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)