QMGreensite_merged

5.7. THEPARTICLEINACLOSEDTUBE 81

Integrateoverx

∫L

0

dxφ∗k(x)ψ(x,0)=

∑∞

n=1

an

∫L

0

dxφ∗k(x)φn(x) (5.121)

andusetheorthogonalityrelation

<φk|φn> =

∫∞

−∞

dxφ∗k(x)φn(x)

=

2

L

∫L

0

dx sin[k

πx

L

]sin[n

πx

L

]

= δkn (5.122)

whereδijistheKroneckerdeltaintroducedinLecture4.
Eq.(5.122)isreferedtoasorthogonalitybecauseitistheexpression,forvectors
inHilbertspace,whichcorrespondstotheorthogonalityrelationforunitvectors%ei
inanN-dimensionalvectorspace

%ei·%ej=δij (5.123)

Inalaterlecture wewill seethatthe setof energy eigenstates{|φn >}can (and
should)beviewedasasetoforthonormalbasisvectorsinHilbertspace.
Applyingtheorthogonalityrelation,eq. (5.121)becomes
∫L

0

dxφ∗k(x)ψ(x,0) =

∑∞

n=1

anδkn

= ak (5.124)

Thisgivesusthecompleteprescriptionforfindingthephysicalstateoftheparticle
atanytimet>0,giventhestateataninitialtimet=0. Onehastocomputethe
setofcoefficients

an = <φn|ψ(t=0)>

=

∫L

0

dxφ∗n(x)ψ(x,0) (5.125)

andsubstitutetheseintoeq. (5.118)togetthewavefunctionψ(x,t).
Theexpectationvaluesofpositionandmomentum,asafunctionoftime,canalso
beexpressedintermsofthecoefficientsan. Fortheposition

<x> =

∫

dxψ∗(x,t)xψ(x,t)

=

∫

dx

{∞

∑

i=1

aiφi(x)e−iEit/ ̄h

}∗

x

∑∞

j=1

ajφj(x)e−iEjt/ ̄h

=

∑∞

i=1

∑∞

j=1

a∗iajei(Ei−Ej)t/ ̄hXij (5.126)