162 CHAPTER10. SYMMETRYANDDEGENERACY
Acompletesetofenergyeigenstatesisthen
{
φnm(x,y)=2
L
sin[nπx
L]sin[mπy
L], Enm=(n^2 +m^2 )̄h^2 π^2
2 mL^2}
(10.62)TheenergiesEnmaretwo-folddegenerateforn+=m,since
Enm=Emn (10.63)SincetheHamiltonianH ̃ isinvariantunderreflections ofthex-axisaroundthe
pointx=L/2,andreflectionsofthey-axisaroundy=L/2, wedefinethecorre-
spondingoperators
Rxf(x,y) = f(L−x,y)
Ryf(x,y) = f(x,L−y) (10.64)ItiseasytoseethatRxandRycommute,
RxRyf(x,y) = RyRxf(x,y)
= f(L−x,L−y))
=⇒[Rx,Ry]= 0 (10.65)andthattheenergy eigenstatesofeq. (10.62)areeigenstates ofbothRxandRy,
witheigenvalues±1,since
sin[
nπ(L−x)
L]
={
1 nodd
− 1 neven
×sin[nπx
L] (10.66)
However,theHamiltonianH ̃ isalsoinvariantunderaninterchangeofthexandy
coordinates
If(x,y)=f(y,x) (10.67)
andthisoperatordoesnotcommutewithRxandRy:
IRxf(x,y) = f(y,L−x)
RxIf(x,y) = f(L−y,x) (10.68)TheeigenvaluesofIaredeterminedbythesamereasoningasinthecaseofparity.
SupposeφβisaneigenstateofIwitheigenvalueβ.Then
IIφβ(x,y)=Iφβ(y,x)=φβ(x,y) (10.69)butalso
IIφβ(x,y)=βIφβ(x,y)=β^2 φβ(x,y) (10.70)