In quantum mechanics there are two types of angular momentum:

• Orbital angular momentum, refers to the movement of the particle in space and is analogous to classical mechanics.
• Spin angular momentum is an intrinsic property of microscopic particles and has no classical analogue.

We are going to build the mechanical-classical operators of angular momentum.

We start from the classical expression $l_x = yp_z - zp_y$ and replace Cartesian coordinates and linear moments with the corresponding quantum operators:

\begin{equation} \hat{l}_x = \hat{y}\left(-i\hbar\frac{\partial}{\partial z}\right)-\hat{z}\left(-i\hbar\frac{\partial}{\partial y}\right)=-i\hbar\left(\hat {y}\frac{\partial}{\partial z}-\hat{z}\frac{\partial}{\partial y}\right) \end{equation}

Analogously:

\begin{equation} \hat{l }_y = -i\hbar\left(\hat{z}\frac{\partial}{\partial x}-\hat{x}\frac{\partial}{\partial z}\right) \end{equation}

\begin{equation} \hat{l}_z = -i\hbar\left(\hat{x}\frac{\partial}{\partial y}-\hat{y}\frac{\partial}{\partial x}\right) \end{equation}