5.5. GAUSSIANWAVEPACKETS 75
andthetimeittakestoexpandtosomeverymuchlargersizea(t)=A>>a
tA=maA
̄h(5.79)
Asanexample,choosetheinitialspreadofthewavepackettobeontheorderofthe
diameterofatomichydrogen
a= 10 −^10 m (5.80)andAtobeamacroscopicdistance,e.g.
A= 10 cm (5.81)Beginbyconsideringanelectron, whichhasamassme= 9. 11 × 10 −^31 kg;and
̄h= 1. 05 × 10 −^34 J-s. Thenthetimerequiredforthewavefunctiontoexpandfrom
a= 10 −^10 mto 2 × 10 −^10 mis
t 2 a=√
3
mea^2
̄h= 1. 5 × 10 −^16 s (5.82)Likewise,thetimerequiredforthewavefunctiontospreadtoA= 10 cmis
tA=meaA
̄h= 8. 7 × 10 −^6 s (5.83)IftheenergyoftheelectronisE= 10 electronvolts(eV),thenthevelocityis
v=√
2 E/m= 1. 87 × 106 m/s (5.84)Therefore,forthewavefunctionofa 10 eVelectrontoexpandfromanatomicdiameter
toawidthof 10 cm,theelectronneedonlytravel
d=vtA= 16. 3 m (5.85)Comparethesefiguresforanelectronwiththecorrespondingresultsforanobject
themassofabaseball,saymB=. 25 kg. Supposingthewidthofthewavefunctionis
alsoinitiallya= 10 −^10 m,thecorrespondingnumbersare
t 2 a = 4. 1 × 1014 s= 1. 3 millionyears
tA = 2. 4 × 1023 s= 1. 8 × 1017 years (5.86)Inotherwords,forallpracticalpurposes,thewavefunctionofanobjectofmacroscopic
massdoesn’texpandatall.