Please help me understand where I went wrong in my conversion from Cartesian to polar coordinates

[shp]

Homework Statement

I need to convert z component of angular momentum operator (in quantum mechanics) Lz from cartesian to polar coordinates

Relevant Equations

L_z = -i \hbar \frac{\partial}{\partial \phi}

I have done this:

I can't clearly understand where I went wrong. Any help would be greatly appreciated. Thanks in advance.

 

[pasmith]

I would start from $$\begin{split}<br /> \frac{\partial}{\partial \rho} &= \frac{\partial x}{\partial \rho}\frac{\partial}{\partial x} + \frac{\partial y}{\partial \rho}\frac{\partial}{\partial y} \\<br /> &= \cos \phi \frac{\partial }{\partial x} + \sin \phi \frac{\partial}{\partial y} \\<br /> \frac{\partial}{\partial \phi} &= \frac{\partial x}{\partial \phi}\frac{\partial}{\partial x} + \frac{\partial y}{\partial \phi}\frac{\partial}{\partial y} \\<br /> &= -\rho \sin \phi \frac{\partial}{\partial x} + \rho \cos \phi \frac{\partial}{\partial y}<br /> \end{split}$$ and solve for $\frac{\partial}{\partial \phi}$ in terms of $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$.

 

[shp]

I would start from $$\begin{split}<br /> \frac{\partial}{\partial \rho} &= \frac{\partial x}{\partial \rho}\frac{\partial}{\partial x} + \frac{\partial y}{\partial \rho}\frac{\partial}{\partial y} \\<br /> &= \cos \phi \frac{\partial }{\partial x} + \sin \phi \frac{\partial}{\partial y} \\<br /> \frac{\partial}{\partial \phi} &= \frac{\partial x}{\partial \phi}\frac{\partial}{\partial x} + \frac{\partial y}{\partial \phi}\frac{\partial}{\partial y} \\<br /> &= -\rho \sin \phi \frac{\partial}{\partial x} + \rho \cos \phi \frac{\partial}{\partial y}<br /> \end{split}$$ and solve for $\frac{\partial}{\partial \phi}$ in terms of $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$.

Thanks for help. But I wanted to know where I went wrong. Seems like (I) and (ii) is wrong. But I can't understand why.

 

[Gavran]

I can't clearly understand where I went wrong. Any help would be greatly appreciated. Thanks in advance.

A partial derivative with respect to rho must also be included into equations.