QMGreensite_merged

78 CHAPTER5. DYNAMICSOFTHEQUANTUMSTATE

Inthecaseofafreeparticle,wehave

vgroup =

(

dω

dk

)

k=k 0

=

(

d(Ep/ ̄h)

d(p/ ̄h)

)

p=<p>

=

(

dEp

dp

)

p=<p>

=

<p>

m

(5.99)

whichisthestandardrelationbetweenvelocityandmomentum. Theconclusionis
that,althoughthephasevelocityofadeBrogliewaveishalftheclassicalvelocity,a
packetofdeBrogliewavestravelscollectivelyatexactlytheclassicalvelocity.

5.7 The Particle in a Closed Tube

Finallywereturnto theexampleofLecture 4,inwhichaparticleismovingina
closedtubeoflengthL;collisionsbetweentheparticleandtheendsofthetubeare
assumedtobeelastic. Sincethepotentialbarrieragainstleavingthetubeis,forall
practicalpurposes,infinite,theparticleismovinginaninfinitepotentialwell

V(x)=

{

0 0 ≤x≤L

∞ otherwise

(5.100)

The problemis tosolvethe time-independent Schrodingerequation (5.35)inthis
potential.
Intheintervalx∈[0,L],thetime-independentSchrodingerequationisthesame
asthatforafreeparticle

−

̄h^2

2 m

∂^2

∂x^2

φ=Eφ (5.101)

andhasthesamesolutionsforE>0,namelyanylinearcombination

φ(x)=c 1 eipx/ ̄h+c 2 e−ipx/ ̄h where p=

√

2 mE (5.102)

Ontheotherhand,intheregionsx< 0 andx>L,wehave
[
−

̄h^2

2 m

∂^2

∂x^2

+∞

]

φ=Eφ (5.103)

Theonlypossiblesolutionofthisequation,forfiniteE,is

φ(x)= 0 x< 0 or x>L (5.104)