QMGreensite_merged

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)