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30 CHAPTER2. ORIGINSOFQUANTUMMECHANICS

passedthroughaprismordiffractiongrating,itissplitintoarelativelysmallnumber
ofbrightlines,knownas”spectrallines”.Itisnothardtounderstandthatanatom,
stimulatedbyanelectriccurrent,mightemitlight;butwhyshoulditonlyemitthat
lightatcertaindefinitefrequencies?Empirically,spectoscopistshadfoundthateach
spectrallineofhydrogencouldbeassociatedwithtwointegers,mandn,suchthat
thewavelengthλofthespectrallinewasgivenby

1

λ

=RH

(

1

n^2

−

1

m^2

)

(2.29)

where
RH= 109677 .576(12) cm−^1 (2.30)

isknownas”Rydberg’sconstant”forHydrogen. ThisformulaworksforHydrogen,
modifiedformsworkforcertainothergases,but,atthetime,nobodycouldexplain
itssuccess.
In1913,NielsBohrdiscoveredthattheformulaforthespectrallinesofHydrogen
could be derived fromonecrucial assumption about electron orbits: the angular
momentumofanelectroncanonlycomeinmultiplesofPlanck’sconstantdividedby
2 π.Inparticular,forcircularorbits,

L=pr=n

h

2 π

(2.31)

wherepistheelectronmomentum,andristheradiusofthe(assumed)circularorbit.
Theexpressionh/ 2 πcomesupsoofteninquantumphysicsthatitisgivenitsown
symbol

̄h≡

h

2 π

(2.32)

pronounced”h-bar”.
Bohrwas ledto the assumption(2.31)bythe following reasoning: Acharged
particle rotatinginacircularorbitwill emitelectromagneticwaves whichpossess
bothenergyandangularmomentum.Suppose∆Eistheenergyofradiationemitted
ina certaintimeinterval ∆t. Then according toMaxwell’s electrodynamics, the
electromagneticradiationalsocontainsacertainamountofangularmomentum∆L,
relatedto∆Ebytheformula
∆E= 2 πf∆L (2.33)

wheref isthefrequencyoftheradiation. Now,accordingtoEinstein,thesmallest
amountofenergyemittedisthatofonephoton,∆E=hf.Thenthesmallestamount
ofangularmomentumthatcouldhavebeenemittedwouldbe

∆L=

h

2 π

(2.34)

Inthecaseof thehydrogenatom, thismeansthatthe angularmomentumof the
electron, upon emittingaphoton, mustchangeby exactlythis amount. Thisled