124 CHAPTER8. RECTANGULARPOTENTIALS
changeinanyofthoseeigenvaluesresultsinanon-physicalstate,theboundstate
energiesalwaysformadiscreteset.
Finally, it should benoted that it isnot alwaysthe case that V(x) → 0 as
x→±∞. Suppose,e.g.,thatV(x)→V∞> 0 asx→±∞. Inthatcase,unbound
stateswouldhaveenergiesintherangeE∈[V∞,∞],whiletheboundstateenergies
wouldbeadiscretesetwithvaluesE<V∞.
8.2 Unbound States and Quantum Scattering
Consideraparticlewhichismovinginatubeclosedatoneend. Thecorresponding
potentialis
V(x)={
0 x< 0
∞ x≥ 0(8.5)
Initially,beforetheparticleencounterstheendofthetube,itsstateisrepresented
bysome wavepacketindicatedinFig. [8.8a],withamomentumexpectationvalue
< p>=p 0. Afterbouncingofftheend of thetube, the ”scattered”wavepacket
willbemovingintheoppositedirectionwith
=−p 0 ,asshowninFig. [8.8c].
Atsomeintermediatetime,thewavepacketisincontactwiththeendofthetube.
Sincethewavepacket isfiniteinextent,thereisgoing tobesome delay fromthe
timethatthefrontofthewavepacketreflectsofftheendof thetube,to thetime
thatthebackofthewavepacketreflectsofftheend. Duringthistimeinterval,part
ofthewavepacketwhichreflectedofftheendofthetubeoverlapswiththerestof
theincomingwavepacket,whichhasnotyetencounteredtheendof thetube(Fig.
[8.8b]).
Nowifthewavepacketisofsomefiniteextent∆x,thenbytheUncertaintyPrin-
ciple∆p∼ ̄h/∆x.Imaginepreparingtheinitialwavepacketwithaverysmalluncer-
tainty∆p. As∆p→0,then∆x→∞,andthetime(andspace)intervalinwhich
theincomingandreflectedwavesoverlapbecomesverylarge.Thelimitingcase,with
∆x=∞,isshowninFig. [8.9]. Inthiscase, theincoming”wavepacket”atany
instantoftimeisaninfiniteplanewave
φinc(x)=Aeip^0 x/ ̄h (8.6)whilethereflectedwavepacketisalsoaninfiniteplanewaveoftheoppositemomentum
φref(x)=Be−ip^0 x/ ̄h (8.7)Theseplanewavesoverlapovertheentirehalf-linex∈[−∞,0],sothetotalwave-
functionisthesuperpositionoftheincomingandreflectedwaves
φ(x) = φinc(x)+φref(x)
= Aeip^0 x/ ̄h+Be−ip^0 x/ ̄h (8.8)