368 CHAPTER24. THEFEYNMANPATHINTEGRAL
Toseehowasumoverpathscomesin,letssupposethatweknewthepropagator,
foranyHamiltonian,whenthetimedifferenceT=!istiny,i.e.
ψ(x,t+!)=
∫
dyG!(x,y)ψ(y,t) (24.5)
Inthat case, we could getthe propagatorforlarge timedifferences by usingthe
propagatorsuccessively,i.e.
ψ(x,t+ 2 !) =
∫
dydx 1 G!(x,x 1 )G!(x 1 ,y)ψ(y,t)
ψ(x,t+ 3 !) =
∫
dydx 2 dx 1 G!(x,x 2 )G!(x 2 ,x 1 )G!(x 1 ,y)ψ(y,t) (24.6)
andbyinduction
ψ(x,t+N!)=
∫
dy
∫ (N∏− 1
i=1
dxn
)
G!(x,xN− 1 )G!(xN− 1 ,xN− 2 )...G!(x 1 ,y)ψ(y,t)
(24.7)
Sothepropagator,forthetimedifferenceT=N!,hastheform
GT(x,y)=
∫ (N∏− 1
i=1
dxn
)
G!(x,xN− 1 )G!(xN− 1 ,xN− 2 )...G!(x 1 ,y) (24.8)
Nowasetofpoints:
y attime t
x 1 attime t+!
x 2 attime t+ 2!
... ........ ....
xN− 1 attime t+(N−1)!
x attime t+T
(24.9)
connectedbystraightlines,asshowninFig.24.1,isapathbetweenpoint(y,t)and
point(x,t+T).Theintegralineq. (24.8)isthereforeasumoverallpathsbetween
thesetwopoints, whichconsist ofN straight-linesegmentsof equaltime-duration
!=T/N. Clearly,asN→∞,anycontinuouspathbetween(y,t)and(x,t+T)can
beapproximatedtoarbitraryprecision.
Itsclear,then,thatapropagatorcanberepresentedbyasumoverpaths. The
problemistofindanexpressionforG!(x,y),whichdoesn’tinvolvefirstsolvingthe
eigenvalueequation(24.2)andpluggingtheresultinto(24.3).Fortunately,if!isvery
small,wecandeduceG!(x,y)fromthetime-dependentSchrodingerequation. First,
approximatethetime-derivativewithafinitedifference
i ̄h
ψ(x,t+!)−ψ(x,t)
!
≈Hψ(x,t) (24.10)