32 CHAPTER2. ORIGINSOFQUANTUMMECHANICS
Thisfinalexpressionnotonlymatchestheformoftheempiricalequationforspectral
lines,butanexplicitevaluationof theconstantfactorshowsthatitisnumerically
equaltotheRydbergconstant
RH=me^4
2 chh ̄^2(2.40)
InadditiontogettingtheHydrogenspectraright,Bohr’squantizationcondition
(2.31)alsoimplies, ifitdoesn’texactly explain,the stability ofthe atom. For if
Bohr’sconditioniscorrect, thenthereisaminimumradiusforthe electronorbit,
namely
r 1 =̄h^2
me^2= 5. 29 × 10 −^11 m (2.41)whichisknownasthe”Bohrradius”. Theelectroncanneitherspiralnorjumpto
aradiussmallerthanthis. Therefore,Bohr’stheoryactuallypredictsthesizeofthe
Hydrogenatom, whichagreeswithempiricalestimatesbasedon,e.g.,the vander
Waalsequationofstate.^1
Afterthesuccessof Bohr’smodel ofthe atom, attemptsweremade to gener-
alizehisworkto ellipticalorbits,to includerelativisticeffects, andto applyitto
multi-electron atoms. The outcomeof agreatdealof effort was this: sometimes
thequantizationconditionworked,andsometimesitdidn’t.Nobodyunderstoodthe
rangeofvalidityofthequantizationcondition,norwhyclassicalmechanicsshouldbe
subjecttosuchadrasticrestriction.Agoodnewideawasneeded.
2.6 De Broglie Waves
Wehavealreadyseenthatforphotons
Ephoton=hf and pphoton=h
λ(2.42)
LouisdeBroglie’sverygreatcontributiontophysics,intheyear1924,wasthesug-
gestionthattheserelationsalsoholdtrueforelectrons,i.e.
Eelectron=hf and pelectron=h
λ(2.43)
Inthecaseoflight,theconceptualdifficultyiswiththeleft-handsideoftheseequa-
tions:howcanlightwaveshaveparticleproperties?Inthecaseofelectrons,itisthe
right-handsideoftheequationswhichisdifficulttounderstand:whatdoesonemean
bythefrequencyandwavelengthofanelectron?
DeBroglie hadthe idea that the wave which is somehow associated withan
electronwouldhavetobeastandingwavealongtheelectronorbitsofthehydrogen
(^1) Ofcourse,thereisahiddenassumptionthatn>0,i.e.therearenoorbitsofradiusr=0.