QMGreensite_merged

15.2. THEHELIUMATOM 245

and

B=

∫

d^3 x 1 d^3 x 2 [φ∗nlm(x 1 )φ 100 (x 1 )]

e^2

r 12

[φnlm(x 2 )φ∗ 100 (x 2 )] (15.55)

TermAisassociatedwithasimple,intuitivepicture.Ifwethinkoftheelectron
instate(100)ashavingachargedensityproportionaltoitsprobabilitydensity

ρ 100 (x)=−e|φ 100 (x)|^2 (15.56)

andlikewisefortheelectroninstate(nlm)

ρnlm(x)=−e|φnlm(x)|^2 (15.57)

thenAissimplytheCouloumbinteractionenergybetweenthosetwochargedistri-
butions,i.e.

A=

∫

d^3 x 1 d^3 x 2

ρ 100 (x 1 )ρnlm(x 2 )

|%x 1 −%x 2 |

(15.58)

Ontheotherhand,thereisnoclassicalpicturefortheoriginoftermB. Interms
oftheFeynmandiagramsofrelativisticquantummechanics(see,e.g. Bjorkenand
Drell,vol. I),onecanvisualizetheinteractionbetweenelectronsasoccuringviathe
transferofaphotonbetweenthem. TheCoulombinteractionAcomesaboutwhen
theelectroninthe(100)stateemitsaphotonatpointx 1 ,whichisabsorbedbythe
electroninthe(nlm)stateatpointx 2. Theexchangeinteractioncomesaboutwhen
theelectroninthe(100)emitsaphotonatpointx 1 andjumpsintothe(nlm)state,
whiletheelectronatpointx 2 absorbsaphoton,anddropsintothe(100)state. In
otherwords, theparticles notonlyexchangea photon,but indoingso they also
exchangestates. ThediagramsareshowninFig. [15.2]andIhopetheygiveahint
ofwhatisgoingon,but,tobehonest,anexplanationofwhatthesediagramsreally
meanwillhavetowaitforacourseinquantumfieldtheory.

BoththequantitiesAandBarepositivenumbers,sowecanconcludethat

E ofParahelium(s=0) > E ofOrthohelium(s=1) (15.59)

Theintuitivereasonforthisfactissomethinglikethis:Thes= 1 stateissymmetric
inspins,itisthereforeantisymmetricinspace. Inspatiallyantisymmetricstates,the
electronstendtobefurtherapart,onaverage,thanforthecorrespondingspatially
symmetricstates. Forexample,inanantisymmetricstatethewavefunctionvanishes
at x 1 = x 2. Beingfurther apart meansthat the repulsive potential issomewhat
smaller,andhencetheenergyofthestateissmaller.Thisisanexampleofarulein
atomicphysicsknownasHund’sFirstRule(seenextsection),whichsaysthat,other
thingsbeingequal,thestateofhighestspinisthestateoflowestenergy.