QMGreensite_merged

256 CHAPTER16. LIVEWIRESANDDEADSTARS

Butalso

ψE(x) = TaT−aψE

= λEλ′EψE

= ψE(x) (16.14)

Thismeansthat λ′E =(λE)−^1. Insert thatfactinto(16.13)andweconcludethat
λ∗E=(λE)−^1 ,i.e.
λE=exp(iKa) (16.15)

forsomeK.Inthiswaywearriveat

Bloch’sTheorem

ForpotentialswiththeperiodicitypropertyV(x+a)=V(x),eachenergyeigen-
stateoftheSchrodingerequationsatisfies

ψ(x+a)=eiKaψ(x) (16.16)

forsomevalueofK.

ItisalsoeasytoworkoutthepossiblevaluesofK,fromthefactthat

ψ(x) = ψ(x+L)

= (Ta)Nψ(x)

= exp[iNKa]ψ(x) (16.17)

whichimplies

K=

2 π

Na

j j= 0 , 1 , 2 ,...,N− 1 (16.18)

ThelimitingvalueofjisN−1,simplybecause

exp[i 2 π

j+N

N

]=exp[i 2 π

j

N

] (16.19)

soj≥Ndoesn’tleadtoanyfurthereigenvalues(jandN−jareequivalent).
AccordingtoBloch’stheorem,ifwecansolve fortheenergy eigenstatesinthe
region 0 ≤x≤a,thenwehavealsosolvedforthewavefunctionatallothervaluesof
x. NowtheperiodicdeltafunctionpotentialV(x)vanishesintheregion 0 <x<a,
sointhisregion(callitregionI)thesolutionmusthavethefree-particleform

ψI(x)=Asin(kx)+Bcos(kx) E=

h ̄^2 k^2

2 m

(16.20)