QMGreensite_merged

246 CHAPTER15. IDENTICALPARTICLES

15.3 The Periodic Table

Onelearnsinabeginningchemistrycoursethatchemicalreactionsamongtheele-
mentsinvolveelectronshoppingfromoneatomtoanother,eitherintheformofan
ionicbond(oneatomcapturestheelectron ofanother,as inNaCl),or acovalent
bond(oneormoreelectronsaresharedbetweenatoms,asinH 2 ). But,whyshould
electronshopbackandforthbetweenatoms? Whydon’ttheysimplysitinthelowest
possibleenergystateofeachatom,andrefuseto budgeunless seriouslymolested?
Thekeytothatquestion,andtounderstandingthecuriousperiodicityofchemical
propertiesamongtheelements,isthePauliExclusionPrinciple.
Inprinciple, everything that canbe knownaboutanatomiscontained inthe
wavefunctionsofitsenergyeigenstates. Forpurposesofchemistryitistheatomic
groundstate,andparticularlytheionizationenergy,whichareimportant(theion-
izationenergyistheminimumenergyrequiredtoremoveasingleelectronfromthe
atom). Againinprinciple,theprocedureforfindingthegroundstateofanatomof
atomicnumberZisverystraightforward: First,writedownSchrodinger’sequation
fortheZ-electronstate
[Z
∑
n=1

{

−

̄h^2

2 m

∇^2 n−

Ze^2

rn

}

+

∑

n>m

e^2

|%xn−%xm|

]

Ψ=EΨ (15.60)

Second,findthesolution,antisymmetricwithrespecttoparticleinterchange,which
hasthelowestenergy. Theantisymmetryrestrictionisveryimportant;thelowest
energyeigenstatesarenotantisymmetricingeneral.
Unfortunately,thesecondpartofthissuggestedprocedureisveryhardcarryout
inpractice. Theproblemisthat,dueto theinteractionsamongelectrons,thereis
nowayofseparatingvariablesinthemany-electronSchrodingerequation. What’s
more,theelectroninteractiontermistoolargetobetreatedasasmallperturbation,
forwhichthereexistsystematicapproximationmethods.
Butinthecasethatthenumberofelectronsisfairlylarge,thetheoreticalphysicist
hasanothertrickuphisor hersleave. Incondensedmatter,nuclear,andparticle
physicsthistrickisknownvariouslyas the”meanfield,”or”self-consistentfield,”
or”independentparticle,”or”randomphase”approximation. Inatomicphysicsthe
methodisknownas the”Hartreeapproximation.”Roughlytheideaisthis: when
eachparticle interactswithmanyotherparticles,the potentialduetothosemany
otherparticlescanbetreated, onaverage, asthoughitwereaclassicalpotential,
inwhicheachparticlemovesindependently. Ifweknewthisaveragepotential,we
couldcalculatethewavefunctionofeachparticlebysolvingaone-particleSchrodinger
equation; and, knowingthe wavefunction of eachparticle, we could calculatethe
averagepotential. Thereforeonecalculatesthewavefunctions”self-consisently,”i.e.
findingsolutions whichleadto apotential whichgivethosesolutions. Thats the
generalidea. Theweaknessoftheapproximationisthatitignoresallcorrellations
inthe positions of the various particles. Still, when eachparticle interacts with