7.5. THETIME-ENERGYUNCERTAINTYRELATION 119
1
̄h
|<ψ|[Q,H]|ψ>|∆t = ∆Q (7.132)
Wenow applythegeneralized uncertaintyprinciple, eq. (7.116)withA= Qand
B=H(notethat∆H≡∆E)
1
̄h
(∆Q∆E)∆t≥
1
̄h
(
1
2
|<ψ|[Q,H]|ψ>|
)
∆t=
1
2
∆Q (7.133)
or,dividingby∆Qonbothsides
∆E∆t≥
̄h
2
(7.134)
whichprovestheenergy-timeuncertaintyrelation,foranyobservableQ.
Asanexampleofthisuncertaintyrelation,considerthetimerequiredforafree
particletomoveadistanceequaltoitsuncertaintyinposition∆x;i.e.for
changeby∆x. Thewavepacketismovingwithagroupvelocity
vg=
<p>
m
(7.135)
sothe timeittakesforthepacketto moveadistanceequalto theuncertaintyin
positionis
∆t=
∆x
vg
=
m∆x
<p>
(7.136)
Ontheotherhand,theuncertaintyinenergyisrelatedtotheuncertaintyinmomen-
tumvia
∆E=∆
(
p^2
2 m
)
=
<p>∆p
m
(7.137)
Then
∆t∆E =
m∆x
<p>
<p>∆p
m
= ∆x∆p
≥
̄h
2
(7.138)
aspredictedbythetime-energyuncertaintyrelation.