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.