θφxyzrWe now proceed to calculate the angular momentum operators in spherical coordinates. The first
step is to write the∂x∂iin spherical coordinates. We use the chain rule and the above transformation
from Cartesian to spherical. We have useddcosθ=−sinθdθanddtanφ=cos^12 φdφ. Ultimately
all of these should be written in the sperical cooridnates but its convenient to usexfor example in
intermediate steps of the calculation.
∂
∂x=
∂r
∂x∂
∂r+
∂cosθ
∂x∂
∂cosθ+
∂tanφ
∂x∂
∂tanφ=x
r∂
∂r+
−xz
r^3− 1
sinθ∂
∂θ−
y
x^2cos^2 φ∂
∂φ= sinθcosφ∂
∂r+
1
r
sinθcosφcosθ1
sinθ∂
∂θ−
1
rsinθsinφ
sin^2 θcos^2 φcos^2 φ∂
∂φ= sinθcosφ∂
∂r+
1
r
cosφcosθ∂
∂θ−
1
rsinφ
sinθ∂
∂φ
∂
∂y=
∂r
∂y∂
∂r+
∂cosθ
∂y∂
∂cosθ+
∂tanφ
∂y∂
∂tanφ=
y
r∂
∂r+
−yz
r^3− 1
sinθ∂
∂θ+
1
xcos^2 φ∂
∂φ= sinθsinφ∂
∂r+
1
rsinθsinφcosθ1
sinθ∂
∂θ+
1
r1
sinθcosφcos^2 φ∂
∂φ= sinθsinφ∂
∂r+
1
rsinφcosθ∂
∂θ+
1
rcosφ
sinθ∂
∂φ
∂
∂z=
∂r
∂z∂
∂r+
∂cosθ
∂z∂
∂cosθ+
∂tanφ
∂z∂
∂tanφ=
z
r∂
∂r+
(
1
r−
z^2
r^3)
− 1
sinθ∂
∂θ= cosθ∂
∂r+
1
r(
1 −cos^2 θ)− 1
sinθ∂
∂θ
= cosθ∂
∂r−
1
rsinθ∂
∂θ