QMGreensite_merged

4.4. EXPECTATION,UNCERTAINTY,ANDTHEQUANTUMSTATE 59

Likewise,theexpectationvaluefortheobservablex^2 is

<x^2 >=

∫∞

−∞

x^2 ψ∗(x,t)ψ(x,t)dx (4.73)

andtheexpectationvalueofthesquaredeviationisgivenby

∆x^2 = <(x−<x>)^2 >

= <(x^2 − 2 x<x>+<x>^2 )>

=

∫∞

−∞

dx[x^2 − 2 x<x>+<x>^2 ]ψ∗(x,t)ψ(x,t) (4.74)

Usingthefactthatisaconstantwhichcanbetakenoutsidetheintegral,and
thenormalizationcondition(3.33),wefind

∆x^2 = <x^2 >− 2 <x><x>+<x>^2

= <x^2 >−<x>^2 (4.75)

asineq. (4.66). Thesquarerootofthisquantity,∆x,isreferredtoastheUncer-
taintyinthepositionoftheparticleinthequantumstate|ψ>.
WeseethatfromtheBorninterpretationwecanextracttwonumbersfromthe
wavefunction, theexpectationvalueandtheUncertainty∆x,whichcan be
compared tothe experimentalvalues of averagepositionx,and root-mean-square
deviationδxrespectively. Experimentally,however,thereareotherobservablesofa
particlewhichcanbemeasured,suchasitsmomentumanditsenergy,andvalues
ofp, δp, E, δEcanbedeterminedfromexperiment. TheBornInterpretationdoes
nottellushowtocomputethesequantities;thiswillrequireafurtherinvestigation
ofthedynamicsofthequantumstate,whichisthetopicofthenextlecture.