72 CHAPTER5. DYNAMICSOFTHEQUANTUMSTATE
Webeginbycomputingtheinitialexpectationvalues 0 andtheinitial (^2) /a 2 and ∫ (^2) /a 2 Theinitialuncertaintyinpositionattimet= 0 istherefore Tofindthetime-evolutionofthewavefunction,wefirstcomputetheinverseFourier Usingtheformulaforgaussianintegration √
uncertainty∆x 0 inposition,attimet=0.First
<x> 0 =
∫
dxxφ∗(x)φ(x)
=
( 1
πa^2
)
) 1 / 2 ∫∞
−∞
dxxe−x
= 0 (5.61)
<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 −
dxx^2 φ∗(x)φ(x)
=
(
1
πa^2
) 1 / 2 ∫∞
−∞
dxx^2 e−x
=
1
2
a^2 (5.63)
∆x 0 =
a
√
2
(5.64)
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)
∫∞
−∞
dze−Az
(^2) −Bz
π
A
eB
(^2) / 4 A
(5.66)