2.5. THEBOHRATOM 29
ofaphysicalstate. Ofcourse,itcouldbethattherelation(2.28)issimplyapracti-
callimitonmeasurement;aparticlemighthaveadefinitepositionandmomentum
despiteourinabilitytomeasurethosequantities simultaneously. Butthedifficulty
couldalsobe muchmoreprofound: ifaphysicalstateissimplythemathematical
representationof theoutcomeof an accuratemeasurementprocess (a viewwhich
wasadvocatedinFig. [1.5]of Lecture1)andifaccuratenumbers(x,p)arenever
anoutcomeofanymeasurement,thenperhapswearemistakeninthinking thata
physicalstatecorresponds todefinitevaluesof(x,p). Inotherwords,theoriginof
theuncertaintycouldbeduetotryingtofitasquarepeg(thetruephysicalstate,
whateverthatmaybe)intoaroundhole(theset(x,p)).Attheveryleast,ifxand
pcannotbemeasuredsimultaneously, thenthereiscertainlynoexperimentalproof
thattheclassicalstateisthetruephysicalstate.Thisviewisobviouslyaveryradical
option;forthemomentweonlyraiseitasapossibility,andturntothemysteryof
thestabilityoftheatom.
2.5 The Bohr Atom
Atomshaveradiiontheorderof 10 −^10 m,andhavemassesontheorderof 10 −^26 kg.
In1911,ErnestRutherfordstudiedtheinternalstructureofatomsbybombarding
gold foilwithα-particlesfromradioactive Cesium. By studyingthe scatteringof
theα-particlesbythegoldatoms(atopicwewillturntoattheendofthecourse),
Rutherfordfoundthatalmostallthemassoftheatomisconcentratedinapositively
chargednucleus,ofradiusontheorderof 10 −^15 m,i.e. 100,000timessmallerthan
theradiusoftheatomitself.Thenucleusissurroundedbymuchlighter,negatively
chargedelectrons,whichcollectivelyaccountforlessthan 1 /2000thofthetotalmass
oftheatom.Atomicstructurewaspicturedasanalogoustothesolarsystem,withthe
nucleusplayingtheroleofthesun,orbitedbymuchlighterelectrons(the”planets”),
boundintheirorbitsbyCoulombattractiontothenucleus.
However,orbitalmotionisaformofaccelleratedmotion,andelectronsarecharged
particles. Accordingtoelectromagnetictheory,anaccelleratingchargedparticlera-
diateselectromagneticwaves. Aselectronsmoveintheirorbits,theyshouldbecon-
stantlyradiatingenergyintheformofelectromagneticwaves,andastheelectrons
loseenergy,theyshouldspiralintothenucleus;aprocesswhichwouldtakeonlya
fraction(about 10 −^10 )ofasecond.Bythisreasoning,atomsshouldbeaboutthesize
ofnuclei,butinfacttheyareroughly100,000timeslarger.Sowhataccountsforthe
stabilityoftheelectronorbits;whydon’telectronsspiralintothenucleus?
Anothermysteryofatomicstructurewastheexistenceofspectrallines. Ifagas
isplacedinadischargetube,withasufficientlylargevoltagedifferencemaintained
atopposite endsofthetube,the gasglows. But,unlikethermalradiation(which
occursduetorandomcollisionsamongatoms)thelightemittedfromthedischarge
tubeisnotspreaddiffuselyoverabroadrangeoffrequencies,butiscomposedinstead
ofdiscrete,verydefinitewavelengths. Whensuchlight(inthevisiblespectrum)is