QMGreensite_merged

106 CHAPTER7. OPERATORSANDOBSERVATIONS

FromthefactthateveryobservableO=x,p,E,...isassociatedwithanHermitian
operatorO ̃ = ̃x,p, ̃H ̃,..., we seethatazero-uncertaintystateφ(x)musthave the
property
(∆O)^2 =<φ|(O−)^2 |φ>= 0 (7.47)

Denotetheexpectationvaluebyλ,anddefine

D ̃≡O ̃−λ (7.48)

SinceO ̃isHermitian,andmultiplicationbyaconstantλisanHermitianoperation,
theoperatorD ̃isalsoHermitian. Then,usingthehermiticityofD ̃

(∆O)^2 = <φ|(D)^2 |φ>

= <φ|D|Dφ>]

= <φ|D|φ′>

= <Dφ|φ′>

= <φ′|φ′>

=

∫

dxφ′∗(x)φ′(x) (7.49)

wherewehavedefined
φ′(x)≡D ̃φ(x) (7.50)

Nowputtingtogether

∆O = 0

(∆O)^2 =

∫

dxφ′∗(x)φ′(x)

φ′∗(x)φ′(x) ≥ 0 forallx (7.51)

wemustconcludethat

0 = φ′(x)

= D ̃φ(x)

= [O ̃−λ]φ(x) (7.52)

or,inotherwords,forazero-uncertaintystate

O ̃φ(x)=λφ(x) (7.53)

Thisaneigenvalueequation,oftheformalreadyseenforposition,momentumand
energyinequation(7.9).Therefore:

I)AnystatewithvanishinguncertaintyintheobservableOisaneigenstate
ofthecorrespondingHermitianoperatorO ̃.