Cartesian and Spherical Polar Forms

It is of value to inspect the angular momentum operator in terms of angles rather than Cartesian coördinates. Remember that

$$\begin{eqnarray*}x = r \sin \theta \cos \phi\\ y=r \sin \theta \sin \phi\\ z = r \cos \theta \end{eqnarray*}$$

and

$$\vec{L} = \left ( \begin{array}{ccc} \hat{i} & \hat{j} & \ha... ... \hat{i} (yp_z-zp_y) + \hat{j}(zp_x-xp_z) + \hat{k}(xp_y-yp_x)$$ (1)

We start with a feast of partial derivatives:

$$\left ( \frac{\partial r}{\partial x} \right )_{y,z} = \sin \theta \cos\phi$$ (2)

$$\left ( \frac{\partial r}{\partial y} \right )_{x,z} = \sin \theta \sin \phi$$ (3)

$$\left ( \frac{\partial r}{\partial z} \right )_{x,y} = \cos \theta$$ (4)

$$\left ( \frac{\partial \theta}{\partial x} \right )_{y,z} = \frac{\cos \theta \cos \phi}{r}$$ (5)

$$\left ( \frac{\partial \theta}{\partial y} \right )_{x,z} = \frac{\cos \theta \sin \phi}{r}$$ (6)

$$\left ( \frac{\partial \theta}{\partial z} \right )_{x,y} = -\frac{\sin \theta} {r}$$ (7)

and finally

$$\left ( \frac{\partial \phi}{\partial x} \right )_{y,z} = -\frac{\sin \phi}{r \sin \theta}$$ (8)

$$\left ( \frac{\partial \phi}{\partial y} \right )_{x,z} = \frac{\cos \phi}{r \sin \theta}$$ (9)

$$\left ( \frac{\partial \phi}{\partial z} \right )_{x,y} = 0$$ (10)

which we employ on the defined x-component of the angular momentum, thus

$$L_x \equiv y p_z - zp_y = -\imath \hbar \sin \theta \sin \phi... ... -\imath \hbar \cos \theta \frac{\partial}{\partial y}\right )$$

where

$$\frac{\partial}{\partial z} = \left ( \frac{\partial r}{\part... ...\phi}{\partial z}\right )_{x,y}\frac{\partial}{\partial \phi}$$

which is

$$\frac{\partial}{\partial z}= \left ( \cos \theta \right )\fra... ...l \theta} + \left ( 0 \right )\frac{\partial}{\partial \phi}$$