# Spherical Coordinates

We now write down the Cartesian components of the angular momentum operator in spherical coordinates. We will make use of this result later in determining the eigenfunctions of the angular momentum squared and of one of its components.

The Cartesian coordinates $x,\!$ $y,\!$ and $z\!$ can be written in terms of the spherical coordinates $r,\!$ $\theta,\!$ and $\phi\!$ as follows:

$x=r\sin\theta\cos\phi,\!$ $y=r\sin\theta\sin\phi,\!$ $z=r\cos\theta\!$

Let us start with the $x\!$ component of the angular momentum, $\hat{L}_x.$ In Cartesian coordinates, this is

$\hat{L}_x=-i\hbar\left (y\frac{\partial}{\partial z}-z\frac{\partial}{\partial y}\right ).$

If we make use of the chain rule, then we obtain

$\hat{L}_{x} = -i\hbar\left( -\sin\phi \frac{\partial}{\partial \theta} \, -\cot\theta\cos\phi \frac{\partial}{\partial\phi} \! \right).$

Similarly, the $y\!$ and $z\!$ components may be found to be

$\hat{L}_{y} = -i\hbar\left(\cos\phi \frac{\partial}{\partial\theta} - \cot\theta\sin\phi \frac{\partial}{\partial\phi} \! \right)$

and

$\hat{L}_{z} = -i\hbar\frac{\partial}{\partial \phi}.$

## Problem

(Richard L. Liboff, Introductory Quantum Mechanics, 2nd Edition, pp. 377-379)

Show, using the above results, that the operator,

$\hat{R}(\Delta\phi)=\exp \left (\frac{i}{\hbar}\Delta\phi\hat{L}_z\right ),$

when applied to a function $f(\phi)\!$ of the azimuthal angle $\phi,\!$ rotates the angle $\phi\!$ to $\phi+\Delta\phi.\!$ That is, show that

$\hat{R}(\Delta\phi)f(\phi)=f(\phi+\Delta\phi).$