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