232 CHAPTER14. THEADDITIONOFANGULARMOMENTUM
isdeterminedbytheorthogonalityandnormalizationconditions
<Φ 10 |Φ 00 > = 0
<Φ 00 |Φ 00 > = 1 (14.71)
Fromtheorthogonalitycondition
0 =
1
√
2
[<χ^1 +χ^2 −|+<χ^1 −χ^2 +|][a|χ^1 +χ^2 −>+b|χ^1 −χ^2 +>]
=
1
√
2
(a+b) (14.72)
whichtellsusthatb=−a.Substitutingintothenormalizationcondition
1 = a^2 [<χ^1 +χ^2 −|−<χ^1 −χ^2 +|][|χ^1 +χ^2 −>−|χ^1 −χ^2 +>]
= 2 a^2 ⇐ a=
1
√
2
(14.73)
wefindthatthespinsingletstateis
Φ 00 =
1
√
2
[χ^1 +χ^2 −−χ^1 −χ^2 +] (14.74)
Thetripletandsingletstates,andprocedureforfindingthem,isshowninFig.[14.4].
Oneofthemanyapplicationsofadditionofspin^12 angularmomentumistothe
so-called”hyperfinesplitting”ofthehydrogenatomgroundstate;thishasimportant
consequences for (of all things!) radio astronomy. Now the ground stateof the
hydrogenatomhaszeroorbitalangularmomentum,sothereisnoL·Ssplittingof
theenergylevel.However,theproton,liketheelectron,isaspin^12 particle,and,like
theelectron,ithasacertainmagneticmoment
μp=
gpe
2 mp
Sp (14.75)
where,experimentally,gp= 5 .59. Fortheelectron, thecorresponding”g-factor”is
veryclosetoge=2,andtheelectronmagneticmomentis
μe=−
e
me
Se (14.76)
Now,accordingtoclassical electrodynamics,themagneticfield dueto amagnetic
dipoleatposition%risgivenby
B%= μ^0
4 πr^3
[3(μ%·%er)%er−%μ]+
2 μ 0
3
%μδ^3 (r) (14.77)