370 CHAPTER24. THEFEYNMANPATHINTEGRAL
SolvingforA,andrecallingη=x−y,wefinallyhavetofirstorderin!,
G!(x,y)=
( m
2 iπ! ̄h
) 1 / 2
exp
[
i
!
̄h
{
1
2
m
(x−y
!
) 2
−V(x)
}]
(24.17)
TheexpressionaboveforG!(x,y)isnotexact;thereareO(!^2 )corrections. Ifwe
insertthisexpressionintoeq. (24.8),theresultforGT(x,y)willnotbeexacteither:
theproductofT/!termswillgiveanoverallerroroforder!. However,wecanget
theexactresultforGT(x,y)bytakingtheN→∞(!=NT →0)limit
GT(x,y)
= lim
N→∞
∫ (N∏− 1
i=1
dxn
)
G!(x,xN− 1 )G!(xN− 1 ,xN− 2 )...G!(x 1 ,y)
= lim
N→∞
∫ (N∏− 1
i=1
dxn
)(
m
2 iπ! ̄h
)N/ 2
exp
[
i
̄h
∑N
n=1
!
(
1
2
m
(xn−xn− 1 )^2
!^2
−V(xn)
)]
(24.18)
wherewehavedefinedx 0 ≡yandxN ≡x. Wenowintroducethenotationforthe
integraloverallpossiblepaths
∫
Dx(t)≡ lim
N→∞
∫ (N∏− 1
i=1
dxi
)(
m
2 iπ! ̄h
)N− 1 / 2
(24.19)
andnotethatinthe!→ 0 limit
lim
N→∞
∑N
n=1
!
(
1
2
m
(xn−xn− 1 )^2
!^2
−V(xn)
)
=
∫t+T
t
dt
( 1
2
mx ̇^2 −V(x)
)
= S[x(t)] (24.20)
whereS[x(t)]istheactionfunctional definedback inLecture1. Atlastwe have
arrivedattheFeynmanPathIntegral
GT(x,y)=
∫
Dx(t)eiS[x(t)]/ ̄h (24.21)
Inwords,thisequationsaysthattheamplitudeforaparticletopropagatefrompoint
yattimettopointxattimet+T isgivenbytheintegraloverallpossiblepaths
runningbetweenthosetwopoints, witheachpathweightedbythe amplitudeeiS,
whereSistheactionofthegivenpath. Ofcourse,theprecisemeaningof
∫
Dx(t),
thesumoverpaths,mustbegivenintermsofalimit,namely,thelimitofamultiple
integral,asthenumberofintegralsgoestoinfinity.Butrecallthattheprecisemeaning
ofanordinaryintegralisalsointermsofalimit: thelimitofasumoververymany
points,asthenumberofpointstendstoinfinity.