154 CHAPTER10. SYMMETRYANDDEGENERACY
figure. Ontheotherhand,supposeeverypoint(x′,y′)ontheoriginal sketchwere
movedtoanewpoint(x′,y′)accordingtotherule
x′=−x y′=y (10.3)Thistransformationreplacestheright-handsideofthefigurebytheleft-handside,
andvice-versa. Thetransformedstick figureisindistinguishablefromtheoriginal
figure,anditisthereforesaidtobesymmetricunderleft-rightreflection.
- Periodicity (Symmetry underFinite Translations) Considernextthe
sin-wave shown inFig. [10.3]. Ingeneral, ifwe transform every point(x,y) →
(x′,y′)accordingtotherule(10.2),thefigurewillendupinadifferentplace,and
thusbe distinguishable fromthe untransformed figure. However, ifwe make the
transformation
x′=x+ 2 πn y′=y (10.4)
wherenisaninteger,thenthenewfigureisthesameastheold,duetotheperi-
odicityofsin-waves.Thesespecialtranslationsarethereforeasymmetryofperiodic
functions,suchassinandcosine.
- RotationSymmetry Asafinalexample,considerthepointsonacircleof
radiusR. Eachpointonthecirclecanbeassignedpolarcoordinates(r,θ),where
r=Rand 0 ≤θ< 2 π. Acircleisobviouslysymmetricunderarbitaryrotations
r′=r θ′=θ+δθ (10.5)whereδθisanyconstantangle.
Symmetriesareimportantinquantummechanics whenevertheHamiltonian is
invariantundersomecoordinatetransformation:
x′ = f(x)
∂
∂x′=
(
∂f
∂x)− 1
∂
∂x(10.6)
wheretheword”invariant”meansthat
H ̃[−ih ̄ ∂
∂x′,x′]=H ̃[−i ̄h∂
∂x,x] (10.7)Supposeφα(x)isaneigenstateofH ̃
H ̃[−i ̄h∂
∂x,x]φα(x)=Eαφα(x) (10.8)