Angular momentum polar coordinates

[johnnyies]

Homework Statement

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)]

to

L_z = -ih(d/dθ)

Homework Equations

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?

 

[Sourabh N]

Since you are transforming from one basis to another, you can use the identity (written for your particular case):

$\frac{∂}{∂θ}$ = $\frac{∂x}{∂θ}$$\frac{∂}{∂x}$ + $\frac{∂y}{∂θ}$$\frac{∂}{∂y}$.

Does that help?