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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)