Schrodinger equation $\hat{p}=-i\hbar\nabla$

$$ \hat{H} = \frac{\hat{p}^2}{2m} + \hat{V}(r) = \frac{(-i\hbar\nabla)^2}{2m}+\hat{V}(r) $$

Generally, Laplacian is

$$ d\vec{r} = h_1 dq_1 \hat{e}_1+ h_2dq_2\hat{e}_2 + h_3dq_3 \hat{e}_3 \\

\nabla = \hat{e}_1 \frac{\partial_1}{h_1} + \hat{e}_2 \frac{\partial_2}{h_2} +\hat{e}_3 \frac{\partial_3}{h_3} \\

\nabla^2 = \frac{1}{h_1h_2h_3}\left[\partial_1\left(\frac{h_2h_3}{h_1} \partial_1\right)+\partial_2\left(\frac{h_1h_3}{h_2} \partial_2\right)+\partial_3\left(\frac{h_1h_2}{h_3} \partial_3\right)\right] $$

In spherical coordinate, Laplacian is

$$ d\vec{r} = dr \hat{r}+r d\theta \hat{\theta}+ r \sin\theta d\phi \hat{\phi} \\

\nabla = \hat{r} \partial_r + \hat{\theta} \frac{\partial_\theta}{r} +\hat{\phi} \frac{\partial_\phi}{r \sin \theta} \\

\nabla^2 = \frac{1}{r^2\sin\theta}\left[\partial_r\left(r^2\sin\theta\partial_r\right)+\partial_\theta\left(\sin\theta \partial_\theta\right)+\partial_\phi\left(\frac{\partial_\phi}{\sin\theta} \right)\right] \\

= \frac{\partial_r\left(r^2\partial_r\right)}{r^2} +\frac{\partial_\theta\left(\sin\theta \partial_\theta\right)}{r^2\sin\theta}+\frac{\partial_\phi^2}{r^2\sin^2\theta} $$

$$

\nabla^2 = \underbrace{\frac{\partial_r\left(r^2\partial_r\right)}{r^2}}{{\rm radial}} \underbrace{-\frac{\hat{L}^2}{\hbar^2 r^2}}{\rm angular} $$