Angular momentum in Cylindrical Coordinates

How to calculate the angular momentum of a particle in a cylindrical coordinates system

$$x_1 = r \cos{\theta}$$ $$x_2 = r \sin{\theta}$$ $$x_3 = z$$

Thanks.

Solution 1:

I assume you are seeking the infinitesimal deformation operators generating rotations, as in quantum mechanics, so, then, $\vec{p}\sim \nabla$.

In that case, since $$ \vec{R}= z \hat z + r \hat r ,\\ \nabla = \hat z \partial_z + \hat r \partial_r + \hat \theta \frac{1}{r} \partial _\theta, $$

$$ \vec L = \vec{R} \times \nabla = (z \hat z + r \hat r ) \times \left ( \hat z \partial_z + \hat r \partial_r + \hat \theta \frac{1}{r} \partial _\theta \right ) \\ = \hat z \partial_\theta- \hat r \frac{z}{r} \partial_\theta +\hat \theta (z\partial_r -r\partial_z). $$ Confirm by limiting cases z=0, θ=0, etc.