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.