgaseswith akinetictheorybasedonRobertBoyle’sgasdatafrom1662.Heassumedthatthe gasparticleswouldallmoveatthesamespeed.DanielBernoulliformulatedakinetic theoryofgases,withthenotionthatthevelocitieswouldbestatisticallydistributed, butdidnotspecifythedistribution,however,heanticipatedtheworkofJamesClerk Maxwellacenturylater.Bernoulli’sworkwasnotwidelyaccepted,inpartbecause

Fig.1.2 Bernoulli’ssketchofgasmoleculesholdingupaweightviatheforceexertedona piston,asstillseentodayinelementarythermodynamicsbooks.(PublicDomain)

conservationofenergyhadnotyetbeenestablishedanditwasnotobviousthatcollisionsbetweenparticlescouldbeperfectlyelastic.RogerBoskovich,aCroatianJesuit, carriedthescientificatomictheoryfurtherbysurmisingthatatomsareinfluencedby interatomicpotentialsmodeledonNewtoniangravity(1758)andprovidedthefirst insightthatcollisionsmightbeelastic.

ItisofinteresttonotethatBenjaminFranklinwasveryinterestedinoilfilms onwater,mostlybecauseoftheirbothanecdotalandactualeffectstoreducethe amplitudeofwind-drivenwaves.Franklinwouldcarryvialsofoilwithhimwhichhe wouldsometimespourontopondsorlakestostudytheeffectsoffilms.Henoticedthat ateaspoonofoliveoilwouldspreadtoanareaofaboutone-halfanacre.Hedidnot estimatethesizeofamoleculebasedonthis,butthisresultimpliesamolecularsize ofaboutananometer.22 Later,AgnesPockels(circa1885)wasthefirsttoestimate thesizeofoilmoleculesbasedonhermeasurementsoffilms.

Experimentswithgasesthatfirstbecamepossibleattheturnofthenineteenth centuryledJohnDalton(1766–1844)in1803toproposethebasisofmodernatomic theorybasedonthefollowingassumptions:

1. Matterismadeupofatomsthatareindivisibleandindestructible.

2. Allatomsofanelementareidentical.

3. Atomsofdifferentelementshavedifferentweightsanddifferentchemicalproperties.

4. Atomsofdifferentelementscombineinsimplewholenumberstoformcompounds.

5. Atomscannotbecreatedordestroyed.Whenacompounddecomposes,theatoms arerecoveredunchanged.

ThisissupplementedwithAvogadro’shypothesis,namedafterAmedeoAvogadro, who,in1812,statedthattwogivensamplesofaperfectgas,withthesamevolume andatthesametemperatureandpressure,containthesamenumberofmolecules.

Atthispoint,thedevelopmentofatomictheoryinphysicsdeviatesfromitsdevelopmentinchemistry.Withfewnotableexceptions,forchemistsatthistimeatoms werebecomingveryrealandbroughtnewunderstandingtochemicalreactionsand compounds.Physicistswereoflessuniformopinionastotherealityofatoms,astate thatpersisteduntilwellintothetwentiethcentury.

In1828thechemistFrederichW¨ohlersynthesizedureafrominorganiccompounds, anddisprovedthevitalisthypothesisthat“organic”compoundscouldbemadeonly bylivingthings.In1855,AugustKekul`eformulatedtheringstructureofBenzene. Whenisomersofdibromobenzenewerenotdiscovered,heproposedthatthedouble bondsintheringsoscillatebetweencarbonatompairs—doesthismarktheinvention ofquantummechanics?

In1869,theRussianchemistDmitriMendeleevdevelopedaframeworkthatwould becomethemodernperiodictable.Whilearrangingtheelementsaccordingtotheir atomicweight,hefoundthattheytendedtofallintocolumnargroupswithsimilar

22See,e.g.,JoostMertens, Oilontroubledwaters:BenjaminFranklinandthehonorofDutch Seamen,PhysicsToday 59,1,36(2006);https://doi.org/10.1063/1.2180175.Also,W.M.Klipstein,J.S.RadnichandS.K.Lamoreaux, Thermallyexcitedliquidsurfacewavesandtheirstudy throughthequasielasticscatteringoflight,AmericanJournalofPhysics64,758(1996);online: https://doi.org/10.1119/1.18174.

Fig.1.3 a.1,2-Dibromobenzenewithfixedbondsinthering,comparedtob.wherethebonds locationsaredelocalized,andconsideredtobethecorrectformulabecausethenon-existence oftwoisomersof1,2-Dibromobenzene.

properties,andheinsertedgapsforelementsthathesuspectedwerenotyetdiscovered. Basedonthepropertiesofagroup,Mendeleevpredictedthepropertiesofsomeundiscoveredelementsandgavethemnamessuchas“eka-aluminum”forananticipated elementwithpropertiessimilartoaluminum.Eka-aluminumwaslaterdiscoveredas gallium.However,discrepanciesremained;thepositionofcertainelements,suchas iodineandtellurium,couldnotbeexplaineduntilthediscoveryofisotopes.

Bythistime,chemicalindustrieswereburgeoning,particularlyinEnglandand theUnitedStates,andespeciallyinGermany.Theacceptanceofatomsandatomism bychemistswasprofitablyproductive;thewealthcondensedintheflasksoforganic chemistshelpedinspireandsecurethefundingofscienceingeneral.Assuch,therejectionofatomsbymanyphysicistsappearsasparticularlyintellectuallyschizophrenic.

1.5 Atomismandphysics

In1860,JamesClerkMaxwell,afterreadingapaperbyClausius23 thatintroducedthe notionofthemeanfreepath,beganhisstudiesofkinetictheoryanddeterminedthe velocityspectrumofspeedsinanidealizedgasbyuseofheuristicmethodsthatwere laterfullydevelopedbyBoltzmann.Atthetime,thenotionofavelocitydistribution wentagainsttheconventionaltheory,whichwasthatarangeofvelocitieswouldbe equalizedbymolecularcollisions.Maxwellalsoinvestigatedkinetictheoryingeneral, anddiscovered“thecuriousresult”thatviscosityisindependentofpressure,which wasunexpected.Hepublishedanestimateofthemeanfreepathbasedonairviscosity measurementthathadbeendonebyStokes.24 MaxwellandKatherineClerkMaxwell (hiswife)madethefirstreliablemeasurementsofthedependenceofgasviscosityon temperatureandpressure.Thesemeasurementswereperformedintheatticoftheir house,withthetemperaturecontrolledbyselectivestokingofthefireplace.Their results,reportedin1866,supportedthekinetictheoryofgasviscosityandprovided thefirstaccuratemeasurementoftheeffectivediameteroftheatomsormoleculesthe gascomprises,basedonLoschmidt’swork,citedbelow.

23Clausius,R.(1857),“UeberdieArtderBewegung,welchewirW¨armenennen,”Annalender Physik,100(3):353-379.Englishtranslation TheNatureoftheMotionwhichwecallHeat,PhilosophicalMagazine,Vol.14,pp.108-27(1857).

24Maxwell,J.C.(1860) Illustrationsofthedynamicaltheoryofgases.PartI.Onthemotionsand collisionsofperfectlyelasticspheres,PhilosophicalMagazine,4thseries,19:19-32.Maxwell,J.C. (1860) Illustrationsofthedynamicaltheoryofgases.PartII.Ontheprocessofdiffusionoftwoor morekindsofmovingparticlesamongoneanother,PhilosophicalMagazine,4thseries,20:21-37.

Inapaperhewrotein1866,Maxwellstatesthat“Loschmidthaddeducedfrom thedynamicaltheorythefollowingremarkableproportion:–Asthevolumeofagasis tothecombinedvolumeofallthemoleculescontainedinit,soisthemeanpathofa moleculetoone-eighthofthediameterofamolecule,”25 whichrelatesthemeanfree pathtothediameteras

8ϵλ = d (1.1)

where ϵ istheratioofacondensedvolumeofgastothevolumeofitsvapor, λ is themeanfreepathinthegas,and d isthemoleculardiameter.Thatis,tosay,by measuringthevolumeofacondensedgas,assumingtheatomsaretightlypacked, andbyuseoftheexpansiononevaporation,thediameterofthemoleculesthegas comprisescanbedetermined.AlthoughLoschmidtdidnotdeterminethenumberof moleculesinaunitvolume(Loschmidt’snumber)thiswasamathematicalstepthat Maxwellprovided,resultinginanumberdifferingbyafactoroftwofromthemodern value.

Shortlythereafter(1867),GeorgeJohnstoneStoneypublishedanestimateofthe numberofmoleculesinavolumeofgasthathehaddeterminedin1860,whichcan beusedtodeterminethenumberofmoleculesinamole.Hesubsequently(1874)inventedtheelectron(hehadvariousnames,electrolion,electrine,andsettledonelectron (Lorentz’spreference))asthechargedvalencyparticleofelectrolysis,anddetermined itselectricchargebydividingFaraday’sconstantbythenumberofmoleculesinamole. Althoughhisestimationoftheelectroncharge e is1/16itspresentvalue,thiserror canbetracedtoanerrorinhisindependentdeterminationofLoschmidt’snumber. Stoneyalsoinventednaturaldimensionlessphysicalunits(whatwenowcallPlanck units),andareinessencethesameasPlanckunitsuptofactorsof √α. 26

Duringthisperiod,blackbodyradiationwasbeingstudiedwithexperimental techniquesofincreasingprecision,andmeasurementbyTyndallledJosephStefanto, in1869,concludethepowerradiatedfromablackbodyscalesasitstemperaturetothe 4thpower.Boltzmannwasabletoderivethisrelationfromthermodynamicprinciples byconsideringanidealheatenginewithelectromagneticradiationastheworkinggas.

1.5.1 Atomismandanti-atomism:theemergenceofatomicphysics

Muchhasbeenwrittenabouttherejectionofatomictheoryinthelatenineteenthcenturywhichcontinuedwellintothetwentiethcentury.27 Theargumentsagainstatomismwerenotputforwardbyaunifiedfront,butbysomevalidandinvalidconcerns

25Loschmidt,J.(1865).“ZurGr¨ossederLuftmole¨ule”.SitzungsberichtederKaiserlichenAkademie derWissenschaftenWien.52(2):395-413.Englishtranslation:J.LoschmidtwithWilliamPorterfield andWalterKruse,trans.(October1995) Onthesizeoftheairmolecules,JournalofChemical Education,72(10):870-875.

26O’Hara,J.G.(1975). GeorgeJohnstoneStoney,F.R.S.,andtheConceptoftheElectron.Notes andRecordsoftheRoyalSocietyofLondon.29(2):265-276.

27Thissectionisbasedonanassemblageofhistoricalbooks,articles,andoriginalpublications, includingDavidLindley, Boltzmann’sAtom (TheFreePress,NewYork,2001);JohnT.Blackmore, ErnstMach,HisWork,Life,andInfluence (Univ.ofCaliforniaPress,Berkeley,1972);E.Broda, The IntellectualTriangle:Mach-Boltzmann-Planck-Einstein CERN81-10,July1981;and TheInteraction ofBoltzmannwithMach,Ostwald,andPlanck,andHisInfluenceonNernstandEinstein,16th InternationCongressontheHistoryofScience,Bucharest,1981;andPullman,loc.cit.

regardingthetheory,whichcomplicatesthediscussion.Forexample,ErnstMach— knownfortheMachnumberforspecifyingsupersonicspeeds,andMach’sprinciple thattheoriginofinertiawasduetothemassdistributionintheuniverse,andwas amongthemostrespectedexperimentalphysicistsinEuropeduringthisperiod—had astronglyheldviewthatatomsareuntestabletheoreticalconstructsthatcannotbe observedandthereforehavenoplaceinscience.Hispositivistviewsappeartohave beenformedinreactiontothedogmaticeducationalsystemoftheAustro-Hungarian empire,thenundertheruleoftheHapsburgs(butthatrulecametoadramaticconclusionwiththeassassinationofDukeFerdinandfollowedbyitssubsequentdissolution attheendofWWI).TheeducationaldogmathatwasinstilledintheGreatUnwashed (Mach’sfamilywasofmeagerexistence)oftheempirewasbasedonScholasticism combinedwithCatholicdoctrineandwasspoonedoutasanintellectualpablumto keepthemassescontentwithwhatMachrealizedwasafalseandarbitraryworldview.

Mach’sinsistencethatsciencebegroundedindirectobservationhadaprofoundly positiveeffectonscience,andinparticularMach’sviewsinfluencedEinsteininhis developmentofrelativity.Whenatomswerebroughtup,Machwasknowntoaskif anyonehaseverseenone.Machwasalignedinhisviewsinmanywayswiththegreat Nobel-prize-winningchemistWilhelmOstwald,whoalsorejectedthenotionofatoms, andputforwardanalternativetheory,energeticism,thatenergywastheultimate entityandmadenoparticularassumptionsaboutthenatureofmatter.Thegoalof energeticismwastounderstandallphysicalprocessesthroughtheconceptofpure energy.Ostwald’sview,beinganexpertoncatalysisandequilibriumphenomena,was thatatomismwastoosimpletobeofuseforsuchcomplicatedproblems,andhewas mostlycorrectonthispoint.OnehastheimpressionthatOstwaldwassearchingfor theconceptoftheGibbsfreeenergywhich,ofcourse,wasbeingdevelopedbyJ.W. GibbsintheUSandwasthenlargelyunknowninEurope.Asanaside,itwasonly later,afterdevelopinghisclassicalnotionsofthermodynamics,includinghisphase changelaws,thatGibbsembarkedonstudiesofstatisticalmechanicsusingatoms andmolecules–butBoltzmannsuspectedthathemighthavehadatomicmodelsinhis mindearlier.

Planckwasanti-atomism,apparentlybecausehedidnotthinkatomswereneeded orusefulforthedevelopmentofthermodynamics.Planckwasalsodeeplyreligious, andalthoughaProtestant,hewasawareofthesufferingofCatholicsunderBismarck’s Kulturkampf(1871–87),whichsubjectedtheRomanCatholicChurchtostatecontrols, andhaditsclimaxin1875,whencivilmarriagewasmadeobligatorythroughout Germany.AtomismwasstillinconflictwithCatholicteachings,andPlanckasaleading scientistandanassociateeditor(from1895–1907,andthenoneoftwoco-editorsuntil 1943)oftheleadingphysicsjournalofthetime, AnnalenderPhysik ,perhapsdidnot wanttopromoteanagendathatcouldbeviewedasanti-Catholicinthelightofthe recentsuffering.Aswehavementioned,Planckhasbeenviewedasbotharevolutionary (Klein)andaconservative(Kuhn).ItshouldbenotedthatPlanckpublishedEinstein’s papers,particularlyonrelativity,andwaswillingtopublishBoltzmann’sworks,but subjecttoaproviso.

Theatomistsandanti-atomistsofthisperiodwere,onapersonallevel,veryfriendly witheachotherdespitetheirheatedandsometimespublicdebateoveratomsand

kinetictheory,withtheexceptionofthechillyrelationshipbetweenBoltzmannand Planck. AnnalenderPhysik wouldnotpublishBoltzmann’spapersonkinetictheory unlesstheycontainedadisclaimerthatatomswereonlyconvenienttheoreticalconstructs.Ontheotherhand,PlanckandBoltzmannteamedupandprovidedaunified frontagainstenergeticism.

AclimacticeventintherelationshipbetweenPlanckandBoltzmannoccurred in1896,regardingBoltzmann’s H-theorem.Boltzmannformulatedthe H-theorem around1871.Briefly, H isdeterminedfromtheenergydistributionfunction f (E,t)dE ofmoleculesattime t.Thevalue f (E,t)dE isthenumberofmoleculesthathave kineticenergybetween E and E + dE H isdefinedas

Foranisolatedidealgas(withfixedtotalenergyandafixedtotalnumberofparticles), H isminimizedwhentheparticleshaveaMaxwell-Boltzmanndistribution.Ifthe particlesaredistributedsomeotherwaye.g.,allhavingthesamekineticenergy,the valueof H willbelarger.Boltzmannshowedthatwhencollisionsbetweenparticles occur,otherdistributionsareunstableandmoveirreversiblytowardtheminimum valueof H,thatis,towardtheMaxwell-Boltzmanndistribution.

Verysoonafterthe H-theoremwaspublished,Loschmidt,whowasBoltzmann’s colleagueatVienna,pointedoutthatitshouldnotbepossibletoproduceanirreversibleprocesswhentheunderlyingdynamicsaretime-symmetric.Alloneneedsto doisreversetimeforanentropy-increasingprocesstoshowthattherearestateswhere H increases(equivalently,entropydecreases)overtime—thisisLoschmidt’sparadox. The H-theoremisbasedontheassumptionof“molecularchaos,”thatinkinetictheoryparticlemotionisbeconsideredindependentanduncorrelated—soitwouldnot generallybepossibletoreversethemotion.BoltzmannconcededtoLoschmidtthat suchstateswerepossible,whilenotingthattheyaresorareandunusualthatinpracticehavenosignificantcontribution.BecauseBoltzmannintroduced H asaproxyfor entropy,the H-theoremwasthefirstattempttousestatisticalmechanics(although thenamestatisticalmechanicswasonlylaterinventedbyGibbs)toderivethesecondlawofthermodynamicsfromclassicalreversibledynamics,however,becauseof theassumptionofmolecularchaos,thereasoningwascircular.Boltzmannworkedto addressthisobjection,leadingtohisentropyformulaof1877,

S = k log W, (1.3)

whichrelatesentropy S tothenumberofpossibleconfigurations W ofthemolecules ofanidealgas.

Asecondcriticismofthe H-theoremwasraisedin1896byErnstZermelo,28 a student/assistantofPlanck.Briefly,Zermoloarguedthataccordingtoanaccepted theoremofHenriPoincar´e,anensembleofparticlesmustaccessalltheaccessible

28Zermelo,E.,“Onalawofdynamicsandthemechanicaltheoryofheat,”Ann.d.Phys.57,485, 1896,(datedDec.1895)EnglishtranslationinBrush,S.G., KineticTheory:SelectedReadingsin Physics,Vol.2,Pergamon,1966

configurations(thatis,sampleallregionsofphasespaceasallowedbyenergyand momentumconservation),andmustthereforereturntothestartingpositioninfinitely manytimes,toanydegreeofprecision.Thismeansthatanisolatedsystemofparticles mustatsometimereturntoastateofdecreasedentropy.Thiscametobeknownas therecurrenceobjection(asnamedbyPaulandTatyanaEhrenfest).Zermolotook thepositionthatthelawofentropyincreaseisabsolutelycorrect(theclassicalthermodynamicistPlanckwouldagreewiththis)andthereforethekinetictheoryofgases mustbeinvalid.AfterZermolo’spublicationin AnnalenderPhysik,Boltzmannwrote abrilliantandastonishinglysarcasticresponse,statingthatZermolowouldlikelysuspectthatthedicewereloadedbecausehefailedtorollasixonethousandtimesin succession,becausethelikelihoodofsuchaneventisnotexactlyzero.29 Boltzmann insistedthathisresponsebepublishedin AnnalenderPhysik withoutchange,andit wasaccepted.30 (Zermeloleftphysicsandmadeseminalcontributionstomathematics, inparticulartosettheory.)

InmorerecentsubsequentstudiesofPoincar´e’sworks,ithasbeendiscoveredthat infact,hedidnotsupporttheuseofhistheoreminthismanner;inessence,itis impossibletomaintainarealphysicalsystemtothedegreeofmathematicalprecision requiredforthetheorem’sstrictvalidity.31 Forexample,itiseasytoimaginethat thewallsofthecontainerholdingagascanpermanentlyabsorbthatgas,orrelease animpuritygastothesample;electricchargesonthewallofthecontainercanmove around,disappear,orbegeneratedbynaturalradioactivityorcosmicrays.Secondly, andperhapsmoreimportantly,Boltzmannhadalreadyaccededthepointthatentropy candecreaseforashorttime,buttheprobabilityissosmallthatinthelongterm, thesecondlawisvalid.Wenowunderstandsuchprocessesthroughthegeneralized fluctuation-dissipationtheorem.

Aswewilldiscussatlengthinanupcomingchapter,in1900PlanckusedBoltzmann’sstatisticalmethods,togetherwiththeintroductionofquantizationthroughhis constant h,toderivetheblackbodyspectrum.

Einstein’s1905 annusmirabilis papersincludesoneonBrownianmotionanddescribestheconsequencesofBoltzmann’sfluctuations.32 Thiswasalsotheyearthat EinsteincompletedhisPh.D.dissertationattheUniversityofZurichentitled“ANew DeterminationofMolecularDimensions.”Hisdissertation,ascant24pages,outlines thetheoryandmeasurementsofdiffusionratesofsugarsolutions;hedetermineda valueofAvogadro’snumbertowithinafactorofthreeofthecurrentlyacceptedvalue. Thisledtohismosthighlycitedpublication.ExperimentalworkbyPerrin,published in1909,confirmedallofEinstein’sBrownianmotionpredictions,andprovidedadeterminationofAvogadro’snumberinconflictwithEinstein’s(recallthatAvogadro’s numberhadalreadybeenobtainedbyLoschmidtand,withlessaccuracy,byStoney 40yearsearlier,andthesizesofmoleculeshadbeendeterminedbyAgnesPockelsin

29K.Mendelssohn, TheWorldofWaltherNernst:TheRiseandFallofGermanScience1864-1941 (London:TheMacmillanPressLtd.,1973)p.115.

30Boltzmann,L., Onenergetics,Ann.d.Phys.,58,595,1896

31Brush,S.G., Poincar´eandCosmicEvolution,PhysicsToday 33,no.3,42(1980).

32Einstein,A., Onthemovementofsmallparticlessuspendedinastationaryliquidrequiredby themolecularkinetictheoryofheat,Ann.d.Phys, 17,549,1905(received11May1905).