72 CHAPTER5. DYNAMICSOFTHEQUANTUMSTATE

Webeginbycomputingtheinitialexpectationvalues 0 ,

0 andtheinitial
uncertainty∆x 0 inposition,attimet=0.First

<x> 0 =

∫

dxxφ∗(x)φ(x)

=

( 1

πa^2

)

) 1 / 2 ∫∞

−∞

dxxe−x

(^2) /a 2

= 0 (5.61)

and

<p> 0 =

∫

dxφ∗(x)

(

−i ̄h

∂

∂x

)

φ(x)

= −

i ̄h

√

πa^2

∫

dxexp

(

−i

p 0 x

̄h

−

x^2

2 a^2

)

∂

∂x

exp

(

i

p 0 x

̄h

−

x^2

2 a^2

)

= −i

i ̄h

√

πa^2

∫

dx

(

i

p 0

̄h

−

x

a^2

)

e−x

(^2) /a 2
= p 0 (5.62)
Thesearetherelevantexpectationvaluesatt= 0 ofpositionandmomentum. Next,
thesquareduncertaintyis
∆x^20 = <x^2 > 0 −^20 =<x^2 > 0

∫
dxx^2 φ∗(x)φ(x)

=

(

1

πa^2

) 1 / 2 ∫∞

−∞

dxx^2 e−x

(^2) /a 2

=

1

2

a^2 (5.63)

Theinitialuncertaintyinpositionattimet= 0 istherefore

∆x 0 =

a

√

2

(5.64)

Tofindthetime-evolutionofthewavefunction,wefirstcomputetheinverseFourier
Transform

f(p) =

∫

dxφ(x)e−ipx/ ̄h

=

(

1

πa^2

) 1 / 4 ∫

dxexp

[

−

x^2

2 a^2

−i

(p−p 0 )

̄h

x

]

(5.65)

Usingtheformulaforgaussianintegration
∫∞

−∞

dze−Az

(^2) −Bz

√
π
A
eB
(^2) / 4 A
(5.66)