Angular Momentum Operator in Quantum Mechanics
Build-Up
The angular momentum operator is naturally derived from the classical definition of angular momentum.
$$ \mathbf{l} = \mathbf{r} \times \mathbf{p} \tag{1} $$
Let’s denote this as $\mathbf{r} = (x, y, z)$ and $\mathbf{p} = (p_{x}, p_{y}, p_{z})$. Then each component of the angular momentum $\mathbf{l} = (l_{x}, l_{y}, l_{z})$ is described as follows.
$$ l_{x} = yp_{z} - zp_{y},\quad l_{y} = zp_{x} - xp_{z},\quad l_{z} = xp_{y} - yp_{x} $$
Following the above equation, the angular momentum operator can be naturally defined using the position operator $X$ and the momentum operator $P$.
Definition
The angular momentum operator $L = (L_{x}, L_{y}, L_{z})$ is defined as follows:
$$ \begin{align*} L_{x} := YP_{z} - ZP_{y} \\ L_{y} := ZP_{x} - XP_{z} \\ L_{z} := XP_{y} - YP_{x} \end{align*} $$
Explanation
Since the momentum operator is specifically $P_{j} = -\i\hbar \dfrac{\partial}{\partial x_{j}}$, the angular momentum operator is as follows:
$$ L = -\i\hbar \mathbf{r} \times \nabla,\qquad \mathbf{r} = (X, Y, Z) $$
At this point, $\nabla = \left( \dfrac{\partial }{\partial x}, \dfrac{\partial }{\partial y}, \dfrac{\partial }{\partial z} \right)$ is the del operator. In three dimensions, the momentum operator is $-\i\hbar \nabla$, so the classical definition of angular momentum is naturally extended to the operator.
Spherical Coordinates
The angular momentum operator is expressed as follows in spherical coordinates:
$$ \begin{align*} L_{x} &= \i\hbar \left(\sin\phi\dfrac{\partial }{\partial \theta} + \cos\phi \cot\theta \dfrac{\partial }{\partial \phi}\right) \\ L_{y} &= -\i\hbar \left( \cos\phi \dfrac{\partial }{\partial \theta} - \sin\phi \cot\theta \dfrac{\partial }{\partial \phi}\right) \\ L_{z} &= -\i\hbar \dfrac{\partial }{\partial \phi} \end{align*} $$
Ladder Operators
The ladder operators of angular momentum are as follows.
$$ L_{+} := L_{x} + \i L_{y} \\ L_{-} := L_{x} - \i L_{y} $$