QMGreensite_merged

118 CHAPTER7. OPERATORSANDOBSERVATIONS

7.5 The Time-Energy Uncertainty Relation

Itwasnoted,attheendofthelastlecture,thatchangeofanykindinaquantum-
mechanicalsystemimpliesanuncertaintyintheenergy ofthatsystem. Wewould
like toquantifythisremark: Howlong doesittakeforaphysical statetochange
substantially,andhowisthatlengthoftimerelatedtotheenergyuncertainty?
Ofcourse,itisnecessarytodefinewhatismeantby a”substantial”changein
thestate. Considerposition,momentum, orany otherobservableQ. Atagiven
moment,Q hasacertainexpectation value< Q >, andanuncertainty ∆Q. As
timepasses, < Q > usuallychanges insome way. Wewill say that< Q > has
changed”substantially”when ithasincreasedordecreasedbyanamountequalto
theuncertainty∆Qoftheobservable. Therelationbetweenthetime∆trequired
forsuchachange,andtheenergy uncertainty ∆E,isexpressed intheformofan
inequalityknownas

TheTime-EnergyUncertaintyRelation

LetQbeanyobservable,andlet∆tbethetimerequiredfortheexpec-
tationvaluetochangebyanamountequaltoitsuncertainty∆Q.
Then

∆t∆E≥

̄h

2

(7.130)

where∆Eistheuncertaintyintheenergyofthesystem.

Toprovethisstatement,letusbeginwithanexpressionfortherateofchangeof
theobservable

d

dt

=

d

dt

∫

dxψ∗(x,t)Q ̃ψ(x,t)

=

∫

dx

[

dψ

dt

∗

Q ̃ψ+ψ∗Q ̃dψ

dt

]

=

∫

dx

[

(

1

i ̄h

H ̃ψ)∗Q ̃ψ+ψ∗Q ̃(^1

i ̄h

H ̃ψ

]

=

1

i ̄h

[−<Hψ|Q|ψ>+<ψ|Q|Hψ>]

=

1

i ̄h

[−<ψ|HQ|ψ>+<ψ|QH|ψ>]

= <ψ|

1

i ̄h

[Q,H]|ψ> (7.131)

Thenthetime∆trequiredfortochangebyanamountequaltoitsuncertainty
∆Qsatisfies
∣∣
∣∣
∣

d<Q>

dt

∣∣

∣∣

∣∆t = ∆Q