QMGreensite_merged

74 CHAPTER5. DYNAMICSOFTHEQUANTUMSTATE

=

1

√

πa^2 (t)

∫

dx(x−v 0 t)^2 exp

[

−

(x−v 0 t)^2

a^2 (t)

]

=

1

√

πa^2 (t)

∫

dx′x′^2 exp

[

−

x′^2

a^2 (t)

]

=

1

2

a^2 (t) (5.73)

sothat

∆x=

a(t)

√

2

=

1

√

2

√

a^2 +

̄h^2

m^2 a^2

t^2 (5.74)

Tocompute

,wemakeuseagainoftherepresentation(5.44)

<p> =

∫

dxψ∗(x,t)

(

−i ̄h

∂

∂x

)

ψ(x,t)

=

∫

dx

{∫

dp 1

2 π ̄h

f(p 1 )ei(p^1 x−Ep^1 t)/ ̄h

}∗(

−i ̄h

∂

∂x

)∫

dp 2

2 πh ̄

f(p 2 )ei(p^2 x−Ep^2 t)/ ̄h

=

∫ dp

1

2 π ̄h

f∗(p 1 )eiEp^1 t/ ̄h

∫ dp

2

2 π ̄h

p 2 f(p 2 )e−iEp^2 t/ ̄h

∫

dxei(p^2 −p^1 )x/ ̄h

=

∫ dp

1

2 π ̄h

f∗(p 1 )eiEp^1 t/ ̄h

∫ dp

2

2 π ̄h

p 2 f(p 2 )e−iEp^2 t/ ̄h 2 πh ̄δ(p 1 −p 2 )

=

∫ dp

2 π ̄h

pf∗(p)f(p) (5.75)

Notethatthetime-dependencehasvanishedcompletelyfromthisequation.Itfollows
that

=

0 =p 0 (5.76)

andtheexpectationvaluesofpositionandmomentumtogetherobey

<x>t = <x> 0 +

<p> 0

m

t

<p>t = p 0 (5.77)

Withthe ”<>”signsremoved,these aretheequations ofaclassicaltrajectoryin
phasespaceforafreeparticleofmassm.
Ontheotherhand,thepositionaluncertainty ∆xofthequantumstate,which
isproportionaltothewidtha(t)ofthegaussian,increasesintime;thisisknownas
the”spreadingofthewavepacket”. AsnotedinLecture3,oneofthereasonsthat
the electroncannot beregarded as being literallyawaveisthat its wavefunction
expandstomacroscopicsizequiterapidly,anditisquitecertainthatelectronsare
notmacroscopicobjects. Letusconsiderthe timet 2 a thatittakesafreeparticle
wavefunctiontodoubleinsize(a(t)= 2 a(0)):

t 2 a=

√

3

ma^2

̄h

(5.78)