160 CHAPTER10. SYMMETRYANDDEGENERACY
reverseorder,firstx′=−xandthenx′′=x′+a,theparticleendsupatx′′=−x+a,
whichisadifferentpoint. Itisnotsurprising,then,thatthemomentumoperatorp ̃
andtheParityoperatorPdon’tcommute:
Ppf ̃(x) = (−p ̃)f(−x)
= −pP ̃ f(x)
+= pP ̃ f(x) (10.48)
Since
[P,p ̃]+= 0 (10.49)
itfollowsthatPandp ̃cannothavethesamesetofeigenstates.Therearetwolessons
tobelearnedfromthis:
- Whentheorderofsymmetrytransformationsisimportant,thentheoperators
associatedwiththosetransformationsdonotcommute.
2.Whentwo(ormore)operatorscommutewiththeHamiltonian,butnotwitheach
other,therearealwaysdegenerateenergyeigenvalues.
10.3 The Particle in a Square
Asasecondexampleofthetwoprinciplesstatedabove,weconsideratwo-dimensional
problem:aparticlemovingfreelyinsideasquareoflengthL,whoselowerleft-hand
cornerislocatedatthepointx= 0 , y=0. Thesymmetriesof asquareinclude:
reflectionofthex-axisaroundthepointx=L/2:
x′=L−x (10.50)
reflectionofthey-axisaroundy=L/ 2
y′=L−y (10.51)
andtheinterchangeofcoordinates
x′=y y′=x (10.52)
whichisequivalenttoareflectionaroundthelinex=y.
TheHamiltonianforaparticleofmassmmovinginthesquareis
H ̃=− ̄h
2
2 m
(
∂^2
∂x^2
+
∂^2
∂y^2
)
+V(x,y) (10.53)