TheSpectrumoftheHydrogenAtom1.PurposeInthisexperimentyouwillobservethediscretelightspectrumfromagasdischargelamp.Youwillfindthatthespectrumconsistsofacollectionofsharpmonochromaticlines.Usingadiffractiongratingspectrometer,youwillbeabletomeasurethewavelengthoftheemittedlighttobetterthanonepartinathousand.Thereforeitiscrucialtomakeallcalculationstofivesignificantfigures.2．IntroductionSpectrumoftheHydrogenAtomWhengasesaresubjectedtolargeappliedvoltages,theytendtoundergodielectricbreakdownandemitbrightlight.Ifoneexaminesthelightfromsuchagasdischargewithaspectrometer1,onefindsthatthelightconsistsmainlyofafewbrightlinesofpurecoloronagenerallydarkbackground.Forexample,excitedhydrogengaswillemitfourvisiblelinesduringbreakdown:red,green,blue,andviolet.Thisphenomenoncontrastssharplywiththecontinuousspectrumofcolorsobservedinlightfromthesunoranincandescentbulb.Thepropertyoflightthatweobserveascolorisactuallyrelatedtoitswavelength?.Inthelatenineteenthcentury,J.J.Ballmerdiscoveredanequationthatcorrectlypredictsthewavelengthsofthevisiblelinesinthehydrogenspectrum:Inthisexpression,ni=3,4,5….,andRistheso-calledRydbergconstant,Otherspectrallinesbeyondthevisiblewavelengthscanbeobservedinhydrogenandothergases;inhydrogen,thesewavelengthsaregivenbythegeneralformulaWherenfandniareintegers.Balmerderivedhisformulaforthehydrogenspectrumempirically;atthetime,hisresultdidnothaveafundamentalexplanationgroundedinclassicalphysics.Infact,classicalelectromagnetismpredictsthathydrogenatomsshouldradiatecontinuously,and,evenworse,thattheyshouldbehighlyunstable.Neitherpredictionisobserved,suggestingdeepflawsintheclassicaldescription.Theoriginofthelinespectrumbecameamajorproblemthatwasnotresolveduntil1913,whenNielsBohrsuggestedanalternativetheoryforatoms.Heproposedthatthevalenceelectroncouldonlyexistincertainenergystates,andcouldonly“jump”betweenthesediscretestatesdiscontinuously.Whena“jump”occurred,theatomwouldemitlighttoconserveenergy.Sincetheenergieswerediscrete,theemittedlightshouldalwayshavethesamefixedsetofcolors(wavelengths).Bohrwasabletoderiveaformulafortheenergyofthehydrogenatom’squantumenergylevelsintermsofthemassmandchargeeoftheelectron,thepermittivityoffreespace20,Planck’sconstanth,andanintegern:ThereforetheenergyemittedbytheatomduringtransitionfromaninitiallevelnitoafinallevelnfisTherelationshipbetweentheenergyEandthewavelength?ofthelightisduetoPlanck:Hence,onecanderiveanexpressionforthewavelengthoflightemittedduringanatomictransition:ResolvingaSpectrumwithaDiffractionGratingInordertodecomposeaspectrum,onecanuseaso-calledtransmissiongrating.Thegratingisnothingmorethanaslabofmaterialwithalargenumberoftinyparallelslits.Transmissiongratingsareoftenmadeoffinelymachinedglassorevencrystals.Thespacingdbetweentheslitsiscalledthe“l(fā)atticeconstant”ofthegrating.Consideracollimated(parallel)lightbeamincidentonagratingfromtheleft,asshowninFig.Eachslitwilldiffractthebeam,andactinturnasanewsourceofwaves.Thewavesallbegininphaseattheslits,butdependingontheanglewithwhichtheyleavethegrating(calledthediffractionangle..Adiffractiongratingwithslitseparationd.Thelocationofthediffractionmaximumontheviewingscreenisdependentonthewavelength?oftheincidentlight.Notethatthehorizontalscaleinthisdiagramhasbeenhighlydistorted,theytraveldifferentpathstotheviewingscreenandmaybeoutofphasebythatpoint.Fromthefigure,thedifferenceinthepathlengthfortwoadjacentslitsisInorderforthegratingtoformamaximumatsomepointontheviewingscreen,thewavesmustbeinphasethere.Thiswilloccurifthepathlengthdifferenceisanintegralmultipleofthewavelength:Therefore,wefindthatmaximawilloccurwheneverWheremiscalledtheorderofthemaxima.Equation(7.2)tellsusthatthemaximaforagiven?willoccuratdifferentangleswithrespecttothelaserbeamdirection.Thisishowthetransmissiongratingdecomposesaspectrumintoitsindividualwavelengths.Whenlightshinesthroughthegrating,particularcolorswillappearatseverallowerandhigheranglesrelativetotheforwarddirectionTheordernumbermreferstotherelativepositionofamaximumwithrespectto3．EXPERIMENTTheequipmentusedinthisexperiment,calledadiffractiongratingspectrometer,isdepictedinFig.Thespectrometercontainsthreemajorcomponents:acollimatortube,arotatingtable,andatelescope.Schematicofthespectrometeryouwillusetoobservethehydrogenspectrum.Thecollimatortubetakeslightfromexcitedhydrogengas,providedherebyanarclamp,andusesalenstocollimatethebeam—thatis,makethelightraysfromthesourceparallel.Figure:Schematicofthespectrometeryouwillusetoobservethehydrogenspectrum.Whentheparallelraysexitthetube,theytraveltothetransmissiongrating,whichismountedinthecenteroftherotatingspectrometerbase.Thelightdiffractedbythegratingmaybeviewedthroughaneyepieceattheendofthetelescopetube.Thetelescopeisabletoswivelwithrespecttothegrating,allowingyoutosweepthroughasetofanglesμandobservetheangledependenceofthevariousspectrallines.youcanthenusethisangletodeterminethewavelengthofeachlineyouobserve.Todeterminetheangles,thespectrometerbasecontainsagraduatedcircleattachedtothetelescope.Asyouturnthetelescope,youcanreadofftheangleusingtheangularscalescoredintothecircle.HowtoReadtheAngularScaleTheangularscaleinthebaseisnotastandardruler,butaVernierruler.Withastandardangularscale,youwouldprobablybeabletoresolveanglesdowntothenearestdegreeorhalfdegree.Inthisexperiment,wewouldlikeconsiderablymoreprecisioninourmeasurements.Therefore,thespectrometercontainsascalethatallowsuserstomeasureangleswithgreataccuracy,tothenearestarcminute:Thedevicecanachievethisprecisionbyhavingtwoscalesratherthanone.Thefirstisastandarddegreescalerunningfrom0±to360±,andthesecondisaVernierscalerunningfrom00to300.Tounderstandhowthesetwoscalesworktogether,consultFig.7.3asyoureadthefollowingprocedure.1.BeginbyfindingthezeromarkerontheVernierscale.SamplereadingfromthedegreeandVernierscalesinthespectrometerbase.2.ScandownfromtheVernierscaletothenextlineonthedegreescale,asreadfromtheleft.Thislineistheangleμ,accuratetothenearesthalfdegree(300).3.Readingfromlefttoright,findthelineontheVernierscalethatbestlinesupwithalineonthedegreescale.Thisvaluemarks,inarcminutes,yourpositionbetweentwoticksonthedegreescale.4.AddthefirstreadingfromthedegreescaletothesecondreadingfromtheVernierscale.Youhavenowmeasuredμtothenearestarcminute.Again,refertoFig.7.3asyoureadthisprocedure.Inthefigure,the0markontheVernierscaleisbetween50.5±(50±300)and51.0±(51±00).Hence,thebasemeasurementis50±300,sincewearereadingfromtheleft.OntheVernierscale,themark13bestmatchesamarkonthedegreescale.Therefore,withinthehalfdegreeinterval,weaddanadditional130.Theresultingmeasurementis4.ProcedureThefirstpartoftheexperimentwillbasicallyincludetheproceduretosetuptheequipment.Thisshouldbedonewithasmuchcareaspossible.Onlythenwillyoubeabletomeasurethewavelengthonthelimitofourapparatus.Ifyoudon’tsetupthespectrometercorrectlyyouwillgetsystematicerrors,skewingyourresults.AdjustingtheSpectrometer?Takethegratingoutoftheholderandclosethegreenknob.?Rotatetheyellowknobsuchthattheslitisabouthalfopen.?Inthestraight-throughposition(180±),lookthroughtheeyepieceandturnthepurplefocusingringuntilyouseeasharpimageoftheslit.?Loosentheredknobandmovethetelescopetubeuntilthecrosshairsareinthemiddleoftheslit.Tightenredknob.?OpenthegreenknobandturnthetabletopsuchthatthezeromarkfromtheVernierscalewiththemagnifyingglassislinedwitheither180±or360±fromtheouterscale.AlwaysuseonlythisVernierscaleanddon’tswitchtotheotheroneinbetween.Closethegreenknobanddon’topenitagainfortherestoftheexperiment!?Nowyoucanfineadjusttherelativepositionoftheinnerandouterscalebyturningtheblueknob.LineupthezeroontheVernierscaleand180±/360±andonthedegreescaleasascarefullyaspossible.NOTE:Forsomeofthespectrometers,thereisasmallmarktotheleftofthezeromarkontheVernierscale.Makesurethatyoulineupthezeromark,andnottheextramark.?Putthegratingintheholdersuchthatitisperpendiculartothetelescopetube-collimatortubeline.Closethewhitescrewtolockthegrating.ObtainingtheGratingLatticeConstantAfteradjustingthespectrometeryouwillmeasuretheyellowlineofaHeliumdischargelamp.Sinceweknowthatthewavelengthofthislightis?=5.8756×10?7m,wecandeterminethelatticeconstantdofthegratingquiteaccurately.Eventhoughthegratinghas600linesmm?1writtenonit,thisisonlyanapproximation.Wewanttoknowthelatticeconstanttofivesignificantdigitsandnotjustthree,?SwitchontheHeliumlampandlinethespectrometerupsuchthatyoucanseetheslitwell-illuminatedbythelampasyoulookthroughthetelescope.?Puttheblackcardboardoverthefrontendofyourcollimatortubeandcoverthespectrometerwithablackpieceofclothtoblocklightfromyoursurroundings(butbecarefulnottoblockthetelescopewiththecloth).?NOTE:thisstepandthenextshouldbeperformedinthedark;thereforeswitchoffthelightandusetheflashlightprovided(theyaresupposedtobequitedim)toreadthescale.?Opentheredknobandmovethetelescopetubetotheleftuntilthecrosshairsareinthecenteroftheyellowline.Youshouldfirstseeafewblueandgreenlines,thentheisolatedyellowline,andthenredlines.Theyellowlineshouldbesomewherearound20±degrees.?Notedowntheangleindegreesandminuteswhereyouseethefirstorderoftheyellowline.Dothesameontherightsideandaveragethesetwonumbers.?Usetheaverageandplugitintothegratingequation(m=1)todeterminethelatticeconstantdtoatleast5significantfigures.?Howmanylinesmm?1