Quantum Mechanics for Mathematicians

We first need to change coordinates from rectangular to spherical in our
expressions forL 3 ,L±. Using the chain rule to compute expressions like

∂

∂r

f(x 1 (r,θ,φ),x 2 (r,θ,φ),x 3 (r,θ,φ))

we find



∂

∂r∂

∂θ∂

∂φ



=





sinθcosφ sinθsinφ cosθ

rcosθcosφ rcosθsinφ −rsinθ

−rsinθsinφ rsinθcosφ 0









∂

∂x 1

∂

∂x 2

∂

∂x 3





so 



∂

1 ∂r

r

∂

1 ∂θ

rsinθ

∂

∂φ



=





sinθcosφ sinθsinφ cosθ

cosθcosφ cosθsinφ −sinθ

−sinφ cosφ 0









∂

∂x 1

∂

∂x∂ 2

∂x 3





This is an orthogonal matrix, so can be inverted by taking its transpose, to get



∂

∂x 1

∂

∂x∂ 2

∂x 3



=





sinθcosφ cosθcosφ −sinφ

sinθsinφ cosθsinφ cosφ

cosθ −sinθ 0









∂

1 ∂r

r

∂

1 ∂θ

rsinθ

∂

∂φ





So we finally have

L 1 =iρ′(l 1 ) =i

(

x 3

∂

∂x 2

−x 2

∂

∂x 3

)

=i

(

sinφ

∂

∂θ

+ cotθcosφ

∂

∂φ

)

L 2 =iρ′(l 2 ) =i

(

x 1

∂

∂x 3

−x 3

∂

∂x 1

)

=i

(

−cosφ

∂

∂θ

+ cotθsinφ

∂

∂φ

)

L 3 =iρ′(l 3 ) =i

(

x 2

∂

∂x 1

−x 1

∂

∂x 2

)

=−i

∂

∂φ

and

L+=iρ′(l+) =eiφ

(

∂

∂θ

+icotθ

∂

∂φ

)

, L−=iρ′(l−) =e−iφ

(

−

∂

∂θ

+icotθ

∂

∂φ

)

Now that we have expressions for the action of the Lie algebra on functions in
spherical coordinates, our two differential equations saying our functionYll(θ,φ)
is of weightland in the highest weight space are

L 3 Yll(θ,φ) =−i

∂

∂φ

Yll(θ,φ) =lYll(θ,φ)

and

L+Yll(θ,φ) =eiφ

(

∂

∂θ

+icotθ

∂

∂φ

)

Yll(θ,φ) = 0

The first of these tells us that

Yll(θ,φ) =eilφFl(θ)