248 CHAPTER15. IDENTICALPARTICLES
commuteswithL^2 ,Lz,andthesolutionscanbelabeledbysamequantumnumbers
astheHydrogenatomwavefunctions,i.e. (nlm). Second,thepotentialisnolonger
exactly 1 /r.Whenagivenelectronisclosetothenucleus,itseesapotential−Ze^2 /r,
duetothefullchargeofthenuclues. However,whentheelectronisneartheedge
oftheatom,itwillonlyseeapotentialofstrength−e^2 /r,becausetheotherZ− 1
electronsareinterposed,andscreenthenuclearcharge. Asaresult,energiesareno
longerindependentofthelquantumnumber.Infact,foragivenn,thelargervalues
oflhaveahigherenergy(i.e. aremoreweaklybound).Thereasonisthatthehigher
angularmomentumstatesarefurther,ontheaverage,fromthenucleus.Thereforean
electroninsuchstatesseesasmallerattractivepotential,andthereforehasasmaller
bindingenergy.^3
Withoutsolvinganything,aground-statewavefunctionintheHartreeapproxima-
tionisdenotedbylistingtheenergylevelsthattheZelectronsarein.Thenotation
foreachlevelis
nlN
n= 1 , 2 , 3 ,... principalquantumno.
l=s,p,d,... spectroscopicsymbolforl= 0 , 1 , 2 ,..,n− 1
N= 1 , 2 , 3 ,... numberofelectronsinthislevel ≤ 2(2l+1)(15.67)
ThePauliprincipletellsusthatthemaximumvalueforN,atanygivenl,is
Nmax(l)=2(2l+1) (15.68)Eachelectronhastobeinadifferentstate,andthereare 2 l+ 1 possiblevaluesofm
foreachl.Thereforetherecanbe 2 l+ 1 electronswithspinup,and 2 l+ 1 electrons
withspindown;atotalof2(2l+1)electronsinall.Whenthereareexactlythismany
electronsinagivenlevel,thelevelissaidtobe”filled.”
Thelastthingneeded,inordertolisttheelectronconfigurationofeachatomic
groundstate,istheactualorderoftheenergies. Inthehydrogenatom,theenergies
depend onlyonthenquantumnumber,but inmulti-electronatoms,asjustmen-
tioned,thebindingenergydecreases,aslincreases.Thislistingcanonlybeobtained
byactuallycarryingouttheHartreeapproximation,andisasfollows:
1 s, 2 s, 2 p, 3 s, 3 p,[4s, 3 d], 4 p,[5s, 4 d], 5 p,[6s, 4 f, 5 d], 6 p,... (15.69)wherethelevelsinbracketsareverycloseinenergy.Notethattherearecases,suchas
4 sand 3 d,wherealevelwithsmallernandlargerlhasahigherenergythanthelevel
withhighernandsmallerl.Again,thisisduetothel-dependenceoftheenergy:the
(^3) YoumightwonderwhyasphericallysymmetricHamiltonianwitha 1 /rpotentialhasagreater
degeneracyofeigenstatesthanHamiltonianswithothercentralpotentialsV(r).Theansweristhat
Hamiltonianswith 1 /rpotentialsactuallyhaveasymmetrywhichincludes,butisgreaterthan,
sphericalsymmetry.Knowledgeofthissymmetryallowsone,inthecaseof 1 /rpotentials,todefine
raisingandloweringoperatorsthatchangelaswellasm.Afulldiscussionofthispoint,however,
isbeyondthescopeofthiscourse.