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76 CHAPTER5. DYNAMICSOFTHEQUANTUMSTATE

5.6 Group Velocity and Phase Velocity

We have seen that the expectation value of position< x > moves at avelocity
v=

/m. ThisisaconsequenceofEhrenfest’sprincipleforafreeparticle. On
theotherhand,thevelocityvphaseofanyplanewaveisgivenbytheusualformula

vphase=λf (5.87)

sothatforaDeBrogliewaveofadefinitemomentump

vphase =

h

p

Ep

h

=

Ep

p

=

p^2 / 2 m

p

=

p

2 m

(5.88)

whichishalfthevelocitythatonemightexpect,sinceaclassicalparticlewithmo-
mentumpmovesatavelocityp/m.Thereisnocontradiction,however. Thevelocity
vphase=λfofaplanewaveisknownasthe”phasevelocity”ofthewave;itrefers
tothespeedatwhichapointofgivenphaseonthewave(acrestoratrough,say),
propagatesthroughspace. However,thatisnotnecessarilythesamespeedatwhich
theenergycarriedbyagroupofwavestravelsthroughspace.Thatvelocityisknown
asthe”groupvelocity”.
Considerawavewhichisasuperpositionoftwoplanewavesoffrequenciesf 1 ,f 2 ,
andwavelengthsλ 1 ,λ 2 ,respectively,andwhichhavethesameamplitudeA. Wewill
assumethatthedifferencebetweenfrequencies∆fismuchsmallerthantheaverage
frequencyf,i.e.
∆f<<f and ∆λ<<λ (5.89)

Bythesuperpositionprincipleofwavemotion

ψ(x,t) = ψ 1 (x,t)+ψ 2 (x,t)

= A

[

ei(k^1 x−ω^1 t)+ei(k^2 x−ω^2 t)

]

= 2 Aei(kx−ωt)cos[

1

2

i(∆kx−∆ωt)] (5.90)

whichistheproductoftwowaves,oneofwhichismovingwiththevelocity

vphase=

ω

k

(5.91)

andtheothermovingwithvelocity

vgroup=

∆ω

∆k

(5.92)