QMGreensite_merged

16.1. THEKRONIG-PENNYMODEL 255

providingthatweimposetheperiodicityrequirementthatx+Ldenotes thesame
pointasx.The“periodicdelta-function”whichincorporatesthisrequirementisgiven
by

δ(x)=

1

2 π

∑

m

exp[2πimx/L] (16.6)

Thetime-independentSchrodingerequationforthemotionofanyoneelectroninthis
potentialhastheusualform

Hψk(x)=

[

−

̄h^2

2 m

∂^2

∂x^2

−g

N∑− 1

n=0

δ(x−na)

]

ψk(x)=Ekψk(x) (16.7)

BecauseV(x)hasthesymmetry(16.5), itisuseful toconsider thetranslation
operatorfirstintroducedinChapter10,

Taf(x) = exp[iap ̃/ ̄h]f(x)

= f(x+a) (16.8)

Likewise,

T−af(x) = exp[−iap ̃/ ̄h]f(x)

= f(x−a) (16.9)

Becausep ̃isanHermitianoperator,itseasytosee(justexpandtheexponentialsin
apowerseries)that

Ta† = T−a

[Ta,T−a] = 0

T−aTa = 1 (16.10)

DuetotheperiodicityofthepotentialV(x),theHamiltoniancommuteswiththe
translationoperators,which,aswe’veseen,alsocommutewitheachother,i.e.

[Ta,H]=[T−a,H]=[Ta,T−a]= 0 (16.11)

Thismeans(seeChapter10)thatwecanchooseenergyeigenstatestoalsobeeigen-
statesofT±a,i.e.

TaψE(x) = λEψE(x)

T−aψE(x) = λ′EψE(x) (16.12)

Therefore,

λE = <ψE|Ta|ψE>

= <Ta†ψE|ψE>

= <T−aψE|ψE>

= (λ′E)∗ (16.13)