164 CHAPTER10. SYMMETRYANDDEGENERACY
andtheLaplacianoperator∇^2 ,intwodimensions,isgiveninpolarcoordinatesby
∇^2 ≡
∂^2
∂x^2
+
∂^2
∂y^2
=
∂^2
∂r^2
+
1
r
∂
∂r
+
1
r^2
∂^2
∂θ^2
(10.78)
Underanarbitraryrotationoftheform(10.76),∂/∂θ′ =∂/∂θ,sotheLaplacianis
invariant.ItfollowsthattheHamiltonianforaparticlemovinginsideacircle
H ̃=− ̄h
2
2 m
(
∂^2
∂r^2
+
1
r
∂
∂r
+
1
r^2
∂^2
∂θ^2
)
+V(r) (10.79)
isinvariantunderrotations.
Asinthepreviousexamples,weintroducearotationoperatorRδθdefinedby
Rδθf(r,θ)=f(r,θ+δθ) (10.80)
and,proceedingasinthecaseofthetranslationoperator
f(r,θ+δθ) = f(r,θ)+
∂f
∂θ
δθ+
1
2
∂^2 f
∂θ^2
+...
= exp[δθ
∂
∂θ
]f(r,θ) (10.81)
sothat
Rδθ=exp[δθ
∂
∂θ
] (10.82)
SincetheHamiltonianisinvariantunderrotations
[Rδθ,H ̃]= 0 (10.83)
itfollowsfrom(10.82)that
[
∂
∂θ
,H ̃]= 0 (10.84)
Now,incartesiancoordinates
∂
∂θ
=
∂x
∂θ
∂
∂x
+
∂y
∂θ
∂
∂y
= −rsinθ
∂
∂x
+rcosθ
∂
∂y
= x
∂
∂y
−y
∂
∂x
=
1
−ih ̄
(xp ̃y−yp ̃x) (10.85)