Question

Write down Lx is term linear angular momentum.

Answer 1

In classical mechanics :

$L = r\times p$

Therefore to pass to quantum mechanics we just replace the physical quantities by the corresponding operators (and sometimes by "hermitianizing" the result). In $r$ representation we have :

$\hat{L} = \hat{r}\times\hat{p}=-i\hbar \hat{r}\times\vec{\nabla}$

And so we have

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

We can verify that the obtained result is an observable (a hermitian operator).