Angular Momentum Operator in spherical cooridnate system - derivation

Classical definition of angular momentum is given by, $$\vec{L} =\vec{r} \times \vec{p} \,\,\,...eq.(1)$$
Writing the components in terms of Cartesian coordinates we get,
$$L_x = yp_z - zp_y \\~\\ L_y =  zp_x -  xp_z \\~\\ L_z = xp_y - yp_x$$

In quantum mechanics, the momentum operator in position basis for x component is given by $ - \hbar \frac{\partial}{\partial{x}} $ using this, we rewrite the above equations as,   $$L_x = -i\hbar ( y\frac{\partial}{\partial{z}} - z\frac{\partial}{\partial{y}}) \\~\\ L_y = -i\hbar ( z\frac{\partial}{\partial{x}} - x\frac{\partial}{\partial{z}}) \\~\\ L_z = -i\hbar ( x\frac{\partial}{\partial{y}} - y\frac{\partial}{\partial{x}})$$

But, spherical symmetric potentials are often useful in quantum mechanics where we might need to express the equations in spherical coordinate system for simplicity. So, we do need to make the coordinate transformation rules to these equations to write this in terms of spherical polar coordinate system.

From our previous post of Cartesian to Spherical Coordinate transformation, we know that $$dx = \,sin\theta \,cos\phi \,dr + r \,cos\theta\, cos\phi \,d\theta - r \,sin\theta \,sin\phi \,d\phi$$
$$dy = \, sin\theta \, sin\phi \, dr + r \, cos\theta \, sin\phi \, d\theta + \, r\, sin\theta\, cos\phi \, d\phi$$  
$$dz = cos\theta \, dr - r\, sin\theta \, d\theta$$
where $$x = \,r \,sin\theta \,cos\phi \\~\\ y = \,r \,sin\theta \,sin\phi \\~\\ z = \,r \,cos\theta\,$$
Solving for $ dr, \,d\theta\,, d\phi $ as follows,
first we solve for $$d\phi = -\frac{1}{r}\frac{sin\phi}{sin\theta} dx + \frac{1}{r}\frac{cos\phi}{sin\theta} dy$$ Now we solve for $$dr = sin\theta\,cos\phi\, dx + sin\theta\, sin\phi\,dy + cos\theta\, dz$$ and finally we get, $$d\theta = \frac{1}{r} cos\theta\, cos\phi\,dx + \frac{1}{r} cos\theta\,sin\phi \, dy - \frac{1}{r}sin\theta\, dz$$
Using the above three equations with chain rule,
$$\frac{\partial}{\partial{x}} = \frac{\partial}{\partial{r}}\frac{\partial{r}}{\partial{x}} +\frac{\partial}{\partial{\theta}}\frac{\partial{\theta}}{\partial{x}} +\frac{\partial}{\partial{\phi}}\frac{\partial{\phi}}{\partial{x}}$$  
or $$\frac{\partial}{\partial{x}} = sin\theta\,cos\phi\, \frac{\partial}{\partial{r}} + \frac{1}{r} cos\theta \,cos\phi\, \frac{\partial}{\partial{\theta}} - \frac{1}{r}\frac{sin\phi}{\sin\theta} \frac{\partial}{\partial{\phi}}$$  Similary for others, $$\frac{\partial}{\partial{y}} = sin\theta\,sin\phi\, \frac{\partial}{\partial{r}} + \frac{1}{r} cos\theta \,sin\phi\, \frac{\partial}{\partial{\theta}} + \frac{1}{r}\frac{cos\phi}{\sin\theta} \frac{\partial}{\partial{\phi}}$$ $$\frac{\partial}{\partial{z}} = cos\theta\,\frac{\partial}{\partial{r}} - \frac{1}{r} sin\theta \, \frac{\partial}{\partial{\theta}}$$
Applying our equations in our definition of angular momentum, the angular momentum operator in terms of spherical coordinate system can be written as,
$$L_x = i\hbar \left( sin\phi \,\frac{\partial}{\partial{\theta}} + cot\theta\,cos\phi\, \frac{\partial}{\partial{\phi}}\right) \\~\\ L_y = i\hbar \left( -cos\phi\, \frac{\partial}{\partial{\theta}} + cot\theta\, sin\phi\,  \frac{\partial}{\partial{\phi}} \right) \\~\\ L_z = -i\hbar \frac{\partial}{\partial{\phi}}$$ and $$\vec{L}^2 = L_x^2 + L_y^2 +L_z^2 = -\hbar^2 \left[ \frac{1}{sin\theta} \frac{\partial}{\partial{\theta}}\left(sin\theta\frac{\partial}{\partial{\theta}}\right) + \frac{1}{sin^2\theta}\frac{\partial^2}{\partial{\phi^2}}\right]$$  

We will be often using this expression in quantum mechanics when we deal hydrogen atom, spherically symmetrical potential problem, etc.