7.3. EIGENSTATESASSTATESOFZEROUNCERTAINTY 105
wherewe have integrated twice by parts. This establishesthe hermiticity ofthe
Hamiltonian.
TheHermitianconjugateO ̃†ofanylinearoperatorO ̃,hermitianornot,hasthe
followingimportantproperty:
<ψ 1 |O|ψ 2 >=<O†ψ 1 |ψ 2 > (7.43)forany|ψ 1 >and|ψ 2 >.Thisisbecause
<ψ 1 |O|ψ 2 > =∫
dx∫
dyψ∗ 1 (x)O(x,y)ψ 2 (y)=∫
dy[∫
dxO(x,y)ψ∗ 1 (x)]
ψ 2 (y)=
∫
dy[∫
dxO∗(x,y)ψ 1 (x)]∗
ψ 2 (y)=∫
dy[∫
dxO†(y,x)ψ 1 (x)]∗
ψ 2 (y)= <O†ψ 1 |ψ 2 > (7.44)Inparticular,foranHermitianoperator
<ψ 1 |O|ψ 2 >=<Oψ 1 |ψ 2 > (7.45)Exercise: Show that the Hermitian conjugate of a product of linear operators
A,B,C,Disgivenby
(ABCD)†=D†C†B†A† (7.46)
7.3 Eigenstates As States of Zero Uncertainty
Whatdoesameasurementdo?
Ameasurementapparatusisdesignedtodeterminethevalueofsomeobservable
O,andbydoingsoitmustleavethesysteminaphysicalstateinwhichthevalue
ofthatobservableisadefinitenumber. Butthatmeansthatthesystemisleftina
stateforwhichtheuncertainty∆Ooftheobservablevanishes,atleastattheinstant
themeasurementisperformed.If,forexample,thepositionofaparticleismeasured
precisely,thentheparticleisknowntobeatthatpositiontheinstantthemeasurement
isperformed;thephysicalstatemustresembleaDiracdeltafunction.Likewise,ifthe
momentumofaparticleismeasured,theparticleisknowntohavethatmomentum,
andthephysicalstatemustresembleaplanewave. Ingeneral,whateverthestateof
thesystemmaybejustbeforethemeasurement,theoutcomeofameasurementisa
stateof”zero-uncertainty”∆O= 0 intheobservableO.Exactlyhowameasurement
apparatusaccomplishesthisfeatisaquestionwhichwillnotconcernusnow,butwill
betakenuplaterinthecourse.