5.5. GAUSSIANWAVEPACKETS 71
Wenowhaveaprescription,giventhewavefunctionatanyinitialtimet=0,for
findingthewavefunctionatanylatertime:Fromφ(x)=ψ(x,0),computetheinverse
Fouriertransformf(p)using(5.54),theninsertf(p)intotherhsofeq. (5.44). The
integralofeq. (5.44)willthengivethewavefunctionψ(x,t)atanylatertime.
5.5 Gaussian Wavepackets
Awavefunctionoftheform
φ(x)=NδL(x) (5.55)goestozero,asL→∞,everywhereexceptatthepointx=0.Attheotherextreme,
aplanewave
φ(x)=Neip^0 x/ ̄h (5.56)hasaconstantmodulus|φ|=N ateverypointalongtheentireline−∞<x<∞.
Awavefunctionwhichinterpolatesbetweenthesetwoextremecasesisthe”gaussian
wavepacket”
φ(x)=Ne−x(^2) / 2 a 2
eip^0 x/ ̄h (5.57)
inwhichagaussiandampingfactorisusedtomodulateaplanewave. Asa→0,
atfixedp 0 , thiswavefunctionisproportionaltoδL(x)(whereL= 1 / 2 a^2 ),while as
a→∞,φ(x)approachesaplanewave.
Gaussianwavepackets provideaveryusefulexampleofhowthewavefunctionof
afreeparticleevolvesintime. Assumethattheinitialwavefunctionψ(x,0)hasthe
formofthegaussianwavepacket(5.57),withsomefixedvalueoftheconstanta.The
firsttaskisto”normalize”thewavefunction,whichmeans:choosetheconstantN
in(5.57)sothatthenormalizationcondition
∫
dxψ∗(x,t)ψ(x,t)= 1 (5.58)
isfulfilledatt= 0 (whichimpliesthatitwillbefulfilledforallt,byconservationof
probability).Theconditionis
1 =
∫
dxψ∗(x,0)ψ(x,0)= N^2∫
dxe−x(^2) /a 2
= N^2
√
πa^2 (5.59)Therefore
N=(
1
πa^2) 1 / 4
(5.60)