14.1. THEGENERALMETHOD 233
SothecontributiontotheHamiltonianfortheelectron,duetotheinteractionofthe
electronmagneticdipole momentwiththemagneticfieldduetotheprotondipole
momentis
H′=
μ 0 gpe^2
8 πmpme
3(Sp·er)(Se·er)−Sp·Se
r^3
+
μ 0 gpe^2
3 mpme
Sp·Seδ^3 (r) (14.78)
TheSpoperatorsinH′actonthespindegreesoffreedomoftheproton.Therefore
wehavetoenlarge,alittlebit,ourexpressionforthehyrogenwavefunction,toinclude
theseextraquantizeddegreesoffreedom. Thegroundstatewavefunction(s),which
areeigenstatesofH 0 ,Se^2 ,Sez,Sp^2 ,Spz,arethefourstates
Rn 0 (r)Y 00 (θ,φ)χe±χp± (14.79)
whereχe±referstothespinstateoftheelectron,andχp±referstothespinstateof
theproton. Wecanalso,followingtheprocedureabove,reorganizethesefourstates
intofourotherstates,whichareeigenstatesofH 0 ,S^2 ,Sz,Se^2 ,S^2 p,whereS%=S%e+S%p,
i.e.
ψstr=1ipl,setz = Rn 0 (r)Y 00 (θ,φ)
χe+χp+ (sz=1)
√^1
2 (χ
e
+χ
p
−+χ
e
−χ
p
+) (sz=0)
χe−χp− (sz=−1)
ψssing=0,sletz=0 = Rn 0 (r)Y 00 (θ,φ)
1
√
2
(χe+χp−−χe−χp+) (14.80)
Then,onceagainmakinguseofaresult(tobeshown)fromfirstorderperturbation
theory
∆E 1 trsiplz et = <ψsstriplz et|H′|ψtrssiplzet>
∆Esinglet = <ψsinglet|H′|ψsinglet> (14.81)
Theexpectationvaluesaboveinvolveanintegrationoveranglesθ,φ. TheY 00 spher-
icalharmonic hasno angulardependence, butthefirstterminH′doeshavesuch
dependence. Whentheintegraloversolidangleiscarriedout,thefirstterminH′
averagestozero.Theintegrationoverr,θ,φforthesecondtermisveryeasilycarried
out,sinceitinvolvesadeltafunction,andwefind
∆E =
μ 0 gpe^2
3 memp
<ψ|δ^3 (r)Sp·Se|ψ>
=
μ 0 gpe^2
3 memp
<ψ|δ^3 (r)
1
2
(S^2 −Sp^2 −Se^2 )|ψ>
=
μ 0 gpe^2
3 memp
1
2
̄h^2 [s(s+1)−
3
4
−
3
4
)<ψ|δ^3 (r)|ψ>
=
μ 0 gpe^2 ̄h^2
3 memp
|ψ(r=0)|^2
1
2
[s(s+1)−