QMGreensite_merged

5.6. GROUPVELOCITYANDPHASEVELOCITY 77

Thecorrespondingexponentialscanbeexpandedintosinesandcosines,theproduct
oftwocosinesisshowninFig. [5.2]. The”packets”ofwavesmovecollectivelywith
thegroupvelocityvgroup;thecrestofawavewithineachpackettravelsatthephase
velocityvphase.
Nowletusconsiderageneralwavepacketoftheform

ψ(x,t)=

∫

dkf(k)ei(kx−ω(k)t) (5.93)

Forafreeparticleinquantummechanics,thewavenumberandangularfrequencyare
relatedtomomentumandenergy

k=

p

̄h

and ω=

Ep

̄h

(5.94)

Suppose,asinthecaseofthegaussianwavepacket(eq. (5.67)),thatf(k)ispeaked
aroundaparticularvaluek 0. Inthatcase,wecanmakeaTaylorexpansionofω(k)
aroundk=k 0 :

ω(k)=ω 0 +

(

dω

dk

)

k 0

(k−k 0 )+O[(k−k 0 )^2 ] (5.95)

Insertingthisexansioninto(5.93),anddroppingtermsoforder(k−k 0 )^2 andhigher,
weget

ψ(x,t) ≈

∫

dkf(k)exp

[

i

(

k 0 x+(k−k 0 )x−ω 0 t−

dω

dk

(k−k 0 )t

)]

≈ ei(k^0 x−ω^0 t)

∫

dkf(k)exp[i(k−k 0 )(x−vgroupt)]

≈ ei(k^0 x−ω^0 t)F[x−vgroupt] (5.96)

where

vgroup=

(

dω

dk

)

k=k 0

(5.97)

Thisisagainaproductoftwowaveforms;aplanewavemovingwithvelocity

vphase=

ω 0

k 0

(5.98)

andawavepulseF[x−vgroupt]movingwithvelocityvgroupofequation(5.97). The
product isindicated schematically inFig. [5.3]. It isclear that the wavepacket
propagatescollectivelyatthevelocityofthepulse,vgroupratherthanthevelocityof
acrestvphase.^2

(^2) Accordingto(5.96)thewavepacketpropagateswithoutchangingitsshape,whichisnotquite
true,aswehaveseeninthecaseofthegaussianwavepacket.The”spreading”ofthewavepacketis
duetotermsoforder(k−k 0 )^2 ,whichweredroppedineq.(5.96).