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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:

1. 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)