QMGreensite_merged

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.forto
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.