QMGreensite_merged

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.