QMGreensite_merged

10.4. THEQUANTUMCORRAL 167

Thisisaboundstatesolution,sincetheparticleisboundbythepotentialinsidea
circle,andweknowthattheenergiesofboundstatesarediscrete. Inthiscase,the
possibleenergiesaredeterminedbytheconditionthat

φn(R,θ)= 0 =⇒φn(R)= 0 =⇒Jn(kR)= 0 (10.105)

Besselfunctionsareoscillatingfunctions(wavefunctionsinclassicallyallowedre-
gionsoscillate),sotheconstantkmustbechosensuchthatkRisa”zero”ofthe
BesselfunctionJn. Denote

Jn(xnj)= 0 (10.106)

where{xn 1 ,xn 2 ,xn 3 ,...}arethezerosofthen-thBesselfunction. Inparticular

x 0 j = 2. 405 , 5. 520 , 8. 654 ,...

x 1 j = 3. 832 , 7. 016 , 10. 173 ,... (10.107)

andthepossibleenergiesmustthereforesatisfy

kR=xnj Enj=

̄h^2 k^2

2 m

(10.108)

Finally,thecompletesetofenergyeigenstatesandeigenvaluesare

{

φnj=Jn(

xnj

R

r)einθ, Enj=

̄h^2 x^2 nj

2 mR^2

, Lz=n ̄h

}

(10.109)

Notethattheenergyeigenvaluesaredegenerate,Enj=E(−n)j. Thismeansthat
theremustbeatleasttwooperatorswhichcommutewiththeHamiltonian,butnot
witheachother. Inthiscase,anoperatorwhichcommuteswhichH ̃,butnotwith
Lz,isthereflectionoperator

Pθf(r,θ)=f(r,−θ) (10.110)

correspondingtothereflectionsymmetryθ′=−θ.
Very recentadvances in technology, specifically, the invention of the scanning
tunnellingmicroscope, andthedevelopmentoftechniquesformanipulatingmatter
attheatomicscale,havemadeitpossibletoactuallytrapelectronsinacircle,and
measurethesquaremodulusofthewavefunction.Theresult,showninFig.[10.4],is
breath-taking.Thepeaksarrangedinacircleshowtheelectroncloudsofironatoms,
whichareusedtoformthe”corral”. Thecircularripplesinsidethe”corral”show
thewavefunction(modulussquared)ofthe trappedelectrons. Inthephotograph,
theelectronsareinanenergyeigenstate,andthewavefunction(modulussquared)is
proportionaltothesquareofaBesselfunction.