QMGreensite_merged

24.4. PATH-INTEGRATIONASQUANTIZATION 377

wherewehave droppedthetermintheexponentproportionaltoV(x),sincethat
termisnegligibleinthe!→ 0 limit.Nowchangevariablestoz=y−x. Then

p ̃ψ(x,t)

= lim

!→ 0

( m

2 πi! ̄h

) 1 / (^2) m
!
∫
dz(−z)exp
[
i
m
2! ̄h
z^2
]
ψ(x+z,t−!)
= lim
!→ 0
(
m
2 πi! ̄h
) 1 / 2
m
!
∫
dz(−z)exp
[
i
m
2! ̄h
z^2
]
ψ(x,t−!)+ψ′(x,t−!)z+O(z^2 )]
= −lim
!→ 0
(
m
2 πi! ̄h
) 1 / 2
m
!
dψ
dx
∫
dzz^2 exp
[
i
m
2! ̄h
z^2
]
(24.55)
Thetermproportionaltozdropsoutintheintegration,becausezisanoddfunction
whereastheexponentiseven. Also,termsoforderz^3 andhigherdisappearinthe
!→ 0 limit. Performingthegaussianintegralandtakingthe!→ 0 limitwefinally
obtain
p ̃ψ(x,t) = −lim
!→ 0
( m
2 πi! ̄h
) 1 / (^2) m
!
dψ
dx
(
−
π
im/(2! ̄h)
) 1 / 2
i! ̄h
m
= −i ̄h
dψ
dx

(24.56)

whichisthesamerulethatwehaddeducedpreviously,inLecture5,fromEhrenfest’s
principle.

Problem- Use thesameanalysistofindtheoperatorcorrespondingtop ̃^2. Note
that,ifyouuseonlyasingleintegralasabove,andjustreplace

m

(x−y)

!

by m^2

(x−y)^2

!^2

(24.57)

somethinggoeswrong!Canyoufigureouthowtofixit?

24.4 Path-Integration as Quantization

Havingderivedthepath-integralfromtheSchrodingerequation,onecan ofcourse
goin the otherdirection, i.e. derive the Schrodingerequation startingfrom the
conceptofanintegrationoverpaths. Wehaveseenthatpath-integralswithgaussian
integrandscanbeevaluatedexactly; integrandswithnon-gaussiantermscanoften
beevaluatedapproximatelyby aperturbationtechnique. Wehave alsoseen that
path-integralsleadtotheusualformofthemomentumoperator.Logically,thepath-
integralapproachisanalternativetocanonicalquantizationbasedoncommutators;
eithermethodcanbeusedtoquantizeaclassicaltheory.