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