70 CHAPTER5. DYNAMICSOFTHEQUANTUMSTATE
Thesetofeigenfunctionsandcorrespondingeigenvalues
{
φp(x)=eipx/ ̄h,Ep=
p^2
2 m}
−∞<p<∞ (5.48)isthereforeacompletesetofsolutionsofthetime-independentSchrodingerequation.
Insertingthesesolutionsintoeq.(5.39),wearriveatthegeneralsolution(5.44).
Itiseasytoshowthateq. (5.44)is,infact,thegeneralsolutionofthefree-particle
Schrodingerwaveequation,inthesensethatanysolutionofthefree-particleequation
canbeputintothisform.Firstofall,thewavefunctionψ(x,t)atalltimesisuniquely
determinedbythewavefunctionatoneinitialtime,sayt=0,denoted
φ(x)≡ψ(x,0) (5.49)ThisisbecausetheSchrodingerequationisfirst-orderinthetimederivative,sothat
givenψ(x,t)atsometimet,thewavefunctionisdeterminedaninfinitesmaltimet+!
laterbytherulegivenineq. (4.59). Next,foranygivenφ(x)wecanalwaysfinda
functionf(p)suchthat
φ(x)=∫∞−∞dp
2 π ̄hf(p)eipx/h ̄ (5.50)Todeterminef(p),multiplybothsidesoftheequationbyexp[−ip′/h ̄],andintegrate
withrespecttox:
∫∞−∞dxφ(x)e−ip′x/ ̄h
=∫∞−∞dx∫∞−∞dp
2 πh ̄f(p)ei(p−p′)x/ ̄h=
∫∞−∞dp
2 π ̄hf(p)∫∞−∞dxexp[ip−p′
h ̄x]=
∫∞−∞dp
2 π ̄hf(p)2πδ(
p−p′
̄h)
(5.51)wherewehave usedthe integral representationof theDiracdeltafunction (4.50).
Usingalsotheidentity(4.52):
δ(
p−p′
̄h)
= ̄hδ(p−p′) (5.52)weget ∫∞
−∞dxφ(x)e−ip
′x/ ̄h
=∫∞−∞dpf(p)δ(p−p′)=f(p′) (5.53)Theconclusionisthatanyφ(x)canbeexpressedintheform(5.50),forf(p)chosen
toequal
f(p)=∫∞−∞dxφ(x)e−ipx/ ̄h (5.54)Theterminologyisthatφ(x)istheFourier Transformoff(p)(eq. (5.50)),and
f(p)istheInverseFourierTransformofφ(x)(eq. (5.54)).