A Very Brief Intro to Quantum Mechanical History

8/3/2019 A Very Brief Intro to Quantum Mechanical History

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ABreifIntroductiontoQuantumMechan icalHistory –ByCiscoJimenez

MaxPlanckdidnotatfirstsubscribetothebeliefthatatomsandmoleculeswere

realentities1.Hewasinfactareservedscientistwhoavoidedwhathetermed“alldoubtful

adventures”.However,hisworkonblackbodyradiationleadtohiseventualconversionto

theatomistdoctrine.

Oneofthemostheatedconflictsbetweenthetwoopposingschoolsofthoughtatthe

timewastheinterpretationofthesecondlawofthermodynamics.Thedebateinitially

manifesteditselfthroughanargumentbetweenLudwigBoltzmannandPlanck’sassistant,

ErnstZemelo,whoattemptedtoshowthatBoltzmann’sinterpretationcontradicteditself.

Boltzmannpointedoutthatthesecondlawwasoftenmisinterpreted,andthatthetrue

decidingfactorwasthatofstatistics.Statisticallyspeaking,itisonlymuchmorelikely to

findasystemfavoringahigherstateofentropy,andthatwhilenot probable,isitstill

possibletoobserveaneventindirectcontradictiontothesecondlawofthermodynamics.

ThisinterpretationofthesecondlawofthermoreallybotheredPlanck,andheset

outtorefuteBoltzmannusingthephysicsinvolvedwithblackbodyradiation2.Muchwork

hadalreadybeendoneatthattimeonblackbodyradiation,includingthatbyGustav

Kirchhoff,whofoundthattheratioofradiationemittedfromanexperimentalradiatorto

thatabsorbedbythesameradiatorhadnothingtodowithsize,shape,normaterial,but

insteadseemedtoimplysomesortoffundamentalpropertyoftheradiationitself.In

addition,WillhelmWeindevisedhisdisplacementlawin1896,establishingamathematical

1Referredtothenastheatomistdoctrine2Atthetime,referredtoascavityradiation



relationshipbetweenradiationfrequencyandexperimentalcavitytemperature.Wein’s

displacementlawwassupportedbyexperimentaldatagatheredbyFreidrichPaschen,but

waslatershowntofallapartatlowerfrequenciesofradiationinexperimentscarriedout

byOttoLummerandErnstPringsheim.

ThisallsetthestageforavisittoPlanckbyphysicistHeinrichRubens,whoshared

withPlancksomeofhislatestexperimentaldata.TheresultshewasshowninspiredPlanck

andhesettowork.UsingWein’slaw,hewasabletoadaptittofitafunctionthatdescribed

thebehavioroftheseradiatorsperfectly,atanyfrequencyofradiation.Bythetimehewas

done,hisequationhadtwofundamentalconstants,relatingtoradiationfrequencyandto

temperature.Theconstantrelatingtoradiationfrequency,orh,eventuallybecameknown

asPlanck’sconstant,anditremainsoneofthemostimportantconstantsusedinquantum

mechanics.

However,eventhoughPlanckwasabletofindaradiationlawthatdescribedall

availabledata,hedidnothaveatheoreticalbasisforthelaw.Planckhadbeeninitially

lookingattheproblemintermsofoscillatorsbeingacteduponbyanelectromagneticfield.

Planckbelievedthattheoscillatorswouldbehaveinwaythatwouldenabletheenergyof

thesystemtobedistributedevenlythroughoutthesystem,whichhadallowedhimto

deriveanexpressionforasingleoscillatorinagivencavitywall,acteduponbyagivenEM

field.However,hisequationhadbeenfoundbymatchingexperimentaldata,andheneeded

todriveasimilarexpressionfrom‘scratch’foranymeaningfulunderstandingofthesource

ofthemathematicsandphysics.UnfortunatelyforPlanck,inordertoachievethis,hefound

himselfdriftingmathematicallytowardthestatisticalmodelsofhismainrival,Boltzmann.





energyasacontinuousfunction,andPlanckhadchangedthisinadvertentlywhenhehad

introducedstatisticstohismethods.Einsteinsawthatthiswasveryimportant.

Strongevidenceexistedatthetimesupportingtheparticulatenatureofmatter,and

thewave-likenatureofradiation.Einsteinalsobelievedthatlight couldpossessparticulate

propertiessimilartothoseofmatter,andinanideahedubbedthe“heuristicprinciple”,he

describedforthefirsttimehisincredibleideaofwave-particleduality.Thiswouldbethe

firsttimeinover200yearsthatlightwouldbethoughtofinthismanner,itwasareturnto

the‘corpuscular’theoryoflightintroducedbyIsaacNewton.Einsteinpostulatedthatlight

couldonlybeemittedandabsorbedasdiscreteenergypackets.However,unlikeNewton,

hewasnotabandoningthewave-likenatureoflight.Ratherhewasinsistingthatthetwo

propertiesoflightbeingexperimentallyobservedweresimplydifferentversionsofthe

samething:wavephenomenaweresimplythetime-averaged,collectivemotionsofmany

light-quantaratherthantheinstantaneoussnapshotofanindividuallightquanta.Thiswas

anincrediblyradicalideaatthetime,andwasnotreadilyacceptedbyEinstein’speers,but

wassomethingthatwasdemonstratedelegantlybytheexperimentaldataobservedinthe

photoelectriceffect.

Thephotoelectriceffectwasaphenomenonthathadbeenunexplainableforsome

time,butwassolvedbyEinsteinbyapplyinghisideasonlightquantization.Experimental

datahadshownthatshininglightonmetalsurfacescausedelectronstobeejectedfromthe

metal.However,contrarytowhatwasexpectedbasedonthewavemodeloflight,the

energyofejectedelectronsdidnotincreasewithanincreaseinintensityoflight.Rather,it

increasedwithanincreaseinlightfrequency.Einsteincorrectlyassumedthatthiswasdue

tothequantizedpacketsofenergyinlight.Increasingintensitydidnotincreasethesizeof



thephotons,butratherincreasedtheirabundance,whichresultedinmoreejected

electrons,butelectronsejectedattheessentiallythesameenergylevel.Einsteintookthisa

stepfurther,relatingPlanck’sequationtoelectromagneticradiationinawaythatmadeit

universallyrelevanttolightinanysituation,notjustappropriatetoblackbodyradiation,

asPlanckhadbelieved.

ThenextmainplayerinourquantumstorywasDanishphysicistNielsBohr.English

physicistJosephJohnThompsonhadshownexperimentallytheexistenceofelectronsin

1897,implyingthatatomscouldnolongerbethoughtofasindivisibleparticles.Further

workbyErnestRutherfordin1909-1911showedtheexistenceofsomethingelsewithin

theatom4,whichledRutherfordtopostulatethatatomsweremostlyemptyspace,with

electronsmovingabout,similartoplanetsorbitingoursun.ThecoreofRutherford’sidea

hasremainedmostlyunchanged,eventothisday.InthemovieMindwalk 5,thetrio

discussedthisideaindetail,givingmanyanalogiestohelpviewerscomprehendthereality

ofthingsvibrating,rotatingandtranslatingallaroundusandevenwithinus.Theplanetary

modelremainsacompellingwaytohelpvisualizetheatomicscale,becauseifonemakesa

directcomparisonbetweenanatomviewedinthissimplifiedmannerandoursolarsystem,

theouterelectronofahydrogenatomwouldbewherePlutois(≈5.9billionkm),andthe

nucleuswouldbeatenththesizeofoursun(≈69,550km).Thatmeanstheouterelectron

wouldbeabout84,892nucleardiametersawayfromthenucleus.

Ofcourse,in1909,whenRutherfordwasperforminghisexperiments,verylittleof

thiswasfullyknownorunderstood.Infact,themaincompetitortoRutherfordatthetime

4UnknownbyRutherfordatthetime,hehadexperimentallyshowntheexistenceoftheatomicnucleus.51990,Starring:LivUllmann,SamWaterson,JohnHeard.Directedby:BerntAmadeusCapra.



wasThompson.Thompson’smodelinvolvedstationary electronsembeddedinpositively

chargedmedium,likeraisinsinplumpudding.Indeed,thisiswhatThompsonismost

widelyrememberedfortoday:his“PlumPudding”modeloftheatom.Themainreason

Thompson,Rutherford,andrestofthegeneralphysicscommunitywerereluctantto

initiallyaccepttheideaofelectronicmotionwithintheatom,wastheproblemofcharged

particlesmovingaboutinanelectromagneticfield.Theverymotionoftheelectrons,

accordingtoclassicalphysics,wouldcausethemtoloseenergy,andbecomelessandless

likelytoresistthepullofthenucleus.Bythelawsofclassicalphysics,atomswithmoving

electronswouldcollapseinonthemselves,effectivelyendingallexistence.

NielsBohrhadinitiallybeenworkingforThompson,butbecamequicklyfrustrated,

asthetwodidnotappeartohavegottenalongsowell.In1911,Bohrwenttoworkfor

Rutherford,withtheapprovalofThompson.Bohreventuallybegantousetheideasof

EinsteinandPlanckinhisapproachtotheproblemofclassicalphysics’failuretoexplain

theatomicmodel.Classicalmechanicswasfailing,soBohrpulledanEinstein—thecurrent

lawswereincapableofdealingwithhisproblem,sohemadehisownrulesbyworking

backwardsfromhisassumption.Bohrassumedthattheorbitsinatomswouldbeordered

inafashionconnectedtotheorderingofenergyinPlanck’scavityoscillators,sohebegan

withthisassumption.Orbitalswouldbespecificallyordered inintegermultiplesofhν.It

wasslowgoing,withmanyproblemsplaguingBohr’smodel,butin1913HansHansen,a

colleagueofBohr’s,sharedwithhimdatafromBalmerandRydbergthatprovidedthefinal

pieceofthepuzzleBohrwouldneed.In1885,JohannJakobBalmer,aSwissmathematician,

hadfoundaformulaforasimplemathematicalpatternhenoticedinhydrogenemission



lines.SwedishphysicistJonhannesRydberghadgeneralizedtheformulain1888,but

neitherBalmernorRydberghadseenthemeaningbehindtheirdatafitting.

However,toBohr,thishydrogenemissionpatternpresentedexactlytheevidence

forwhichhehadbeensearching.Bohrbelievedthemathematicalpatternsinthehydrogen

spectralineswerethedirectresultofthediscreteorderingofatomicorbitals.Infact,ittied

directlyinwithPlanck’sideas,ashehadbelieved.Anytimeanelectron“jumped”fromone

orbitaltoanother,lowerorbital,itreleasedacorresponding fixed amountofradiation,

whichgeneratedtheprecisespectradocumentedbysomany.ThisfitsowellwithBohr’s

ideas,thathewasabletoderiveresultsinfullagreementwiththeBalmerandPaschen

seriespurelytheoretically,aswellaspredictseveralasyetunseenemissionseriesinthe

UVandIRregionsoftheEMspectrum.Inaddition,Bohrwasalsoabletocalculateand

demonstratethesourceoftheRydbergconstantusingonlyfundamentalphysical

constants,includingPlanck’sconstant,andwasalsoabletoexplaindataobtainedby

EdwardCharlesPickering,whichinvolvedhalf-integernumbersnotallowedinBohr’s

theory.Bohrre-wrotethenumbersaswholeintegersandshowedthattheywerefrom

spectrageneratedbyionizedheliumratherthanhydrogen.Anoteworthyconsequenceof

Bohr’stheorywasthatalongwithfixedandstableelectronorbitalscamefixedangular

momentum,whichwasfoundtobeh/2π.

DespitetheapparentsuccessofBohr’stheory,therewerestillsomeproblemsand

unexplainedphenomena.Howdidthetransitionfromoneorbittoanotherhappen?Itcould

onlybepossibleifdoneinstantaneously.Ifthetransitionweremadeslowly,atremendous

amountofenergywouldbeinvolved,astheresultoftheclassicalmechanicsdescribe

above.Inaddition,ifthistransitionweretobeassumedtobeinstantaneous,howwould



theelectronmakingthejumpemittheexactlycorrectamountofradiationtoendupinthe

correctdestinationorbital?Justwheredidthequantumnumberscomefrom?Thenextbig

pieceofthequantumpuzzlewastobefoundbyPrinceLouisDeBroglie,ofFrance.

Bohrhadshownthatelectronsdoindeedmoveaboutwithinanatom,andthatthey

dosoonlyinspecific,stableorbits,withfixedangularmomentum.However,itwouldbe

DeBroglie,in1923,whowoulduseideasfromEinstein’stheoriestomakethenext

quantumleap,andshowfromwhencethequantumnumbersoriginate.DeBrogliedecided

thatEinstein’stheoriesofrelativityandlight-quantumcouldbeandshouldbeextendedto

applytoallmatter,especiallyelectrons.Thiswasreallythefirsttimeanelectron,aformof

matter,hadbeenthoughtofasbothawaveandaparticle.DeBrogliewrote:“Thatwhich

makesanelectronanatom6ofenergyisnotitssmallvolumethatitoccupiesinspace,I

repeat:itoccupiesallspace,butthefactthatitisundividable,thatitconstitutesaunit.”

Thiswasanincrediblethoughtinandofitself,butDeBrogliewasfarfromdone.He

connectedtheequationrelatingenergyandfrequency(ε=hν)withtheequationrelating

energyandmass(ε=mc 2),andreducedthistohisnowfamousrelationship:λ=h/p,where p

representsthemomentumofaparticle.DeBrogliepostulatedthatonlytheextremelysmall

sizeofPlanck’sconstantiswhatlimitswhathedescribedas“matterwaves”or“phase

waves”frombeingvisibleinlarger,moreclassicalsystems.

ThiswasanideathatIbelieveistiedintoBohr’scorrespondenceprinciple.Ibelieve

thattheswitchingpointbetweenclassicalandquantummechanicsrevolvessomehow

aroundPlanck’sconstant.Isaythisonlybecauseitrearsitselfeverytimeabridging

betweenclassicalandquantumtheoryismade.Eitherway,DeBrogliealsosaidthatthese

6Inthiscase,IbelieveDeBrogliewasusingthewordatomwithrespecttoitsoriginalGreekmeaning:atomos( ἄτομος) ,orindivisible.



matterwavescouldnotbesimilartoclassicalwavesintermsofenergyandvelocity,forif

suchwerethecase,theywouldexceedthespeedoflight.DeBrogliebelievedthemtobe

representativeofamatterdistributioninspace,anditwashisloveofmusicthateventually

helpedhimtomakehisnextmajorbreakthrough.Musicalnotesandpitchesareexplained

byclassicalphysicsasoriginatingfromstandingwaves.Thesearesinusoidalpatterns

producedbyvibrationsofa fixed lengthofstringorafixedsizeresonator,likeahuman

voicebox.Thekeyisthatthewavesareonlyfreetovibrateonapre-definedregion,and

theendsofthewavesare“tieddown”andmustalwaysbezero.HereiswhereDeBroglie

usedmusicastheinspirationforsomethingincredible:thequantumnumbersfromBohr’s

atomicmodelweremathematicallyinherentiftheelectronsweretreatedasstanding

wavesconfinedtospecificcircularorbits.Ifhisideaweretrue,itwouldrequireelectron

matterwavestohaveharmonicresonancethatfitpreciselywithinthecircumferenceofa

givenorbit.Theelectronmatterwavecouldonly existifthebeginningandending

amplitudesforonecircularorbitwereidentical:ithadtoendexactly whereitstarted7,orit

wouldcancelitselfoutdestructively.DeBrogliedescribedthisastheresonancecondition,

anditbeautifullyexplainedthestabilityofBohr’squantumnumberedelectronorbitals.

Whilethisideawasincredible,itwasalsoveryradical,andmostofthescientific

communityatthetimewasunwillingtoacceptit.DeBrogliehadmadeatheoretical

connection,buthadnotbeenabletomathematicallyderiveanyproofofhisideas.In

addition,DeBroglie’sideastilldidnotexplainthecausalityorthenatureofquantumjumps.

WhatDeBrogliedidsaywasthatthecurrentquantummechanicalmodelwasnolonger

sufficienttoexplaintheinnerworkingsoftheatom.Somethingnewwasneeded.Einstein,

7Thisissimilartotheendsofastandingwavebeing“tieddown”atbothends.



whowasalsobecomingincreasinglyconcernedwiththeinabilityofquantumtheoryto

explainanever-increasingamountofexperimentaldata,sharedDeBroglie’sfeelings.There

werethoseinthescientificcommunitywhofelttheoppositehowever,andcontinuedtotry

anduseahybridclassical-quantumtheory,despiteitsincreasingfailure.Mostnotablyof

thesewasNielsBohr,whoalongwithHendrikKramersandJohnC.Slaterproposeda

theoryindirectoppositiontoEinstein’slight-quantumtheory.TheBKS8theory,proposed

inearly1924soughttoreplaceEinstein’slight-quantumtheorywithoneinvolving

statisticalaveraging.Thetheoryproveddisastroushowever,andwasdisprovedmy

experimentaldataby1925.Classical-quantumhybridtheorywasjustnotadequate,andin

thatsameyear,itwasWernerHeisenbergwhowouldrealizethismostclearly,andattempt

tobuildanewsetofrulesfromthegroundup.

Heisenbergrealizedthatahybridclassical-quantummechanicalmodelwas

insufficienttoexplaintheinternalworkingsofatoms,andhedecidedthatinordertomake

anyprogress,itmustbediscardedcompletely.Heisenberg’sfirstdecisionwastoavoid

usinganythingthatwasnotdirectlyobservable.Thislefthimwithfewchoices,andhe

chosetoworkwiththeintensitiesandfrequenciesoflineemissionspectra,theobservable

propertiesoriginatingfromquantumjumpsbetweenatomicorbitals.UsingFourierinfinite

series,heconstructedanelaboratetableofharmonicmatterwaveswitheachtermofthe

serieshavinganamplitudeandafrequencythatcorrespondedtooneofthequantum

jumpsofanelectronbetweenorbitals.Heisenbergbecameexcitedbecausewhenhehad

finished,hiscalculationsforenergywereallworkingfordifferentorbits,andwerein

agreementwithconservationofenergylaws.Therewasoneproblemwithhiscalculations,

8ThisisshortforBohr-Kramers-Slater



however:hewasonlyabletocalculateonephysicalpropertyatatimeforcertain

situations,meaningattimes xy≠ yx .Thistroubledhim,buthesharedhisworkwithMax

Born,whomhehadbeenworkingwithpreviouslyinGöttingen.Bornrecognizedwhat

Heisenberghadunknowinglydone:hehadbeenusingaruleformultiplicationofmatrices,

whichexplainedtheoddsituationsinwhichpropertiesdidnotcommute.

BorntookHeisenberg’sworkandbegantoworkonitinearnestwithhisstudent,

PascualJordan.Theybegantoapplyrulesofmatrixmultiplicationandhadsomevery

interestingresults.Energymatriceswerealldiagonalized,andfurtherthematrix

calculationsforpositionandmomentumdidnotcommute.9Heisenberg,Born,andJordan

publishedapaperontheirfindingsinNovember1925.Thepapereventuallyfounditsway

intothehandsofPAMDirac,whosawtheimplicationsofthesenewmatrixmechanics.

Usingthisnewapproach,Diracwasabletoindependentlycalculatetheposition-

momentumfactor9,provethatenergyconservationwasupheld,andderivedfrompure

theorythequantumnumberscorrespondingtostableatomicorbitalsthatBohrhadgotten

fromexperimentaldatabutneverderived.Thiswasanincredibleaccomplishment.Even

thoughthisnewtheoryofmatrixmechanicswasinitsinfancy,itshowedgreatpromise.Of

courseaswitheveryotherradicalnewidea,mostofthescientificcommunityremained

unimpressed.

9Thedifferencebetweenthemomentummatrixp,andthepositionmatrixqwasfoundtobeequalto–ih/2π.



Author’sNote:

3,623Words

62Adverbs

0.0171[adverb:word]Ideallyitshouldbesomewherearoundh.

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A Very Brief Intro to Quantum Mechanical History