QMGreensite_merged

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.