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376 CHAPTER24. THEFEYNMANPATHINTEGRAL

andtendtocanceleachotherout.Theexceptionisforpathsinthevicinityofapath
wherethephaseS/ ̄hisstationary. For pathsinthatvicinity,thecontributionsto
thefunctionalintegralhavenearlythesamephase,andhencesumupcontructively.
Butthepathwherethephaseisstationaryisthepathxcl(t′)suchthat
(
δS
δx(t)

)

x=xcl

= 0 (24.51)

Aswehaveseen,thisisjustthecondition thatthepathxcl(t)isasolution ofthe
classical equations of motion. Therefore, for S >> ̄h, the path integralis domi-
natedbypathsintheimmediatevicinityoftheclassicaltrajectoryxcl(t).Inthelimit
̄h→0,onlytheclassicalpath(andpathsinfinitesmallyclosetotheclassicalpath)
contributes. The”semiclassical” orWKBapproximationtotheFeynmanpathin-
tegralis,infact, theapproximationof evaluatingtheintegralintheneighborhood
singleconfigurationxcl(t),i.e.

GT(x,y) =

∫

Dx(t)eiS[x(t)]/ ̄h

≈ prefactor×eiS[xcl(t)]/ ̄h (24.52)

wherethe prefactorisa numerical termwhichcomes fromintegratingoversmall
variationsδxaroundxcl(t).

Problem - Forsome problems, the WKB approximation works extremely well,
evenifS[x,t]isnotsolargecomparedto ̄h. Applythisapproximationtofind the
propagatorofafreeparticle,andcompareittotheexactresult.

24.3 Operators from Path Integrals

Givenatrajectory,x(t),onecanalwaysdefineamomentum,mx ̇(t).Thenanatural
waytodefineamomentumoperatoractingonawavefunctionattimetf isinterms
ofthepath-integral

p ̃ψ(xf,tf)≡

∫

dy

∫

Dx(t)mx ̇(tf)eiS/ ̄hψ(y,t 0 ) (24.53)

wherethepathsrunfrom(x 0 ,t 0 )to(xf,tf),andψ(x,t 0 )isthewavefunctionatany
earliertimet 0 <tf. Letustaket=tfandt 0 =t−!,andthengotothelimit!→0.
Inthatcase

p ̃ψ(x,t) ≡ lim

!→ 0

∫

dym

(x−y)

!

G!(x,y)ψ(y,t−!)

= lim

!→ 0

∫

dym

(x−y)

!

( m

2 πi! ̄h

) 1 / 2

exp

[

i

m

2! ̄h

(x−y)^2 −i

!

h ̄

V(x)

]

ψ(y,t−!)

= lim

!→ 0

∫

dym

(x−y)

!

(

m

2 πi! ̄h

) 1 / 2

exp

[

i

m

2! ̄h

(x−y)^2

]

ψ(y,t−!) (24.54)