10.4. THEQUANTUMCORRAL 163
Therefore
β^2 = 1 =⇒ β=± 1 (10.71)Becauseφnm(x,y)andφmn(x,y)havethesameenergy,sodoesanylinearcombi-
nation
Φ(x,y)=aφnm(x,y)+bφmn(x,y) (10.72)RequiringΦtobeaneigenstateofIwitheigenvalueβ= 1 meansthat φ(x,y)=
φ(y,x),or
asin[nπy
L]sin[mπx
L]+bsin[mπy
L]sin[nπx
L]
= asin[nπx
L]sin[mπy
L]+bsin[mπx
L]sin[nπy
L] (10.73)
whichissatisfiedfora=b. Likewise,forΦaneigenstateofIwitheigenvalueβ=−1,
whichmeansthatφ(x,y)=−φ(y,x)weneeda=−b. Finally,thecompletesetof
energyeigenstateswhicharealsoeigenstatesofthex−yinterchangeoperatorIare
{
Φnm+=√^12 [φnm(x,y)+φnm(y,x)]
Φnm−=√^12 [φnm(x,y)−φnm(y,x)]}
(10.74)Onceagain,weseethattherearesymmetryoperationswhichdo notcommute
withoneanother,andthattheenergyeigenvaluesaredegenerate. Thechoiceofa
completesetofenergyeigenstates,(10.62)or(10.74),isdeterminedbyrequiringthat
theenergyeigenstates,inadditiontobeingeigenstatesoftheHamiltonian,arealso
eigenstatesofoneofthesymmetryoperators.
10.4 The Quantum Corral
The”quantumcorral”refersto aparticlemovingintheinterior ofacircle. The
potential,inpolarcoordinates,isgivenby
V(r)={
0 r<R
∞ r>R(10.75)
which,sinceitdependsonlyontheradialcoordinate,isobviouslyinvariantunder
arbitraryrotations
r′=r θ′=θ+δ′θ (10.76)Polarcoordinatesarerelatedtocartesiancoordinatesby
x=rcosθ y=rsinθ (10.77)