24.1. THEFREEPARTICLEPROPAGATOR 371

24.1 The Free Particle Propagator

Onemightwonderifthepath-integralformulationisuseful: Isitreallypossibleto
carryoutaninfinitemultipleintegral?Theanswerisyes,iftheintegrandisgaussian.
Anygaussianintegralcanbedone,nomatterwhetheritisasingleintegral,amultiple
integral,orapathintegral.
Asanexample,wewillworkoutthepropagatorforafreeparticlebytwomethods:
first,byusingeq. (24.3),and,second,fromtheFeynmanpathintegral. Forafree
particle

φp(x) =

1

√

2 π ̄h

eipx/ ̄h

Ep =

p^2

2 m

(24.22)

soeq. (24.3)becomes

GT(x,y)=

∫ dp

2 π ̄h

exp

[

−

iT

2 m ̄h

p^2 +

i(x−y)

̄h

p

]

(24.23)

Usingthegaussianintegralformula
∫
dpe−ap

(^2) +bp

√
π
a
eb
(^2) / 4 a
(24.24)
wefind
GT(x,y) =

1

2 π ̄h

(

π

iT/(2m ̄h)

) 1 / 2

exp

[

−

(x−y)^2 / ̄h^2

4 iT/(2m ̄h)

]

=

(

m

2 iπ ̄hT

) 1 / 2

exp

[

i

m(x−y)^2

2 ̄hT

]

(24.25)

Nowletstrytocomputethesamequantitybythepath-integralmethod.Wehave

GT(x,y)= lim

N→∞

B−N

∫

dxN− 1 dxN− 2 ...dx 1 exp

[

−

m

2 ih ̄!

∑N

n=1

(xn−xn− 1 )^2

]

(24.26)

where

B=

(

2 πi! ̄h

m

) 1 / 2

(24.27)

Factorslike(xk−xk− 1 )^2 coupletogethertheintegralstogether. Whatweneedisa
changeofvariableswhichwilluncouplethem, sothatwecandotheintegralsone-
by-one.Thisisaccomplishedbythechangeofvariables

zk = xk−xk− 1

N∑− 1

n=1

zk = xN− 1 −y (24.28)