from the cartesian definition of angular momentum, derive the operator for the z component in polar coordinates
L_z = -ih[x(d/dy) - y(d/dx)]
L_z = -ih(d/dθ)
x = rcosθ
y = rsinθ
r^2 = x^2 + y^2
r = (x^2 + y^2)^1/2
The Attempt at a Solution
first of all I'm not sure how this is even possible. Every derivation of the angular momentum operator I've seen requires spherical coordinates, not polar.
I tried taking the derivative of r with respect to x to get cosθ and with respect to y to get sinθ
and dx/dθ = -rsinθ dy/dθ = rcosθ but it's not getting me anywhere.
is there something i should be rewriting d/dx and d/dy as?