QMGreensite_merged

5.7. THEPARTICLEINACLOSEDTUBE 79

whichsimplymeansthattheprobabilityoffindingtheparticleoutsidethetubeis
zero.
Thesolutionofadifferentialequationmustbeacontinuousfunction.Continuity
ofthewavefunctionatthepointsx= 0 andx=Lgivestwoboundaryconditions

0 =φ(0) = c 1 +c 2

0 =φ(L) = c 1 eipL/ ̄h+c 2 e−ipL/ ̄h (5.105)

Thefirstconditiongivesc 2 =−c 1 ,andthenthesecondconditionbecomes

2 ic 1 sin[

pL

̄h

]= 0 (5.106)

Thesecondequationcanberecognizedastheconditionforastandingwaveinan
intervaloflengthL,i.e.
sin(kL)= 0 (5.107)

whichissatisfiedforwavenumbers

k=

nπ

L

(5.108)

or,intermsofwavelengthsλ= 2 π/k,

L=n

λ

2

(5.109)

Inthecaseofaparticleinatube,thewavenumberkisthesameasfordeBroglie
waves

k=

2 π

λ

=

p

̄h

(5.110)

andthestandingwaverequirementsin(kL)= 0 implies

pL

̄h

= nπ

⇒ pn = n

πh ̄

L

(5.111)

whereasubscriptnhasbeenaddedtoindicatethateachp=

√

2 mEisassociated
withapositiveintegern= 1 , 2 , 3 ,.... Theenergyeigenstatesaretherefore

φn(x)=

{

Nsin

[

nπ

Lx

]

0 ≤x≤L

0 otherwise

(5.112)

whereN= 2 ic 1 ,eachwithacorrespondingeigenvalue

En=

p^2 n

2 m

=n^2

π^2 ̄h^2

2 mL^2

(5.113)