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)