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)