3.1. WAVEEQUATIONFORDEBROGLIEWAVES 37
Thesameresultisobtained,ofcourse,forthecosine.Thisexpressionfortheenergy
ofanon-relativisticparticleintermsofitsmomentumissimplywrong.Foraparticle
ofmassm,thecorrectexpressionis
E=
p^2
2 m
(3.10)
Inordertorecoverthisexpression,weneedawaveequationwithonetimederivative
(whichbringsdownonefactorofE),andtwospacederivatives(whichbringdowna
factorofp^2 ),i.e.
∂ψ
∂t
=α
∂^2 ψ
∂x^2
(3.11)
Asin-functionwillnotsatisfythisequation,however,sinceweendupwithacosine
ontheleft-handside,andasinontheright-handside;andvice-versaforthecosine
wavefunction. Thewavefunctionwhichdoesworkisthecomplexform(3.6),which,
wheninsertedinto(3.11)gives
−iE
̄h
ei(px−Et)/ ̄h=α
−p^2
̄h^2
ei(px−Et)/ ̄h (3.12)
SettingE=p^2 / 2 m,wecansolveforαtofind
∂ψ
∂t
=
i ̄h
2 m
∂^2 ψ
∂x^2
(3.13)
or,equivalently,
i ̄h
∂ψ
∂t
=−
̄h^2
2 m
∂^2 ψ
∂x^2
(3.14)
ThisisthewaveequationfordeBrogliewavesmovinginonedimension. Thegener-
alizationtothreedimensionsisawavefunctionoftheform
ψ(%x,t)=ei(
"k·"x−ωt)
=ei("p·"x−Et)/ ̄h (3.15)
whichsatisfiesthewaveequation
i ̄h
∂ψ
∂t
=−
̄h^2
2 m
∇^2 ψ (3.16)
Itisimportanttorealizethatincontrasttowavesinclassicalphysics,theamplitude
ofdeBrogliewavesisnecessarilycomplex.
Problem:Considerawavefunctionoftheform
ψ(x,t)=Asin
(
px−Et
̄h
)
+Bcos
(
px−Et
̄h
)
(3.17)
whereAandB arecomplexnumbers. Assumingthatthiswavefunctionsolvesthe
waveequation(3.11),showthatitmustbeproportionaltoacomplexexponential.