QMGreensite_merged

88 CHAPTER6. ENERGYANDUNCERTAINTY

6.2 The Heisenberg Uncertainty Principle

Letususe(6.16)tocomputethemomentumuncertainty∆pforthegaussianwavepacket

φ(x)=

1

(πa^2 )^1 /^4

e−x

(^2) / 2 a 2
eip^0 x/ ̄h (6.18)
whichwealreadyknowhasamomentumexpectationvalue

=p 0 (6.19)
andapositionuncertainty
∆x=
a
√
2

(6.20)

Thesquaredmomentumuncertaintyis

∆p^2 = <p^2 >−<p>^2 =<p^2 >−p^20

= − ̄h^2

∫

dxφ∗(x)

∂^2

∂x^2

φ(x) − p^20

= −

̄h^2

√

πa^2

∫

dxexp

(

−i

p 0 x

̄h

−

x^2

2 a^2

)

×

[

(i

p 0

̄h

−

x

a^2

)^2 −

1

a^2

]

exp

(

i

p 0 x

̄h

−

x^2

2 a^2

)

− p^20

=

̄h^2

2 a^2

(6.21)

or

∆p=

̄h

√

2 a

(6.22)

Multiplying∆xand∆p,theproductofuncertaintiesissimply

∆x∆p=

̄h

2

(6.23)

Wewill show inthe next chapter(the sectionon the generalized Uncertainty
Principle)that ̄h/ 2 isthesmallestvaluefortheproductof∆x∆pthatcanbeobtained
foranywavefunction;wavefunctionswhichdifferfromthegaussianwavepackethave
productsgreaterthan ̄h/2. Therefore,foranyphysicalstate,theproductofposition
andmomentumuncertaintiesobeystheinequality

∆x∆p≥

̄h

2

(6.24)

Acorollaryofthisfactis