QMGreensite_merged

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)−

3

2

] (14.82)